For the better part of my life, I have found great interest in nature and the unique occurrences within our environment such as the beautiful tiger stripes,flower petals which are delicately set, among others.Recently however,it dawned on me that It is onIy when we take time to analyze all components of our environment that we will vividly understand the extent to which Math is connected to important aspect in our lives as it plays a huge role in making up almost everything in our environs. Such may be explained by various concepts in the subject.When it comes to different mathematical concepts, it is without any doubt that there are daily advancements with numerous new discoveries and application to these concepts. Recently, air was cleared concerning an issue that had been termed a mathematical curiosity, know as the black hole. Scientists were able to make a remarkable achievement by being able to take a picture or create an illusion of the black hole, with the help of mathematical tools.A recent report from the‘University of Bath’ and‘Yale university‘ dated 5th September 2019 states that we can be able to see or perceive the beauty present in complex mathematical arguments just in the very same way that we can see and appreciate the beauty of a beautiful image such as a landscape painting or a piano sonata. Though not related to the topic of discussion, this research may have implications for the teaching of mathematics in different school levels and even encourage lower level students who do not view mathematics as beautiful or do not see any beauty in math. Hence encouraging more people to join mathematical fields in the future.Math is all around us, in everything we do, it is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports ( J. Hom, 2013).In his book‘ Mathematics in Nature’, Adam vividly explains numerous phenomena which can not be explained or accounted for in any other way other than application of mathematics. He makes mention of examples and qualitative discussions of patterns in nature, examples of the power of arithmetic in solving fermi problems,meteorological optics, linear waves, the Fibonacci sequence and the golden ratio among other important mathematical concepts. However, in one part of his text, he makes mention of a question posed by Peter S Stephens in his book,‘Patterns in Nature‘ where he asks why nature appears to make use of just but a few fundamental forms but applies them in different contexts. He questions the similarity in appearance of the branching of trees, the arteries and the branching of rivers. Moreover, he wonders why crystal grains look like soap bubbles and the plates of a tortoise shell.According to his analysis, Stephens came up with a conclusion that nature is made up of one hundred and thirty seven point five degree angles, branchings, hexagons, spirals and meanders.Of the various phenomena that I was able to read on from this book, such as the ones mentioned above,the scattering of light and the application of mathematical principles in plants and animals seemed to be of the most interest to me. Light, because it is a very important aspect in our lives. Life without the blue sky above us, the beautiful sunrise and sunset views which we so much love to enjoy their sights,the rainbow ; which many people may believe serves as an obvious symbol of calmness and security and the different range of colors we can see seems rather uninteresting compared to how it is in the present, obviously with the mentioned things. Therefore,understanding how these things come about is rather interesting as they are occurrences or phenomena which we come across in our every day lives.Plants and the characteristics that they show in the arrangement of their leaves also largely illustrate the concept of the “golden number “ (1.618). There are various phenomena which exhibit different types of spirals. Such are the rolling up of a chameleons tail, the spiral arrangement of daisy seeds in its head,the curling up of a feen that is in the process of drying up, among other occurrences.When it comes to animals and insects,patterns on their coats in animals such as tigers and cheetahs just to mention but a few and wing markings, for example in moths and butterflies can easily be studied with the use of mathematics by applying “reaction-diffusion equations”The question of “What Is School For” is one with a complex history and Americans have still not agreed upon one answer to this question. While a popular opinion is that schools are supposed to prepare students to become functioning members of society as a whole, the content and principles that are drilled into students is a topic of controversy. I believe school is for exposing children and young adults to a wide variety of subjects, ideas, viewpoints, and experiences. This exposure to different schools of thought should lead to citizens being educated, aware, and knowledgeable about themselves.The different levels of education, from elementary to middle to high school, all serve out the goals of what a school should help a student with, but they do it to varying degrees. As a young child, school is a prime setting for social and emotional learning. This could be a child’s first time spending an extended amount of time with those other than their family, which doubles as an opportunity to see the similarities and differences between themself and others. Different schools can encourage this exposure in different ways. Natureschooling, which is a nature-centered homeschool education, can emphasize the variety of ways we interact with the Earth and how we interact with others. Public schools and private schools alike expose students who don’t look like them or think like them. As students progress and move to middle school, the social aspect is still very much present, but the academic aspect gains a heightened importance. Students continue to learn about themselves and others, and this social awareness is key to being a functioning member of society. As students experience high school, and possibly higher education, the process of learning never stops, both academically and socially. With age comes new relationships, changing interactions, and a need for adaptability. School is the best vehicle to provide this knowledge and self-awareness. Without the exposure to those unlike ourselves, the United States cannot thrive. As a country that is supposedly built on the value of equality, the ability to have complex thought is necessary to make progress.Without school, Americans would not be able to communicate and learn from others. Without being exposed to views unlike your own in a humanizing way would breed adults with closed minds and stubborn hearts. Progress cannot be made without the introduction of new ideas, and if no one will hear the ideas in the first place, it is impossible for societal change to occur. While school is traditionally thought of as a place for academic learning through memorization and tests, the real value of school in the United States is the social learning that occurs. Learning how to interact with others with the guidance of teachers who act as mentors and help students fix their mistakes is an invaluable experience. An education filled with new experiences is essential to the development of active and engaged citizens.hat is the Purpose of School?What is the purpose of school? Neil Postman said that “without a purpose, schools are houses of detention, not attention” (1995, p. 7). Most countries have systems of formal education and many of these are compulsory. Although the names of schools differ, most include a primary school for young children and a secondary school for teenagers (Roser & Oritz-Ospina, 2019).Objectives and Key TermsIn this chapter, readers will…Understand the basic purposes of schoolDescribe several different understandings of the concept of “school”Define the nature of school for each level: elementary, middle, and high schoolsKey terms in the chapter are…CompulsoryFormal EducationInformal EducationVocationalDefining SchoolBefore landing upon a definition for school, it is important to delineate the differences between education and schooling.Education is a process of learning and growing as one gains understanding about the surrounding world. This is a lifelong process. It is, as John Dewey (1916) put it, a social process – ‘a process of living and not a preparation for future living.’Schooling can often look like an institution with a very specific motive – drill learning into people according to some plan often drawn up by others. Paulo Friere (1973) famously called this banking – making deposits of knowledge. This type of “schooling” treats learners like objects.School CultureWhat makes a good school culture? Shafer (2018) noted that it is all about connections. She describes five interwoven elements that support school structure, 1) Fundamental beliefs, 2) Shared values, 3) Norms (how people believe they should act), 4) Patterns and behaviors, and 5) Tangible evidence. To read more about those elements, you can find the article here:“In a strong culture, there are many, overlapping, and cohesive interactions among all members of the organization.”– Leah ShaferPurposes of SchoolIs School for Knowledge?If asked, most people would say that the purpose of school is to provide knowledge, but the question becomes what knowledge and who should decide. Is learning for the sake of learning what school is about? Learning expands the mind and school is a way for students to be exposed to different ideas and concepts. Knowledge obtained through school can provide students with a sense of personal fulfillment (Education). “It seems to me, that education has a two-fold function to perform in the life of man and in society: the one is utility and the other is culture. Education must enable a man to become more efficient, to achieve with increasing facility the legitimate goals of his life.”–Martin Luther King Jr (1947) It is argued that anything learned in school could be learned on your own (Gatto, 2005). In the modern-day of the Internet and with vast libraries of knowledge available to us, this is very true. There then becomes a problem of motivation. What would make someone want to learn math or science? Does a child just decide someday that they want to learn all about Chemistry? (Postman, 1995) It is not an issue of what information is necessary, but an issue of exposing students to different ideas that they can choose to grow and build on. It is teaching them how to learn. Education should expose students to information and teach them how to think, not tell them what to think. Martin Luther King Jr said, “Education must enable one to sift and weigh evidence, to discern the true from the false, the real from the unreal, and the facts from the fiction” (1947).Is School for Getting a Job?Not everyone has the opportunities or wants to go to college. Therefore, the purpose of school must be to give students the skills to get a job. This means that education is a way for anyone to support him or herself and economically contribute to society (Education). Some of these skills are taught in many of the basic classes: reading, writing, and arithmetic. There is also vocational education, which is extremely important to the lives of students who do not enjoy academia. Just because a student does not like school does not mean that the school should ignore them. It is the school’s responsibility to educate all students and prepare them for their future.Is School for Socialization?It is argued that any of the above items can be learned on your own (Gatto, 2005). As stated earlier, the issue of motivation and outside circumstances does provide a problem with this theory, but what can replace the socialization that a student receives in school?“Schooling at its best can be about how to make a life, which is quite different from how to make a living” (Postman, 1995, p. x). Einstein said that the school’s responsibility is to educate the individual as a free individual but to also educate them to be part of society (Haselhurst, 2007). Students are around hundreds of people their own age and this teaches them how to act in society and how to communicate. This is helpful no matter what they do with their future and nothing can replace those skills. Being in a school with that many people also exposes the student to people who are different from him or herself and this is extremely helpful in anyone’s development as a human being and a better member of society (Postman, 1995). Professor Nel Noddings said that the school’s aim is “to produce competent, caring, loving, and lovable people” (Kohn, 2004, p.2).Types of SchoolsIn most states, the school year is 180 days. School days often last a total of six and a half hours. This means that a child may spend more than 1,000 hours in school each year. In elementary school, how are these hours typically spent? In these sections, we will discuss the teacher’s role, what students experience in elementary, middle school, and high school.ElementaryAn elementary school is the main point of delivery of primary education for children between 5-11. In elementary school, children are exposed to a broad range of topics and often remain together in one classroom. School districts and the state determine the curriculum, but generally, a student learns basic arithmetic, English proficiency, social studies, science, physical development, and fine arts.The Role of the TeacherAn elementary school teacher is trained with an emphasis on human cognitive and psychological development as well as the principles of curriculum development and instruction. Teachers earn either a Bachelors or Master’s degree in Early Childhood or Elementary Education.The public elementary teacher typically instructs between twenty and thirty students of diverse learning needs. These teachers use a variety of ways to teach, with a focus on student engagement (getting a student’s attention).What Students ExperienceOriginally, an elementary school was synonymous with primary education. Many students prior to World War I did not attend school past Grade 8. Over the past few decades, schools in the USA have seen numbers of high school graduates rise and with it, changes in what students experience in school. An elementary school typically contained one-teacher, one-class models, but this has been changing over time. Multi-age programs, where children in different grades share the same classroom and teachers. Another alternative is that children might have a main class and go to another teacher’s room for one subject. This could be called a rotation and it is similar to the concept of teams found in junior high school.Middle SchoolWatch the following video from the perspective of a middle schooler. What would you point out as part of her environment at school? Is there any evidence of her relationship with what she is learning or her relationship with educators at her school?A YouTube element has been excluded from this version of the text. You can view it online here: SchoolWhile there is no set standard for an American high school, some generalizations can be made about the majority. Schools are managed by local, elected school districts. Students ages 14-18 participate in four years of school. School years are normally around nine months and are broken up into quarters or semesters. The High School curriculum is defined in terms of Carnegie Units, which approximate 120 class contact hours within a year. No two schools will be the same, and no two students will have the same classes. There are some general core subjects, but electives will vary by school.Activity:Fill out the following to highlight was is important in each level: the following video with this question in mind: According to this author, what do effective schools do differently? Does this align more closely with the notion of “schooling” or “education”?or other uses, see Mathematics (disambiguation). "Math" and "Maths" redirect here. For other uses, see Math (disambiguation).Euler's identity, sometimes called the most beautiful theorem of mathematics[1]MathematicshideAreasNumber theoryGeometryAlgebraCalculus and analysisDiscrete mathematicsLogic and set theoryProbabilityStatistics and decision scienceshideRelationship with other fieldsPhysicsComputationBiologyLinguisticsEconomicsPhilosophyEducationPortalvteMathematics (from Ancient Greek μάθημα; máthēma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory,[2] algebra,[3] geometry,[2] and analysis,[4][5] respectively.Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated with certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered as true starting points of the theory under consideration.[6]Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model.Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. The fundamental truths of mathematics are independent from any scientific experimentation, although mathematics is extensively used for modeling phenomena. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other mathematical areas are developed independently from any application (and are therefore called pure mathematics), but practical applications are often discovered later.[7][8] A fitting example is the problem of integer factorization, which goes back to Euclid, but which had no practical application before its use in the RSA cryptosystem (for the security of computer networks).Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.[9] Since its beginning, mathematics was essentially divided into geometry, and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra[a] and infinitesimal calculus were introduced as new areas of the subject. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both.[10] At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method. This gave rise to a dramatic increase in the number of mathematics areas and their fields of applications. This can be seen, for example, in the contemporary Mathematics Subject Classification, which lists more than 60 first-level areas of mathematics.Contents1Etymology2Areas of mathematics2.1Number theory2.2Geometry2.3Algebra2.4Calculus and analysis2.5Discrete mathematics2.6Mathematical logic and set theory2.7Statistics and other decision sciences2.8Computational mathematics3History3.1Ancient3.2Medieval and later4Symbolic notation and terminology5Relationship with science5.1Pure and applied mathematics5.2Unreasonable effectiveness6Philosophy6.1Reality6.2Proposed definitions6.3Logic and rigor7Psychology (aesthetic, creativity and intuition)8Education9Awards and prize problems10See also11Notes12References13Bibliography13.1Further readingEtymologyThe word mathematics comes from Ancient Greek máthēma (μάθημα), meaning "that which is learnt,"[11] "what one gets to know," hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.[12] Its adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious," which likewise further came to mean "mathematical." In particular, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art."Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle this meaning was fully established.[13]In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.[14]The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά), used by Aristotle (384–322 BC), and meaning roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from Greek.[15] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.[16]Areas of mathematicsBefore the Renaissance, mathematics was divided into two main areas: arithmetic — regarding the manipulation of numbers, and geometry — regarding the study of shapes.[17] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.[18]During the Renaissance, two more areas appeared. Mathematical notation led to algebra, which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields infinitesimal calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities (variables). This division into four main areas–arithmetic, geometry, algebra, calculus[19]–endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were often then considered as part of mathematics, but now are considered as belonging to physics. Some subjects developed during this period predate mathematics and are divided into such areas as probability theory and combinatorics, which only later became regarded as autonomous areas.[citation needed]At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics.[20] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas.[21] Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century (for example category theory; homological algebra, and computer science) or had not previously been considered as mathematics, such as Mathematical logic and foundations (including model theory, computability theory, set theory, proof theory, and algebraic logic).Number theoryMain article: Number theoryThis is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.Number theory began with the manipulation of numbers, that is, natural numbers {\displaystyle (\mathbb {N} ),} and later expanded to integers {\displaystyle (\mathbb {Z} )} and rational numbers {\displaystyle (\mathbb {Q} ).} Formerly, number theory was called arithmetic, but nowadays this term is mostly used for numerical calculations.[22] The origin of number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid and Diophantus.[23] The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.[24]Many easily-stated number problems have solutions that require sophisticated methods from across mathematics. One prominent example is Fermat's last theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory and homological algebra.[25] Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven to this day despite considerable effort.[26]Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented).GeometryMain article: GeometryGeometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.[27]A fundamental innovation was the introduction of the concept of proofs by ancient Greeks, with the requirement that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or they are a part of the definition of the subject of study (axioms). This principle, which is foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.[28][29]The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the (three-dimensional) Euclidean space.[b][27]Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This was a major change of paradigm, since instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates (which are numbers). This allows one to use algebra (and later, calculus) to solve geometrical problems. This split geometry into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.[30]Analytic geometry allows the study of curves that are not related to circles and lines. Such curves can be defined as graph of functions (whose study led to differential geometry). They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider spaces of higher than three dimensions.[27]In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning the truth of that postulate, this discovery has been viewed as joining Russel's paradox in revealing the foundational crisis of mathematics.This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem.[31] In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that are invariant under specific transformations of the space.[32]In the present day, the subareas of geometry include:Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.Affine geometry, the study of properties relative to parallelism and independent from the concept of length.Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functionsManifold theory, the study of shapes that are not necessarily embedded in a larger spaceRiemannian geometry, the study of distance properties in curved spacesAlgebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomialsTopology, the study of properties that are kept under continuous deformationsAlgebraic topology, the use in topology of algebraic methods, mainly homological algebraDiscrete geometry, the study of finite configurations in geometryConvex geometry, the study of convex sets, which takes its importance from its applications in optimizationComplex geometry, the geometry obtained by replacing real numbers with complex numbers Examples of shapes encountered in geometryPythagorean theorem Conic Sections Elliptic curve Triangle on a paraboloid Torus FractalAlgebraMain article: AlgebraAlgebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra.[33][34] The first one solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. The second one introduced systematic methods for transforming equations (such as moving a term from a side of an equation into the other side). The term algebra is derived from the Arabic word al-jabr meaning "the reunion of broken parts"[35] that he used for naming one of these methods in the title of his main treatise.The quadratic formula, which concisely expresses the solutions of all quadratic equationsAlgebra became an area in its own right only with François Viète (1540–1603), who introduced the use of letters (variables) for representing unknown or unspecified numbers.[36] This allows mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term that is still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid.[37] The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. Due to this change, the scope of algebra grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether.[38] (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)Rubik's cube: the study of its possible moves is a concrete application of group theorySome types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:group theory;field theory;vector spaces, whose study is essentially the same as linear algebra;ring theory;commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry;homological algebra;Lie algebra and Lie group theory;Boolean algebra, which is widely used for the study of the logical structure of computers.The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory.[39] The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.[40]Calculus and analysisA Cauchy sequence consists of elements that become arbitrarily close to each other as the sequence progresses (from left to right).Main articles: Calculus and Mathematical analysisCalculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz.[41] It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results.[42] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:Multivariable calculusFunctional analysis, where variables represent varying functions;Integration, measure theory and potential theory, all strongly related with Probability theory;Ordinary differential equations;Partial differential equations;Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.Discrete mathematicsMain article: Discrete mathematicsDiscrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers.[43] Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.[c] Algorithms – especially their implementation and computational complexity – play a major role in discrete mathematics.[44]The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century.[45] The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.[46]Discrete mathematics includes:Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapesGraph theory and hypergraphsCoding theory, including error correcting codes and a part of cryptographyMatroid theoryDiscrete geometryDiscrete probability distributionsGame theory (although continuous games are also studied, most common games, such as chess and poker are discrete)Discrete optimization, including combinatorial optimization, integer programming, constraint programmingMathematical logic and set theoryThe Venn diagram is a commonly used method to illustrate the relations between sets.Main articles: Mathematical logic and set theoryThe two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.[47][48] Before this period sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.[49]Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets,[50] but by showing that this implies different sizes of infinity (see Cantor's diagonal argument) and the existence of mathematical objects that cannot be computed, or even explicitly described (for example, Hamel bases of the real numbers over the rational numbers).[citation needed] This led to the controversy over Cantor's set theory.In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.This became the foundational crisis of mathematics.[51] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have.[20] For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning.[52] The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs.This approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.[53] This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.[54]These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories.[55]Statistics and other decision sciencesWhatever the form of a random population distribution (μ), the sampling mean (x̄) tends to a Gaussian distribution and its variance (σ) is given by the central limit theorem of probability theory.[56]Main article: StatisticsThe field of statistics is a type of mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments.[57] The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical models and the theory of inference, using model selection and estimation. The models and consequential predictions should then be tested against new data.[d]Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.[58] Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.[59]Computational mathematicsMain article: Computational mathematicsComputational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.HistoryMain article: History of mathematicsAncientThe history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,[60] was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are two of them. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years.[61][62]The Babylonian mathematical tablet Plimpton 322, dated to 1800 BCEvidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[63] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a agesimal numeral system which is still in use today for measuring angles and time.[64]Euclid holding a compass, as imagined by Raphael in this detail from The School of Athens[e]In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right.[65] Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.[66] His book, Elements, is widely considered the most successful and influential textbook of all time.[67] The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse.[68] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.[69] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),[70] trigonometry (Hipparchus of Nicaea, 2nd century BC),[71] and the beginnings of algebra (Diophantus, 3rd century AD).[72]The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century ADThe Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.Medieval and laterA page from al-Khwārizmī's AlgebraLeonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system.[73] Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.During the early modern period, mathematics began to develop at an accelerating pace in Western Europe. The development of calculus by Isaac Newton and Gottfried Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved.Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[74]Symbolic notation and terminologyMain articles: Mathematical notation and Language of mathematicsLeonhard Euler created and popularized much of the mathematical notation used today.Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous and accurate way.Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas.More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and include often subscripts. Operation and relations are generally represented by specific glyphs, such as + (plus), × (multiplication), {\textstyle \int } (integral), = (equal), < (less than). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of noun phrases and formulas play the role of clauses.Many technical terms used in mathematics are often neologisms, such as polynomial and homeomorphism. Many other technical terms are words of the common language that are used in an accurate meaning that may differs slightly from their common meaning. For example, in mathematics, "or" means "one, the other or both", while, in common language, it is either amiguous or means "one or the other but not both" (in mathematics, the latter is called "exclusive or").Also, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module is flat" and "a field is always a ring".Relationship with scienceCarl Friedrich Gauss, known as the prince of mathematiciansMathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model.There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often obtained from experimentation.[75] In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss (one of the greatest mathematicians of the 19th century) once replied "durch planmässiges Tattonieren" (through systematic experimentation).[76] However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence.[77][78][79][80]What precedes is only one aspect of the relationship between mathematics and other sciences. Other aspects are considered in the next subsections.Pure and applied mathematicsMain articles: Applied mathematics and Pure mathematicsIsaac Newton (left) and Gottfried Wilhelm Leibniz developed infinitesimal calculus.Until the end of the 19th century, the development of mathematics was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics. For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians. However, a notable exception occurred in Ancient Greece; see Pure mathematics § Ancient Greece.In the second half ot the 19th century, new mathematical theories were introduced which were not related with the physical world (at least at that time), in particular, non-Euclidean geometries and Cantor's theory of transfinite numbers. This was one of the starting points of the foundational crisis of mathematics, which was eventually solved by the systematization of the axiomatic method for defining mathematical structures.So, many mathematicians focused their research on internal problems, that is, pure mathematics, and this led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value.During the second half of the 20th century, it appeared that many theories issued from applications are also interesting from the point of view of pure mathematics, and that many results of pure mathematics have applications outside mathematics (see next section); in turn, the study of these applications may give new insights on the "pure theory". An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is definitely impossible to implement, because of a computational complexity that is much too high. For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.So, the distinction between pure and applied mathematics is presently more a question of personal research aim of mathematicians than a division of mathematics into broad areas. The Mathematics Subject Classification does not mention "pure mathematics" nor "applied mathematics". However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge.Unreasonable effectivenessThe unreasonable effectiveness of mathematics[8] is a phenomenon that was named and first made explicit by physicist Eugene Wigner. It is the fact that many mathematical theories, even the "purest" have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.A famous example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem.Another historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.In the 19th century, the internal development of geometry (pure mathematics) lead to define and study non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of the special relativity is a non-Euclidean space of dimension four, and spacetime of the general relativity is a (curved) manifold of dimension four.Similar examples of unexpected applications of mathematical theories can be found in many areas of mathematics.Another striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon {\displaystyle \Omega ^{-}.} In both cases, the equations of the theories had unexplained solutions, which led to conjecture the existence of a unknown particle, and to search these particles. In both cases, these particles were discovered a few years later by specific experiments.[81]PhilosophyMain article: Philosophy of mathematicsRealityThe connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.[82]Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views.[81]Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.[83] Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ...Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics.Proposed definitionsMain article: Definitions of mathematicsThere is no general consensus about a definition of mathematics or its epistemological status—that is, its place among other human activities.[84][85]A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[84] There is not even consensus on whether mathematics is an art or a science.[85] Some just say, "mathematics is what mathematicians do".[84] This makes sense, as there is a strong consensus among them about what is mathematics and what is not.Most proposed definitions try to define mathematics by its object of study.Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart.[86]In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given.[87] With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task.Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems —assertions whose validity relies on a proof, that is, a purely-logical deduction.Logic and rigorSee also: LogicMathematicians strive to develop their results with systematic reasoning in order to avoid mistaken "theorems". These false proofs often arise from fallible intuitions and have been common in mathematics' history. To allow deductive reasoning, some basic assumptions need to be admitted explicitly as axioms. Traditionally, these axioms were selected on the grounds of common-sense, but modern axioms typically express formal guarantees for primitive notions, such as simple objects and relations.The validity of a mathematical proof is fundamentally a matter of rigour, and misunderstanding rigor is a notable cause for some common misconceptions about mathematics. Mathematical language may give more precision than in everyday speech to ordinary words like or and only. Other words such as open and field are given new meanings for specific mathematical concepts. Sometimes, mathematicians even coin entirely new words (e.g. homeomorphism). This technical vocabulary is both precise and compact, making it possible to mentally process complex ideas. Mathematicians refer to this precision of language and logic as "rigor".The rigor expected in mathematics has varied over time: the ancient Greeks expected detailed arguments, but in Isaac Newton's time, the methods employed were less rigorous (not because of a different conception of mathematics, but because of the lack of the mathematical methods that are required for reaching rigor). Problems inherent in Newton's approach were solved only in the second half of the 19th century, with the formal definitions of real numbers, limits and integrals. Later in the early 20th century, Bertrand Russell and Alfred North Whitehead would publish their Principia Mathematica, an attempt to show that all mathematical concepts and statements could be defined, then proven entirely through symbolic logic. This was part of a wider philosophical program known as logicism, which sees mathematics as primarily an extension of logic.Despite mathematics' concision, many proofs require hundreds of pages to express. The emergence of computer-assisted proofs has allowed proof lengths to further expand. Assisted proofs may be erroneous if the proving software has flaws.[f][88] On the other hand, proof assistants allow for the verification of details that cannot be given in a hand-written proof, and provide certainty of the correctness of long proofs such as that of the 255-page Feit–Thompson theorem.[g]Psychology (aesthetic, creativity and intuition)The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process. An extreme example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.Creativity and rigor are not the only psychological aspects of the activity of mathematicians.Many mathematicians see their activity as a game, more specifically as solving puzzles. This aspect of mathematical activity is emphasized in recreational mathematics.Many mathematicians give also an aesthetic value to mathematics. Like beauty, it is hard to define, it is commonly related to elegance, which involves qualities like simplicity, symmetry, completeness, and generality. G. H. Hardy in A Mathematician's Apology expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic.[89]Paul Erdős expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book Proofs from THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform for harmonic analysis.Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts.[90] One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are created (as in art) or discovered (as in science).[81] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as intuitionism, which primarily identifies mathematics with certain creative processes in the mind.[91] Intuitionism is in turn one flavor of a stance known as constructivism, which only considers a mathematical object valid if it can be directly constructed, not merely guaranteed by logic indirectly. This leads committed constructivists to reject certain results, particularly arguments like existential proofs based on the law of excluded middle.[92]In the end, neither constructivism nor intuitionism displaced classical mathematics or achieved mainstream acceptance. However, these programs have motivated specific developments, such as intuitionistic logic and other foundational insights, which are appreciated in their own right.[92]EducationMain article: Mathematics educationThis section needs expansion with: more aspects of mathematics in society such as education, math as a career, popular culture, etc. You can help by adding to it. (June 2022)Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum. While the content of courses varies, many countries in the world teach mathematics to students for significant amounts of time.[93]Awards and prize problemsMain category: Mathematics awardsThe front side of the Fields MedalThe most prestigious award in mathematics is the Fields Medal,[94][95] established in 1936 and awarded every four years (except around World War II) to up to four individuals.[96][97] It is considered the mathematical equivalent of the Nobel Prize.[97]Other prestigious mathematics awards include:The Abel Prize, instituted in 2002[98] and first awarded in 2003[99]The Chern Medal for lifetime achievement, introduced in 2009[100] and first awarded in 2010[101]The Wolf Prize in Mathematics, also for lifetime achievement,[102] instituted in 1978[103]A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.[104] This list has achieved great celebrity among mathematicians[105][unreliable source?], and, as of 2022, at least thirteen of the problems (depending how some are interpreted) have been solved.[106]A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.[107] To date, only one of these problems, the Poincaré conjecture, has been solved.[108]A school is an educational institution designed to provide learning spaces and learning environments for the teaching of students under the direction of teachers. Most countries have systems of formal education, which is sometimes compulsory.[2] In these systems, students progress through a series of schools. The names for these schools vary by country (discussed in the Regional terms section below) but generally include primary school for young children and secondary school for teenagers who have completed primary education. An institution where higher education is taught is commonly called a university college or university.In addition to these core schools, students in a given country may also attend schools before and after primary (elementary in the U.S.) and secondary (middle school in the U.S.) education.[3] Kindergarten or preschool provide some schooling to very young children (typically ages 3–5). University, vocational school, college or seminary may be available after secondary school. A school may be dedicated to one particular field, such as a school of economics or dance. Alternative schools may provide nontraditional curriculum and methods.Non-government schools, also known as private schools,[4] may be required when the government does not supply adequate or specific educational needs. Other private schools can also be religious, such as Christian schools, gurukula (Hindu schools), madrasa (Arabic schools), hawzas (Shi'i Muslim schools), yeshivas (Jewish schools), and others; or schools that have a higher standard of education or seek to foster other personal achievements. Schools for adults include institutions of corporate training, military education and training and business schools.Critics of school often accuse the school system of failing to adequately prepare students for their future lives,[5] of encouraging certain temperaments while inhibiting others,[6] of prescribing students exactly what to do, how, when, where and with whom, which would suppress creativity,[7] and of using extrinsic measures such as grades and homework, which would inhibit children's natural curiosity and desire to learn.[8]In homeschooling and distance education, teaching and learning take place independent from the institution of school or in a virtual school outside a traditional school building, respectively. Schools are organized in several different organizational models, including departmental, small learning communities, academies, integrated, and schools-within-a-school.Contents1Etymology2History and development3Regional terms3.1United Kingdom and Commonwealth of Nations3.2India3.3Europe3.4North America and the United States3.5Africa4Ownership and operation5Components of most schools5.1Education facilities in low-income countries6Security7Health services8Online schools and classes9Stress10Discipline towards students11See also12References13Further reading14Sources15External linksEtymologyThe word school derives from Greek σχολή (scholē), originally meaning "leisure" and also "that in which leisure is employed", but later "a group to whom lectures were given, school".[9][10][11]History and developmentSee also: History of educationPlato's academy, mosaic from PompeiiThe concept of grouping students together in a centralized location for learning has existed since Classical antiquity. Formal schools have existed at least since ancient Greece (see Academy), ancient Rome (see Education in Ancient Rome) ancient India (see Gurukul), and ancient China (see History of education in China). The Byzantine Empire had an established schooling system beginning at the primary level. According to Traditions and Encounters, the founding of the primary education system began in 425 AD and "... military personnel usually had at least a primary education ...". The sometimes efficient and often large government of the Empire meant that educated citizens were a must. Although Byzantium lost much of the grandeur of Roman culture and extravagance in the process of surviving, the Empire emphasized efficiency in its war manuals. The Byzantine education system continued until the empire's collapse in 1453 AD.[12]In Western Europe, a considerable number of cathedral schools were founded during the Early Middle Ages in order to teach future clergy and administrators, with the oldest still existing, and continuously operated, cathedral schools being The King's School, Canterbury (established 597 CE), King's School, Rochester (established 604 CE), St Peter's School, York (established 627 CE) and Thetford Grammar School (established 631 CE). Beginning in the 5th century CE, monastic schools were also established throughout Western Europe, teaching religious and secular subjects.Mental calculations. In the school of S. Rachinsky by Nikolay Bogdanov-Belsky. Russia, 1895.In Europe, universities emerged during the 12th century; here, scholasticism was an important tool, and the academicians were called schoolmen. During the Middle Ages and much of the Early Modern period, the main purpose of schools (as opposed to universities) was to teach the Latin language. This led to the term grammar school, which in the United States informally refers to a primary school, but in the United Kingdom means a school that selects entrants based on ability or aptitude. The school curriculum has gradually broadened to include literacy in the vernacular language and technical, artistic, scientific, and practical subjects.Obligatory school attendance became common in parts of Europe during the 18th century. In Denmark-Norway, this was introduced as early as in 1739–1741, the primary end being to increase the literacy of the almue, i.e., the "regular people".[13] Many of the earlier public schools in the United States and elsewhere were one-room schools where a single teacher taught seven grades of boys and girls in the same classroom. Beginning in the 1920s, one-room schools were consolidated into multiple classroom facilities with transportation increasingly provided by kid hacks and school buses.Islam was another culture that developed a school system in the modern sense of the word. Emphasis was put on knowledge, which required a systematic way of teaching and spreading knowledge and purpose-built structures. At first, mosques combined religious performance and learning activities. However, by the 9th century, the madrassa was introduced, a school that was built independently from the mosque, such as al-Qarawiyyin, founded in 859 CE. They were also the first to make the Madrassa system a public domain under Caliph's control.Under the Ottomans, the towns of Bursa and Edirne became the main centers of learning. The Ottoman system of Külliye, a building complex containing a mosque, a hospital, madrassa, and public kitchen and dining areas, revolutionized the education system, making learning accessible to a broader public through its free meals, health care, and sometimes free accommodation.Regional termsThis section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2021) (Learn how and when to remove this template message)The term school varies by country, as do the names of the various levels of education within the country.United Kingdom and Commonwealth of NationsIn the United Kingdom, the term school refers primarily to pre-university institutions, and these can, for the most part, be divided into pre-schools or nursery schools, primary schools (sometimes further divided into infant school and junior school), and secondary schools. Various types of secondary schools in England and Wales include grammar schools, comprehensives, secondary moderns, and city academies. While they may have different names in Scotland, there is only one type of secondary school. However, they may be funded either by the state or independently funded. Scotland's school performance is monitored by Her Majesty's Inspectorate of Education. Ofsted reports on performance in England and Estyn reports on performance in Wales.In the United Kingdom, most schools are publicly funded and known as state schools or maintained schools in which tuition is provided for free. There are also private schools or independent schools that charge fees. Some of the most selective and expensive private schools are known as public schools, a usage that can be confusing to speakers of North American English. In North American usage, a public school is publicly funded or run.In much of the Commonwealth of Nations, including Australia, New Zealand, India, Pakistan, Bangladesh, Sri Lanka, South Africa, Kenya, and Tanzania, the term school refers primarily to pre-university institutions.IndiaLoyola School, Chennai, India – run by the Catholic Diocese of Madras. Christian missionaries played a pivotal role in establishing modern schools in India.In ancient India, schools were in the form of Gurukuls. Gurukuls were traditional Hindu residential learning schools, typically the teacher's house or a monastery. Schools today are commonly known by the Sanskrit terms Vidyashram, Vidyalayam, Vidya Mandir, Vidya Bhavan in India.[14][15] In southern languages, it is known as Pallikoodam or PaadaSaalai. During the Mughal rule, Madrasahs were introduced in India to educate the children of Muslim parents. British records show that indigenous education was widespread in the 18th century, with a school for every temple, mosque, or village in most regions. The subjects taught included Reading, Writing, Arithmetic, Theology, Law, Astronomy, Metaphysics, Ethics, Medical Science, and Religion.A school building in Kannur, IndiaUnder British rule, Christian missionaries from England, the United States, and other countries established missionary and boarding schools in India. Later as these schools gained popularity, more were started, and some gained prestige. These schools marked the beginning of modern schooling in India. The syllabus and calendar they followed became the benchmark for schools in modern India. Today most schools follow the missionary school model for tutoring, subject/syllabus, and governance, with minor changes.Schools in India range from large campuses with thousands of students and hefty fees to schools where children are taught under a tree with a small / no campus and are free of cost. There are various boards of schools in India, namely Central Board for Secondary Education (CBSE), Council for the Indian School Certificate Examinations (CISCE), Madrasa Boards of various states, Matriculation Boards of various states, State Boards of various boards, Anglo Indian Board, among others. Today's typical syllabus includes Language(s), Mathematics, Science – Physics, Chemistry, Biology, Geography, History, General Knowledge, and Information Technology/Computer Science. Extracurricular activities include physical education/sports and cultural activities like music, choreography, painting, and theatre/drama.EuropeAlbert Bettannier's 1887 painting La Tache noire depicts a child being taught about the "lost" province of Alsace-Lorraine in the aftermath of the Franco-Prussian War – an example of how European schools were often used in order to inoculate Nationalism in their pupils.In much of continental Europe, the term school usually applies to primary education, with primary schools that last between four and nine years, depending on the country. It also applies to secondary education, with secondary schools often divided between Gymnasiums and vocational schools, which again, depending on country and type of school, educate students for between three and six years. In Germany, students graduating from Grundschule are not allowed to progress into a vocational school directly. Instead, they are supposed to proceed to one of Germany's general education schools such as Gesamtschule, Hauptschule, Realschule or Gymnasium. When they leave that school, which usually happens at age 15–19, they may proceed to a vocational school. The term school is rarely used for tertiary education, except for some upper or high schools (German: Hochschule), which describe colleges and universities.In Eastern Europe modern schools (after World War II), of both primary and secondary educations, often are combined. In contrast, secondary education might be split into accomplished or not. The schools are classified as middle schools of general education. For the technical purposes, they include "degrees" of the education they provide out of three available: the first – primary, the second – unaccomplished secondary, and the third – accomplished secondary. Usually, the first two degrees of education (eight years) are always included. In contrast, the last one (two years) permits the students to pursue vocational or specialized educations.North America and the United StatesOne-room school in 1935, AlabamaIn North America, the term school can refer to any educational institution at any level and covers all of the following: preschool (for toddlers), kindergarten, elementary school, middle school (also called intermediate school or junior high school, depending on specific age groups and geographic region), high school (or in some cases senior high school), college, university, and graduate school.In the United States, school performance through high school is monitored by each state's department of education. Charter schools are publicly funded elementary or secondary schools that have been freed from some of the rules, regulations, and statutes that apply to other public schools. The terms grammar school and grade school are sometimes[why?] used to refer to a primary school. In addition, there are tax-funded magnet schools which offer different programs and instruction not available in traditional schools.AfricaIn West Africa, "school" can also refer to "bush" schools, Quranic schools, or apprenticeships. These schools include formal and informal learning.Bush schools are training camps that pass down cultural skills, traditions, and knowledge to their students. Bush schools are semi-similar to traditional western schools because they are separated from the larger community. These schools are located in forests outside of the towns and villages, and the space used is solely for these schools. Once the students have arrived in the forest, they cannot leave until their training is complete. Visitors are prohibited from these areas.[16]Instead of being separated by age, Bush schools are separated by gender. Women and girls cannot enter the boys' bush school territory and vice versa. Boys receive training in cultural crafts, fighting, hunting, and community laws among other subjects.[17] Girls are trained in their own version of the boys' bush school. They practice domestic affairs such as cooking, childcare, and being a good wife. Their training is focused on how to be a proper woman by societal standards.A madrasah in the GambiaQur'anic schools are the principal way of teaching the Quran and knowledge of the Islamic faith. These schools also fostered literacy and writing during the time of colonization. Today, the emphasis is on the different levels of reading, memorizing, and reciting the Quran. Attending a Qur'anic school is how children become recognized members of the Islamic faith. Children often attend state schools and a Qur'anic school.In Mozambique, specifically, there are two kinds of Qur'anic schools. They are the tariqa based and the Wahhabi-based schools. What makes these schools different is who controls them. Tariqa schools are controlled at the local level. In contrast, the Wahhabi are controlled by the Islamic Council.[18] Within the Qur'anic school system, there are levels of education. They range from a basic level of understanding, called chuo and kioni in local languages, to the most advanced, which is called ilimu.[19]In Nigeria, the term school broadly covers daycares, nursery schools, primary schools, secondary schools and tertiary institutions. Primary and secondary schools are either privately funded by religious institutions and corporate organisations or government-funded. Government-funded schools are commonly referred to as public schools. Students spend six years in primary school, three years in junior secondary school, and three years in senior secondary school. The first nine years of formal schooling is compulsory under the Universal Basic Education Program (UBEC).[20] Tertiary institutions include public and private universities, polytechnics, and colleges of education. Universities can be funded by the federal government, state governments, religious institutions, or individuals and organisations.Ownership and operationMany schools are owned or funded by states. Private schools operate independently from the government. Private schools usually rely on fees from families whose children attend the school for funding; however, sometimes such schools also receive government support (for example, through School vouchers). Many private schools are affiliated with a particular religion; these are known as parochial schools.Components of most schoolsThis section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2021) (Learn how and when to remove this template message)See also: Learning environment and Learning spaceA school entrance building in AustraliaSchools are organized spaces purposed for teaching and learning. The classrooms where teachers teach and students learn are of central importance. Classrooms may be specialized for certain subjects, such as laboratory classrooms for science education and workshops for industrial arts education.Typical schools have many other rooms and areas, which may include:Cafeteria (Commons), dining hall or canteen where students eat lunch and often breakfast and snacks.Athletic field, playground, gym, or track place where students participating in sports or physical education practiceSchoolyards, all-purpose playfields typically in elementary schools, often made of concrete.Auditorium or hall where student theatrical and musical productions can be staged and where all-school events such as assemblies are heldOffice where the administrative work of the school is doneLibrary where students ask librarians reference questions, check out books and magazines, and often use computersComputer labs where computer-based work is done and the internet accessedCultural activities where the students uphold their cultural practice through activities like games, dance, and musicEducation facilities in low-income countriesIn low-income countries, only 32% of primary, 43% of lower secondary and 52% of upper secondary schools have access to electricity.[21] This affects access to the internet, which is just 37% in upper secondary schools in low-income countries, as compared to 59% in those in middle-income countries and 93% in those in high-income countries.[21]Access to basic water, sanitation and hygiene is also far from universal. Among upper secondary schools, only 53% in low-income countries and 84% in middle-income countries have access to basic drinking water. Access to water and sanitation is universal in high-income countries.[21]SecurityMain article: School securityTo curtail violence, some schools have added CCTV surveillance cameras. This is especially common in schools with gang activity or violence.The safety of staff and students is increasingly becoming an issue for school communities, an issue most schools are addressing through improved security. Some have also taken measures such as installing metal detectors or video surveillance. Others have even taken measures such as having the children swipe identification cards as they board the school bus. These plans have included door numbering to aid public safety response for some schools.[clarification needed]Other security concerns faced by schools include bomb threats, gangs, and vandalism.[22] In recognition of these threats, the United Nations Sustainable Development Goal 4 advocates for upgrading education facilities to provide a safe, non-violent learning environment.[23]Health servicesMain article: School health servicesSchool health services are services from medical, teaching and other professionals applied in or out of school to improve the health and well-being of children and, in some cases, whole families. These services have been developed in different ways around the globe. However, the fundamentals are constant: the early detection, correction, prevention, or amelioration of disease, disability, and abuse from which school-aged children can suffer.Online schools and classesMain article: Virtual schoolSome schools offer remote access to their classes over the internet. Online schools also can provide support to traditional schools, as in the case of the School Net Namibia. Some online classes also provide experience in a class. When people take them, they have already been introduced to the subject and know what to expect. Classes provide high school/college credit, allowing students to take the classes at their own pace. Many online classes cost money to take, but some are offered free.ESL online learningInternet-based distance learning programs are offered widely through many universities. Instructors teach through online activities and assignments. Online classes are taught the same as in-person, with the same curriculum. The instructor offers the syllabus with their fixed requirements like any other class. Students can virtually turn their assignments in to their instructors according to deadlines. This being through via email or on the course webpage. This allows students to work at their own pace yet meet the correct deadlines. Students taking an online class have more flexibility in their schedules to take their classes at a time that works best.Conflicts with taking an online class may include not being face to face with the instructor when learning or being in an environment with other students. Online classes can also make understanding the content challenging, especially when unable to get in quick contact with the instructor. Online students have the advantage of using other online sources with assignments or exams for that specific class. Online classes also have the advantage of students not needing to leave their house for a morning class or worrying about their attendance for that class. Students can work at their own pace to learn and achieve within that curriculum.[24]The convenience of learning at home has been an attraction point for enrolling online. Students can attend class anywhere a computer can go – at home, in a library, or while traveling internationally. Online school classes are designed to fit a student's needs while allowing students to continue working and tending to their other obligations.[25] Online school education is divided into three subcategories: Online Elementary School, Online Middle School, Online High school.StressSee also: Cram schoolAs a profession, teaching has levels of work-related stress (WRS)[26] that are among the highest of any profession in some countries, such as the United Kingdom and the United States.[27] The degree of this problem is becoming increasingly recognized and support systems are being put into place.[28][29]Stress sometimes affects students more severely than teachers, up to the point where the students are prescribed stress medication. This stress is claimed to be related to standardized testing, and the pressure on students to score above average.[30][31]According to a 2008 mental health study by the Associated Press and mtvU,[32] eight in 10 U.S. college students said they had sometimes or frequently experienced stress in their daily lives. This was an increase of 20% from a survey five years previously. Thirty-four percent had felt depressed at some point in the past three months, 13 percent had been diagnosed with a mental health condition such as an anxiety disorder or depression, and 9 percent had seriously considered suicide.[32]Discipline towards studentsMain article: School disciplineSchools and their teachers have always been under pressure – for instance, pressure to cover the curriculum, perform well compared to other schools, and avoid the stigma of being "soft" or "spoiling" toward students. Forms of discipline, such as control over when students may speak, and normalized behaviour, such as raising a hand to speak, are imposed in the name of greater efficiency. Practitioners of critical pedagogy maintain that such disciplinary measures have no positive effect on student learning. Indeed, some argue that disciplinary practices detract from learning, saying that they undermine students' dignity and sense of self-worth – the latter occupying a more primary role in students' hierarchy of needs.In the United States education system (and other countries), social studies is the integrated study of multiple fields of social science and the humanities, including history, culture, geography, and political science. The term was first coined by American educators around the turn of the twentieth century as a catch-all for these subjects, as well as others which did not fit into the traditional models of lower education in the United States, such as philosophy and psychology.[1] One of the purposes of social studies, particularly at the level of higher education, is to integrate several disciplines, with their unique methodologies and special focuses of concentration, into a coherent field of subject areas that communicate with each other by sharing different academic "tools" and perspectives for deeper analysis of social problems and issues.[2] Social studies aims to train students for informed, responsible participation in a diverse democratic society. The content of social studies provides the necessary background knowledge in order to develop values and reasoned opinions, and the objective of the field is civic competence.[3]Contents1History of social studies2Subject fields3Teaching social studies4Ten themes of social studies4.1Culture4.2Time, continuity, and change4.3People, places, and environment4.4Individual development and identity4.5Individuals, groups, and institutions4.6Power, authority, and governance4.7Production, distribution, and consumption4.8Science, technology, and society4.9Global connections4.10Civic ideals and practices5References6External linksHistory of social studies[edit]The original onset of the social studies field emerged in the 19th century and later grew in the 20th century. Those foundations and building blocks were put into place in the 1820s in the country of Great Britain before being integrated into the United States. The purpose of the subject itself was to promote social welfare and its development in countries like the United States and others.[4]An early concept of social studies is found in John Dewey's philosophy of elementary and secondary education. Dewey valued the subject field of geography for uniting the study of human occupations with the study of the earth. He valued inquiry as a process of learning, as opposed to the absorption and recitation of facts, and he advocated for greater inquiry in elementary and secondary education, to mirror the kind of learning that takes place in higher education. His ideas are manifested to a large degree in the practice of inquiry-based learning and student-directed investigations implemented in contemporary social studies classrooms. Dewey valued the study of history for its social processes and application to contemporary social problems, rather than a mere narrative of human events. In this view, the study of history is made relevant to the modern student and is aimed at the improvement of society.[5]In the United States through the 1900s, social studies revolved around the study of geography, government, and history. In 1912, the Bureau of Education (not to be confused with its successor agency, the United States Department of Education) was tasked by then Secretary of the Interior Franklin Knight Lane with completely restructuring the American education system for the twentieth century. In response, the Bureau of Education, together with the National Education Association, created the Commission on the Reorganization of Secondary Education. The commission was made up of 16 committees (a 17th was established two years later, in 1916), each one tasked with the reform of a specific aspect of the American Education system. Notable among these was the Committee on Social Studies, which was created to consolidate and standardize various subjects that did not fit within normal school curricula into a new subject, to be called "the social studies."[6]In 1920, the work done by the Committee on Social Studies culminated in the publication and release of Bulletin No. 28 (also called "The Committee on Social Studies Report, 1916").[6] The 66-page bulletin, published and distributed by the U.S. Bureau of Education, is believed to be the first written work dedicated entirely to the subject. It was designed to introduce the concept to American educators and serve as a guide for the creation of nationwide curricula based around social studies. The bulletin proposed many ideas that were considered radical at the time, and it is regarded by many educators as one of the most controversial educational resources of the early twentieth century.[7][8]In the years after its release, the bulletin received criticism from educators on its vagueness, especially in regards to the definition of Social Studies itself.[7] Critics often point to Section 1 of the report, which vaguely defines Social Studies as "understood to be those whose subject matter relates directly to the organization and development of human society, and to man as a member of social groups."[6]The changes to the field of study never fully materialized until the 1950s, when changes occurred at the state and national levels that dictated the curriculum and the preparation standards of its teacher. This led to a decrease in the amount of factual knowledge being delivered instead of focusing on key concepts, generalizations, and intellectual skills. Eventually, around the 1980s and 1990s, the development of computer technologies helped grow the publishing industry. Textbooks were created around the curriculum of each state and that coupled with the increase in political factors from globalization and growing economies lead to changes in the public and private education system. Now came the influx of national curriculum standards, from the increase of testing to the accountability of teachers and school districts shifting the social study education system to what it is today.[9][10]Subject fields[edit]Social studies is not a subject, instead functioning as a field of study that incorporates many different subjects. It primarily includes the subjects of history, geography, economics, civics, and sociology. Through all of that, the elements of ethics, psychology, philosophy, anthropology, art, and literature are incorporated into the subject field itself. The field of study itself focuses on human beings and their respective relationships. With that, many of these subjects include some form of social utility that is beneficial to the subject field itself.[11]Teaching social studies[edit]To teach social studies in the United States, one must obtain a valid teaching certification to teach in that given state and a valid subject specific certification in social studies. The social studies certification process focuses on the core areas: history, geography, economics, civics, and political science. Each state has specific requirements for the certification process and the teacher must follow the specific guidelines of the state they wish to teach.[12]Ten themes of social studies[edit]According to the National Council for the Social Studies, there are ten themes that represent the standards about human experience that is constituted in the effectiveness of social studies as a subject study from pre-K through 12th grade.[13]Culture[edit]The study of culture and diversity allows learners to experience culture through all stages from learning to adaptation, shaping their respective lives and society itself.[13] This social studies theme includes the principles of multiculturalism, a field of study in its own right that aims to achieve greater understanding between culturally diverse groups of students as well as including the experiences of culturally diverse learners in the curriculum.[3]Time, continuity, and change[edit]Learners examine the past and the history of events that lead to the development of the current world. Ultimately, the learners will examine the beliefs and values of the past to apply them to the present. Learners build their inquiry skills in the study of history.[13]People, places, and environment[edit]Learners will understand who they are and the environment and places that surround them. It gives spatial views and perspectives of the world to the learner.[13] This theme is largely contained in the field of geography, which includes the study of humanity's connections with resources, instruction in reading maps and techniques and perspectives in analyzing information about human populations and the Earth's systems.[14]Individual development and identity[edit]Learners will understand their own personal identity, development, and actions. Through this, they will be able to understand the influences that surround them.[13]Individuals, groups, and institutions[edit]Learners will understand how groups and institutions influence people's everyday lives. They will be able to understand how groups and institutions are formed, maintained, and changed.[13]Power, authority, and governance[edit]Learners will understand the forms of power, authority, and governance from historical to contemporary times. They will become familiar with the purpose of power, and with the limits that power has on society.[13]Production, distribution, and consumption[edit]Learners will understand the organization of goods and services, ultimately preparing the learner for the study of greater economic issues.[13] The study of economic issues, and with it, financial literacy, is intended to increase students' knowledge and skills when it comes to participating in the economy as workers, producers, and consumers.[15]Science, technology, and society[edit]Learners will understand the relationship between science, technology, and society, understanding the advancement through the years and the impacts they have had.[13]Global connections[edit]Learners will understand the interactive environment of global interdependence and will understand the global connections that shape the everyday world.[13]Civic ideals and practices[edit]Learners will understand the rights and responsibilities of citizens and learn to grow in their appreciation of active citizenship. Ultimately, this helps their growth as full participants in society.[13] Some of the values that civics courses strive to teach are an understanding of the right to privacy, an appreciation for diversity in American society, and a disposition to work through democratic procedures. One of the curricular tools used in the field of civics education is a simulated congressional hearing.[16] Social studies educators and scholars distinguish between different levels of civic engagement, from the minimal engagement or non-engagement of the legal citizen to the most active and responsible level of the transformative citizen. Within social studies, the field of civics aims to educate and develop learners into transformative citizens who not only participate in a democracy, but challenge the status quo in the interest of social justice.[3]Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.[1][2]Science may be as old as the human species,[3] and some of the earliest archeological evidence for scientific reasoning is tens of thousands of years old.[4] The earliest written records in the history of science come from Ancient Egypt and Mesopotamia in around 3000 to 1200 BCE.[5][6] Their contributions to mathematics, astronomy, and medicine entered and shaped Greek natural philosophy of classical antiquity, whereby formal attempts were made to provide explanations of events in the physical world based on natural causes.[5][6] After the fall of the Western Roman Empire, knowledge of Greek conceptions of the world deteriorated in Western Europe during the early centuries (400 to 1000 CE) of the Middle Ages,[7] but was preserved in the Muslim world during the Islamic Golden Age[8] and later by the efforts of Byzantine Greek scholars who brought Greek manuscripts from the dying Byzantine Empire to Western Europe in the Renaissance.The recovery and assimilation of Greek works and Islamic inquiries into Western Europe from the 10th to 13th century revived "natural philosophy",[7][9] which was later transformed by the Scientific Revolution that began in the 16th century[10] as new ideas and discoveries departed from previous Greek conceptions and traditions.[11][12] The scientific method soon played a greater role in knowledge creation and it was not until the 19th century that many of the institutional and professional features of science began to take shape;[13][14] along with the changing of "natural philosophy" to "natural science".[15]Modern science is typically divided into three major branches:[16] natural sciences (e.g., biology, chemistry, and physics), which study the physical world; the social sciences (e.g., economics, psychology, and sociology), which study individuals and societies;[17][18] and the formal sciences (e.g., logic, mathematics, and theoretical computer science), which study formal systems, governed by axioms and rules.[19][20] There is disagreement whether the formal sciences are science disciplines,[21][22][23] because they do not rely on empirical evidence.[24][22] Applied sciences are disciplines that use scientific knowledge for practical purposes, such as in engineering and medicine.[25][26][27]New knowledge in science is advanced by research from scientists who are motivated by curiosity about the world and a desire to solve problems.[28][29] Contemporary scientific research is highly collaborative and is usually done by teams in academic and research institutions,[30] government agencies, and companies.[31][32] The practical impact of their work has led to the emergence of science policies that seek to influence the scientific enterprise by prioritizing the ethical and moral development of commercial products, armaments, health care, public infrastructure, and environmental protection.Contents1Etymology2History2.1Earliest roots2.2Classical antiquity2.3Middle Ages2.4Renaissance2.5Age of Enlightenment2.619th century2.720th century2.821st century3Branches3.1Natural science3.2Social science3.3Formal science3.4Applied science3.5Interdisciplinary science4Scientific research4.1Scientific method4.2Scientific literature4.3Challenges5Philosophy of science6Scientific community6.1Scientists6.2Learned societies6.3Awards7Society7.1Funding and policies7.2Education and awareness7.3Anti-science attitudes7.4Politics8See also9Notes10ReferencesEtymologyLook up science in Wiktionary, the free dictionary.The word science has been used in Middle English since the 14th century in the sense of "the state of knowing". The word was borrowed from the Anglo-Norman language as the suffix -cience, which was borrowed from the Latin word scientia, meaning "knowledge, awareness, understanding". It is a noun derivative of the Latin sciens meaning "knowing", and undisputedly derived from the Latin sciō, the present participle scīre, meaning "to know".[33]There are many hypotheses for science's ultimate word origin. According to Michiel de Vaan, Dutch linguist and Indo-Europeanist, sciō may have its origin in the Proto-Italic language as skije- or skijo- meaning "to know", which may originate from Proto-Indo-European language as skh1-ie, skh1-io, meaning "to incise". The Lexikon der indogermanischen Verben proposed sciō is a back-formation of nescīre, meaning "to not know, be unfamiliar with", which may derive from Proto-Indo-European sekH- in Latin secāre, or skh2-, from sḱʰeh2(i)- meaning "to cut".[34]In the past, science was a synonym for "knowledge" or "study", in keeping with its Latin origin. A person who conducted scientific research was called a "natural philosopher" or "man of science".[35] In 1833, William Whewell coined the term scientist and the term first appeared in literature one year later in Mary Somerville's On the Connexion of the Physical Sciences, published in the Quarterly Review.[36]HistoryMain article: History of scienceEarliest rootsMain article: History of science in early culturesThe Plimpton 322 tablet by the Babylonians records Pythagorean triples, written in about 1800 BCEScience has no single origin. Rather, scientific methods emerged gradually over the course of thousands of years, taking different forms around the world, and few details are known about the very earliest developments. Some of the earliest evidence for scientific reasoning is tens of thousands of years old,[4] and women likely played a central role in prehistoric science,[37] as did religious rituals.[38] Some Western authors have dismissed these efforts as "protoscientific".[39]Direct evidence for scientific processes becomes clearer with the advent of writing systems in early civilizations like Ancient Egypt and Mesopotamia.[5] Although the words and concepts of "science" and "nature" were not part of the conceptual landscape at the time, the ancient Egyptians and Mesopotamians made contributions that would later find a place in Greek and medieval science: mathematics, astronomy, and medicine.[40][5] From the 3rd millennium BCE, the ancient Egyptians developed a decimal numbering system,[41] solved practical problems using geometry,[42] and developed a calendar.[43] Their healing therapies involved drug treatments and the supernatural, such as prayers, incantations, and rituals.[5]The ancient Mesopotamians used knowledge about the properties of various natural chemicals for manufacturing pottery, faience, glass, soap, metals, lime plaster, and waterproofing.[44] They studied animal physiology, anatomy, behavior, and astrology for divinatory purposes.[45] The Mesopotamians had an intense interest in medicine[44] and the earliest medical prescriptions appeared in Sumerian during the Third Dynasty of Ur.[46] They seem to study scientific subjects which have practical or religious applications and have little interest of satisfying curiosity.[44]Classical antiquityMain article: History of science in classical antiquityPlato's Academy mosaic, made between 100 BCE to 79 AD, shows many Greek philosophers and scholarsIn classical antiquity, there is no real ancient analog of a modern scientist. Instead, well-educated, usually upper-class, and almost universally male individuals performed various investigations into nature whenever they could afford the time.[47] Before the invention or discovery of the concept of phusis or nature by the pre-Socratic philosophers, the same words tend to be used to describe the natural "way" in which a plant grows,[48] and the "way" in which, for example, one tribe worships a particular god. For this reason, it is claimed that these men were the first philosophers in the strict sense and the first to clearly distinguish "nature" and "convention".[49]The early Greek philosophers of the Milesian school, which was founded by Thales of Miletus and later continued by his successors Anaximander and Anaximenes, were the first to attempt to explain natural phenomena without relying on the supernatural.[50] The Pythagoreans developed a complex number philosophy[51]: 467–68  and contributed significantly to the development of mathematical science.[51]: 465  The theory of atoms was developed by the Greek philosopher Leucippus and his student Democritus.[52][53] The Greek doctor Hippocrates established the tradition of systematic medical science[54][55] and is known as "The Father of Medicine".[56]A turning point in the history of early philosophical science was Socrates' example of applying philosophy to the study of human matters, including human nature, the nature of political communities, and human knowledge itself. The Socratic method as documented by Plato's dialogues is a dialectic method of hypothesis elimination: better hypotheses are found by steadily identifying and eliminating those that lead to contradictions. The Socratic method searches for general commonly-held truths that shape beliefs and scrutinizes them for consistency.[57] Socrates criticized the older type of study of physics as too purely speculative and lacking in self-criticism.[58]Aristotle in the 4th century BCE created a systematic program of teleological philosophy.[59] In the 3rd century BCE, Greek astronomer Aristarchus of Samos was the first to propose a heliocentric model of the universe, with the Sun at the center and all the planets orbiting it.[60] Aristarchus's model was widely rejected because it was believed to violate the laws of physics,[60] while Ptolemy's Almagest, which contains a geocentric description of the Solar System, was accepted through the early Renaissance instead.[61][62] The inventor and mathematician Archimedes of Syracuse made major contributions to the beginnings of calculus.[63] Pliny the Elder was a Roman writer and polymath, who wrote the seminal encyclopedia Natural History.[64][65][66]Middle AgesMain article: History of science § Middle AgesThe first page of Vienna Dioscurides depicts a peacock, made in the 6th centuryDue to the collapse of the Western Roman Empire, the 5th century saw an intellectual decline in western Europe.[67]: 307, 311, 363, 402  During the period, Latin encyclopedists such as Isidore of Seville preserved the majority of general ancient knowledge.[68] In contrast, because the Byzantine Empire resisted attacks from invaders, they were able to preserve and improve prior learning. John Philoponus, a Byzantine scholar in the 500s, started to question Aristotle's teaching of physics, noting its flaws.[67]: 307, 311, 363, 402  His criticism served as an inspiration to medieval scholars and Galileo Galilei, who ten centuries later extensively cited his works.[67][69]During late antiquity and the early Middle Ages, natural phenomena were mainly examined via the Aristotelian approach. The approach includes Aristotle's four causes: material, formal, moving, and final cause.[70] Many Greek classical texts were preserved by the Byzantine empire and Arabic translations were done by groups such as the Nestorians and the Monophysites. Under the Caliphate, these Arabic translations were later improved and developed by Arabic scientists.[71] By the 6th and 7th centuries, the neighboring Sassanid Empire established the medical Academy of Gondeshapur, which is considered by Greek, Syriac, and Persian physicians as the most important medical center of the ancient world.[72]The House of Wisdom was established in Abbasid-era Baghdad, Iraq,[73] where the Islamic study of Aristotelianism flourished[74] until the Mongol invasions in the 13th century. Ibn al-Haytham, better known as Alhazen, began experimenting as a means to gain knowledge[75][76] and disproved Ptolemy's theory of vision[77]: Book I, [6.54]. p. 372  Avicenna's compilation of the Canon of Medicine, a medical encyclopedia, is considered to be one of the most important publications in medicine and was used until the 18th century.[78]By the eleventh century, most of Europe had become Christian,[7] and in 1088, the University of Bologna emerged as the first university in Europe.[79] As such, demand for Latin translation of ancient and scientific texts grew,[7] a major contributor to the Renaissance of the 12th century. Renaissance scholasticism in western Europe flourished, with experiments done by observing, describing, and classifying subjects in nature.[80] In the 13rd century, medical teachers and students at Bologna began opening human bodies, leading to the first anatomy textbook based on human dissection by Mondino de Luzzi.[81]RenaissanceMain articles: Scientific Revolution and Science in the RenaissanceDrawing of the heliocentric model as proposed by the Copernicus's De revolutionibus orbium coelestiumNew developments in optics played a role in the inception of the Renaissance, both by challenging long-held metaphysical ideas on perception, as well as by contributing to the improvement and development of technology such as the camera obscura and the telescope. At the start of the Renaissance, Roger Bacon, Vitello, and John Peckham each built up a scholastic ontology upon a causal chain beginning with sensation, perception, and finally apperception of the individual and universal forms of Aristotle.[77]: Book I  A model of vision later known as perspectivism was exploited and studied by the artists of the Renaissance. This theory uses only three of Aristotle's four causes: formal, material, and final.[82]In the sixteenth century, Nicolaus Copernicus formulated a heliocentric model of the Solar System, stating that the planets revolve around the Sun, instead of the geocentric model where the planets and the Sun revolve around the Earth. This was based on a theorem that the orbital periods of the planets are longer as their orbs are farther from the center of motion, which he found not to agree with Ptolemy's model.[83]Johannes Kepler and others challenged the notion that the only function of the eye is perception, and shifted the main focus in optics from the eye to the propagation of light.[82][84] Kepler is best known, however, for improving Copernicus' heliocentric model through the discovery of Kepler's laws of planetary motion. Kepler did not reject Aristotelian metaphysics and described his work as a search for the Harmony of the Spheres.[85] Galileo had made significant contributions to astronomy, physics and engineering. However, he became persecuted after Pope Urban VIII sentenced him for writing about the heliocentric model.[86]The printing press was widely used to publish scholarly arguments, including some that disagreed widely with contemporary ideas of nature.[87] Francis Bacon and René Descartes published philosophical arguments in favor of a new type of non-Aristotelian science. Bacon emphasized the importance of experiment over contemplation, questioned the Aristotelian concepts of formal and final cause, promoted the idea that science should study the laws of nature and the improvement of all human life.[88] Descartes emphasized individual thought and argued that mathematics rather than geometry should be used to study nature.[89]Age of EnlightenmentMain article: Science in the Age of EnlightenmentTitle page of the 1687 first edition of Philosophiæ Naturalis Principia Mathematica by Issac NewtonAt the start of the Age of Enlightenment, Isaac Newton formed the foundation of classical mechanics by his Philosophiæ Naturalis Principia Mathematica, greatly influencing future physicists.[90] Gottfried Wilhelm Leibniz incorporated terms from Aristotelian physics, now used in a new non-teleological way. This implied a shift in the view of objects: objects were now considered as having no innate goals. Leibniz assumed that different types of things all work according to the same general laws of nature, with no special formal or final causes.[91]During this time, the declared purpose and value of science became producing wealth and inventions that would improve human lives, in the materialistic sense of having more food, clothing, and other things. In Bacon's words, "the real and legitimate goal of sciences is the endowment of human life with new inventions and riches", and he discouraged scientists from pursuing intangible philosophical or spiritual ideas, which he believed contributed little to human happiness beyond "the fume of subtle, sublime or pleasing [speculation]".[92]Science during the Enlightenment was dominated by scientific societies[93] and academies, which had largely replaced universities as centers of scientific research and development. Societies and academies were the backbones of the maturation of the scientific profession. Another important development was the popularization of science among an increasingly literate population.[94] Enlightenment philosophers chose a short history of scientific predecessors – Galileo, Boyle, and Newton principally – as the guides to every physical and social field of the day.[95]The 18th century saw significant advancements in the practice of medicine[96] and physics;[97] the development of biological taxonomy by Carl Linnaeus;[98] a new understanding of magnetism and electricity;[99] and the maturation of chemistry as a discipline.[100] Ideas on human nature, society, and economics evolved during the Enlightenment. Hume and other Scottish Enlightenment thinkers developed A Treatise of Human Nature, which was expressed historically in works by authors including James Burnett, Adam Ferguson, John Millar and William Robertson, all of whom merged a scientific study of how humans behaved in ancient and primitive cultures with a strong awareness of the determining forces of modernity.[101] Modern sociology largely originated from this movement.[102] In 1776, Adam Smith published The Wealth of Nations, which is often considered the first work on modern economics.[103]19th centuryMain article: 19th century in scienceThe first diagram of an evolutionary tree made by Charles Darwin in 1837During the nineteenth century, many distinguishing characteristics of contemporary modern science began to take shape. These included the transformation of the life and physical sciences, frequent use of precision instruments, emergence of terms such as "biologist", "physicist", "scientist", increased professionalization of those studying nature, scientists gained cultural authority over many dimensions of society, industrialization of numerous countries, thriving of popular science writings and emergence of science journals.[104] During the late 19th century, psychology emerged as a separate discipline from philosophy when Wilhelm Wundt founded the first laboratory for psychological research in 1879.[105]During the mid-19th century, Charles Darwin and Alfred Russel Wallace independently proposed the theory of evolution by natural selection in 1858, which explained how different plants and animals originated and evolved. Their theory was set out in detail in Darwin's book On the Origin of Species, published in 1859.[106] Separately, Gregor Mendel presented his paper, "Experiments on Plant Hybridization" in 1865,[107] which outlined the principles of biological inheritance, serving as the basis for modern genetics.[108]Early in the 19th century, John Dalton suggested the modern atomic theory, based on Democritus's original idea of indivisible particles called atoms.[109] The laws of conservation of energy, conservation of momentum and conservation of mass suggested a highly stable universe where there could be little loss of resources. However, with the advent of the steam engine and the industrial revolution there was an increased understanding that not all forms of energy have the same energy qualities, the ease of conversion to useful work or to another form of energy.[110] This realization led to the development of the laws of thermodynamics, in which the free energy of the universe is seen as constantly declining: the entropy of a closed universe increases over time.[a]The electromagnetic theory was established in the 19th century by the works of Hans Christian Ørsted, André-Marie Ampère, Michael Faraday, James Clerk Maxwell, Oliver Heaviside, and Heinrich Hertz. The new theory raised questions that could not easily be answered using Newton's framework. The discovery of X-rays inspired the discovery of radioactivity by Henri Becquerel and Marie Curie in 1896,[113] Marie Curie then became the first person to win two Nobel prizes.[114] In the next year came the discovery of the first subatomic particle, the electron.[115]20th centuryFirst global view of the ozone hole in 1983, using a space telescopeMain article: 20th century in scienceIn the first half of the century, the development of antibiotics and artificial fertilizers improved human living standards globally.[116][117] Harmful environmental issues such as ozone depletion, ocean acidification, eutrophication and climate change came to the public's attention and caused the onset of environmental studies.[118]During this period, scientific experimentation became increasingly larger in scale and funding.[119] The extensive technological innovation stimulated by World War I, World War II, and the Cold War led to competitions between global powers, such as the Space Race[120] and nuclear arms race.[121] Substantial international collaborations were also made, despite armed conflicts.[122]In the late 20th century, active recruitment of women and elimination of discrimination greatly increased the number of women scientists, but large gender disparities remained in some fields.[123] The discovery of the cosmic microwave background in 1964[124] led to a rejection of the steady-state model of the universe in favor of the Big Bang theory of Georges Lemaître.[125]The century saw fundamental changes within science disciplines. Evolution became a unified theory in the early 20th-century when the modern synthesis reconciled Darwinian evolution with classical genetics.[126] Albert Einstein's theory of relativity and the development of quantum mechanics complement classical mechanics to describe physics in extreme length, time and gravity.[127][128] Widespread use of integrated circuits in the last quarter of the 20th century combined with communications satellites led to a revolution in information technology and the rise of the global internet and mobile computing, including smartphones. The need for mass systematization of long, intertwined causal chains and large amounts of data led to the rise of the fields of systems theory and computer-assisted scientific modeling.[129]21st centuryRadio light image of M87 black hole, made by the earth-spanning Event Horizon Telescope array in 2019Main article: 21st century § Science and technologyThe Human Genome Project was completed in 2003 by identifying and mapping all of the genes of the human genome.[130] The first induced pluripotent human stem cells were made in 2006, allowing adult cells to be transformed into stem cells and turn to any cell type found in the body.[131] With the affirmation of the Higgs boson discovery in 2013, the last particle predicted by the Standard Model of particle physics was found.[132] In 2015, gravitational waves, predicted by general relativity a century before, were first observed.[133][134] In 2019, the international collaboration Event Horizon Telescope presented the first direct image of a black hole's accretion disk.[135]BranchesModern science is commonly divided into three major branches: natural science, social science, and formal science.[16] Each of these branches comprises various specialized yet overlapping scientific disciplines that often possess their own nomenclature and expertise.[136] Both natural and social sciences are empirical sciences,[137] as their knowledge is based on empirical observations and is capable of being tested for its validity by other researchers working under the same conditions.[138]Natural scienceNatural science is the study of the physical world. It can be divided into two main branches: life science and physical science. These two branches may be further divided into more specialized disciplines. For example, physical science can be subdivided into physics, chemistry, astronomy, and earth science. Modern natural science is the successor to the natural philosophy that began in Ancient Greece. Galileo, Descartes, Bacon, and Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way. Still, philosophical perspectives, conjectures, and presuppositions, often overlooked, remain necessary in natural science.[139] Systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, animals, minerals, and so on.[140] Today, "natural history" suggests observational descriptions aimed at popular audiences.[141]Social scienceSupply and demand curve in economics, crossing over at the optimal equilibriumSocial science is the study of human behavior and functioning of societies.[17][18] It has many disciplines that include, but are not limited to anthropology, economics, history, human geography, political science, psychology, and sociology.[17] In the social sciences, there are many competing theoretical perspectives, many of which are extended through competing research programs such as the functionalists, conflict theorists, and interactionists in sociology.[17] Due to the limitations of conducting controlled experiments involving large groups of individuals or complex situations, social scientists may adopt other research methods such as the historical method, case studies, and cross-cultural studies. Moreover, if quantitative information is available, social scientists may rely on statistical approaches to better understand social relationships and processes.[17]Formal scienceFormal science is an area of study that generates knowledge using formal systems.[142][19][20] A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules.[143] It includes mathematics,[144][145] systems theory, and theoretical computer science. The formal sciences share similarities with the other two branches by relying on objective, careful, and systematic study of an area of knowledge. They are, however, different from the empirical sciences as they rely exclusively on deductive reasoning, without the need for empirical evidence, to verify their abstract concepts.[24][146][138] The formal sciences are therefore a priori disciplines and because of this, there is disagreement on whether they constitute a science.[21][147] Nevertheless, the formal sciences play an important role in the empirical sciences. Calculus, for example, was initially invented to understand motion in physics.[148] Natural and social sciences that rely heavily on mathematical applications include mathematical physics,[149] chemistry,[150] biology,[151] finance,[152] and economics.[153]Applied scienceA steam turbine with the case opened, such turbines produce most of the electricity used todayApplied science is the use of the scientific method and knowledge to attain practical goals and includes a broad range of disciplines such as engineering and medicine.[154][27] Engineering is the use of scientific principles to invent, design and build machines, structures and technologies.[155] Science may contribute to the development of new technologies.[156] Medicine is the practice of caring for patients by maintaining and restoring health through the prevention, diagnosis, and treatment of injury or disease.[157][158] The applied sciences are often contrasted with the basic sciences, which are focused on advancing scientific theories and laws that explain and predict events in the natural world.[159][160]Computational science applies computing power to simulate real-world situations, enabling a better understanding of scientific problems than formal mathematics alone can achieve. The use of machine learning and artificial intelligence is becoming a central feature of computational contributions to science for example in agent-based computational economics, random forests, topic modeling and various forms of prediction. However, machines alone rarely advance knowledge as they require human guidance and capacity to reason; and they can introduce bias against certain social groups or sometimes underperform against humans.[161][162]Interdisciplinary scienceInterdisciplinary science involves the combination of two or more disciplines into one,[163] such as bioinformatics, a combination of biology and computer science.[164] The concept has existed since the ancient Greek and it became popular again in the 20th century.[165]Scientific researchScientific research can be labeled as either basic or applied research. Basic research is the search for knowledge and applied research is the search for solutions to practical problems using this knowledge. Most understanding comes from basic research, though sometimes applied research targets specific practical problems. This leads to technological advances that were not previously imaginable.[166]Scientific methodA diagram variant of scientific method represented as an ongoing processScientific research involves using the scientific method, which seeks to objectively explain the events of nature in a reproducible way.[167] Scientists usually take for granted a set of basic assumptions that are needed to justify the scientific method: there is an objective reality shared by all rational observers; this objective reality is governed by natural laws; these laws were discovered by means of systematic observation and experimentation.[168] Mathematics is essential in the formation of hypotheses, theories, and laws, because it is used extensively in quantitative modeling, observing, and collecting measurements.[169] Statistics is used to summarize and analyze data, which allows scientists to assess the reliability of experimental results.[170]In the scientific method, an explanatory thought experiment or hypothesis is put forward as an explanation using parsimony principles and is expected to seek consilience – fitting with other accepted facts related to an observation or scientific question.[171] This tentative explanation is used to make falsifiable predictions, which are typically posted before being tested by experimentation. Disproof of a prediction is evidence of progress.[167]: 4–5 [172] Experimentation is especially important in science to help establish causal relationships to avoid the correlation fallacy, though in some sciences such as astronomy or geology, a predicted observation might be more appropriate.[173]When a hypothesis proves unsatisfactory, it is modified or discarded.[174] If the hypothesis survived testing, it may become adopted into the framework of a scientific theory, a logically reasoned, self-consistent model or framework for describing the behavior of certain natural events. A theory typically describes the behavior of much broader sets of observations than a hypothesis; commonly, a large number of hypotheses can be logically bound together by a single theory. Thus a theory is a hypothesis explaining various other hypotheses. In that vein, theories are formulated according to most of the same scientific principles as hypotheses. Scientists may generate a model, an attempt to describe or depict an observation in terms of a logical, physical or mathematical representation and to generate new hypotheses that can be tested by experimentation.[175]While performing experiments to test hypotheses, scientists may have a preference for one outcome over another.[176][177] Eliminating the bias can be achieved by transparency, careful experimental design, and a thorough peer review process of the experimental results and conclusions.[178][179] After the results of an experiment are announced or published, it is normal practice for independent researchers to double-check how the research was performed, and to follow up by performing similar experiments to determine how dependable the results might be.[180] Taken in its entirety, the scientific method allows for highly creative problem solving while minimizing the effects of subjective and confirmation bias.[181] Intersubjective verifiability, the ability to reach a consensus and reproduce results, is fundamental to the creation of all scientific knowledge.[182]Scientific literatureMain articles: Scientific literature and Lists of important publications in scienceCover of the first issue of Nature, November 4, 1869Scientific research is published in a range of literature.[183] Scientific journals communicate and document the results of research carried out in universities and various other research institutions, serving as an archival record of science. The first scientific journal, Journal des sçavans by Philosophical Transactions, began publication in 1665. Since that time the total number of active periodicals has steadily increased. In 1981, one estimate for the number of scientific and technical journals in publication was 11,500.[184]Most scientific journals cover a single scientific field and publish the research within that field; the research is normally expressed in the form of a scientific paper. Science has become so pervasive in modern societies that it is considered necessary to communicate the achievements, news, and ambitions of scientists to a wider population.[185]ChallengesThe replication crisis is an ongoing methodological crisis that affects parts of the social and life sciences. In subsequent investigations, the results of many scientific studies are proven to be unrepeatable.[186] The crisis has long-standing roots; the phrase was coined in the early 2010s[187] as part of a growing awareness of the problem. The replication crisis represents an important body of research in metascience, which aims to improve the quality of all scientific research while reducing waste.[188]An area of study or speculation that masquerades as science in an attempt to claim a legitimacy that it would not otherwise be able to achieve is sometimes referred to as pseudoscience, fringe science, or junk science.[189][190] Physicist Richard Feynman coined the term "cargo cult science" for cases in which researchers believe and at a glance looks like they are doing science, but lack the honesty allowing their results to be rigorously evaluated.[191] Various types of commercial advertising, ranging from hype to fraud, may fall into these categories. Science has been described as "the most important tool" for separating valid claims from invalid ones.[192]There can also be an element of political or ideological bias on all sides of scientific debates. Sometimes, research may be characterized as "bad science," research that may be well-intended but is incorrect, obsolete, incomplete, or over-simplified expositions of scientific ideas. The term "scientific misconduct" refers to situations such as where researchers have intentionally misrepresented their published data or have purposely given credit for a discovery to the wrong person.[193]Philosophy of scienceFor Kuhn, the addition of epicycles in Ptolemaic astronomy was "normal science" within a paradigm, whereas the Copernican revolution was a paradigm shift.There are different schools of thought in the philosophy of science. The most popular position is empiricism, which holds that knowledge is created by a process involving observation; scientific theories generalize observations.[194] Empiricism generally encompasses inductivism, a position that explains how general theories can be made from the finite amount of empirical evidence available. Many versions of empiricism exist, with the predominant ones being Bayesianism[195] and the hypothetico-deductive method.[194]Empiricism has stood in contrast to rationalism, the position originally associated with Descartes, which holds that knowledge is created by the human intellect, not by observation.[196] Critical rationalism is a contrasting 20th-century approach to science, first defined by Austrian-British philosopher Karl Popper. Popper rejected the way that empiricism describes the connection between theory and observation. He claimed that theories are not generated by observation, but that observation is made in the light of theories: that the only way theory A can be affected by observation is after theory A were to conflict with observation, but theory B were to survive the observation.[197] Popper proposed replacing verifiability with falsifiability as the landmark of scientific theories, replacing induction with falsification as the empirical method.[197] Popper further claimed that there is actually only one universal method, not specific to science: the negative method of criticism, trial and error,[198] covering all products of the human mind, including science, mathematics, philosophy, and art.[199]Another approach, instrumentalism, emphasizes the utility of theories as instruments for explaining and predicting phenomena. It views scientific theories as black boxes with only their input (initial conditions) and output (predictions) being relevant. Consequences, theoretical entities, and logical structure are claimed to be something that should be ignored.[200] Close to instrumentalism is constructive empiricism, according to which the main criterion for the success of a scientific theory is whether what it says about observable entities is true.[201]Thomas Kuhn argued that the process of observation and evaluation takes place within a paradigm, a logically consistent "portrait" of the world that is consistent with observations made from its framing. He characterized normal science as the process of observation and "puzzle solving" which takes place within a paradigm, whereas revolutionary science occurs when one paradigm overtakes another in a paradigm shift.[202] Each paradigm has its own distinct questions, aims, and interpretations. The choice between paradigms involves setting two or more "portraits" against the world and deciding which likeness is most promising. A paradigm shift occurs when a significant number of observational anomalies arise in the old paradigm and a new paradigm makes sense of them. That is, the choice of a new paradigm is based on observations, even though those observations are made against the background of the old paradigm. For Kuhn, acceptance or rejection of a paradigm is a social process as much as a logical process. Kuhn's position, however, is not one of relativism.[203]Finally, another approach often cited in debates of scientific skepticism against controversial movements like "creation science" is methodological naturalism. Naturalists maintain that a difference should be made between natural and supernatural, and science should be restricted to natural explanations.[204] Methodological naturalism maintains that science requires strict adherence to empirical study and independent verification.[205]Scientific communityThe scientific community is a network of interacting scientists who conducts scientific research. The community consists of smaller groups working in scientific fields. By having peer review, through discussion and debate within journals and conferences, scientists maintain the quality of research methodology and objectivity when interpreting results.[206]ScientistsMarie Curie was the first person to be awarded two Nobel Prizes: Physics in 1903 and Chemistry in 1911.[114]Scientists are individuals who conduct scientific research to advance knowledge in an area of interest.[207][208] In modern times, many professional scientists are trained in an academic setting and upon completion, attain an academic degree, with the highest degree being a doctorate such as a Doctor of Philosophy or PhD.[209] Many scientists pursue careers in various sectors of the economy such as academia, industry, government, and nonprofit organizations.[210][211][212]Scientists exhibit a strong curiosity about reality and a desire to apply scientific knowledge for the benefit of health, nations, the environment, or industries. Other motivations include recognition by their peers and prestige. In modern times, many scientists have advanced degrees[213] in an area of science and pursue careers in various sectors of the economy such as academia, industry, government, and nonprofit environments.[214][215]Science has historically been a male-dominated field, with notable exceptions. Women in science faced considerable discrimination in science, much as they did in other areas of male-dominated societies. For example, women were frequently being passed over for job opportunities and denied credit for their work.[216] The achievements of women in science have been attributed to the defiance of their traditional role as laborers within the domestic sphere.[217] Lifestyle choice plays a major role in female engagement in science; female graduate students' interest in careers in research declines dramatically throughout graduate school, whereas that of their male colleagues remains unchanged.[218]Learned societiesPicture of scientists in 200th anniversary of the Prussian Academy of Sciences, 1900Learned societies for the communication and promotion of scientific thought and experimentation have existed since the Renaissance.[219] Many scientists belong to a learned society that promotes their respective scientific discipline, profession, or group of related disciplines.[220] Membership may either be open to all, require possession of scientific credentials, or conferred by election.[221] Most scientific societies are non-profit organizations, and many are professional associations. Their activities typically include holding regular conferences for the presentation and discussion of new research results and publishing or sponsoring academic journals in their discipline. Some societies act as professional bodies, regulating the activities of their members in the public interest or the collective interest of the membership.The professionalization of science, begun in the 19th century, was partly enabled by the creation of national distinguished academies of sciences such as the Italian Accademia dei Lincei in 1603,[222] the British Royal Society in 1660,[223] the French Academy of Sciences in 1666,[224] the American National Academy of Sciences in 1863,[225] the German Kaiser Wilhelm Society in 1911,[226] and the Chinese Academy of Sciences in 1949.[227] International scientific organizations, such as the International Science Council, are devoted to international cooperation for science advancement.[228]AwardsScience awards are usually given to individuals or organizations that have made significant contributions to a discipline. They are often given by prestigious institutions, thus it is considered a great honor for a scientist receiving them. Since the early Renaissance, scientists are often awarded medals, money, and titles. The Nobel Prize, a widely regarded prestigious award, is awarded annually to those who have achieved scientific advances in the fields of medicine, physics, and chemistry.[229]Society"Science and society" redirects here. Not to be confused with Science & Society or Sociology of scientific knowledge.Funding and policiesBudget of NASA as percentage of United States federal budget, peaking at 4.4% in 1966 and slowly declining sinceScientific research is often funded through a competitive process in which potential research projects are evaluated and only the most promising receive funding. Such processes, which are run by government, corporations, or foundations, allocate scarce funds. Total research funding in most developed countries is between 1.5% and 3% of GDP.[230] In the OECD, around two-thirds of research and development in scientific and technical fields is carried out by industry, and 20% and 10% respectively by universities and government. The government funding proportion in certain fields is higher, and it dominates research in social science and humanities. In the lesser-developed nations, government provides the bulk of the funds for their basic scientific research.[231]Many governments have dedicated agencies to support scientific research, such as the National Science Foundation in the United States,[232] the National Scientific and Technical Research Council in Argentina,[233] Commonwealth Scientific and Industrial Research Organization in Australia,[234] National Centre for Scientific Research in France,[235] the Max Planck Society in Germany,[236] and National Research Council in Spain.[237] In commercial research and development, all but the most research-oriented corporations focus more heavily on near-term commercialization possibilities rather than research driven by curiosity.[238]Science policy is concerned with policies that affect the conduct of the scientific enterprise, including research funding, often in pursuance of other national policy goals such as technological innovation to promote commercial product development, weapons development, health care, and environmental monitoring. Science policy sometimes refers to the act of applying scientific knowledge and consensus to the development of public policies. In accordance with public policy being concerned about the well-being of its citizens, science policy's goal is to consider how science and technology can best serve the public.[239] Public policy can directly affect the funding of capital equipment and intellectual infrastructure for industrial research by providing tax incentives to those organizations that fund research.[185]Education and awarenessMain articles: Public awareness of science and Science journalismDinosaur exhibit in the Houston Museum of Natural ScienceScience education for the general public is embedded in the school curriculum, and is supplemented by online pedagogical content (for example, YouTube and Khan Academy), museums, and science magazines and blogs. Scientific literacy is chiefly concerned with an understanding of the scientific method, units and methods of measurement, empiricism, a basic understanding of statistics (correlations, qualitative versus quantitative observations, aggregate statistics), as well as a basic understanding of core scientific fields, such as physics, chemistry, biology, ecology, geology and computation. As a student advances into higher stages of formal education, the curriculum becomes more in depth. Traditional subjects usually included in the curriculum are natural and formal sciences, although recent movements include social and applied science as well.[240]The mass media face pressures that can prevent them from accurately depicting competing scientific claims in terms of their credibility within the scientific community as a whole. Determining how much weight to give different sides in a scientific debate may require considerable expertise regarding the matter.[241] Few journalists have real scientific knowledge, and even beat reporters who are knowledgeable about certain scientific issues may be ignorant about other scientific issues that they are suddenly asked to cover.[242][243]Science magazines such as New Scientist, Science & Vie, and Scientific American cater to the needs of a much wider readership and provide a non-technical summary of popular areas of research, including notable discoveries and advances in certain fields of research.[244] Science fiction genre, primarily speculative fiction, can transmit the ideas and methods of science to the general public.[245] Recent efforts to intensify or develop links between science and non-scientific disciplines, such as literature or poetry, include the Creative Writing Science resource developed through the Royal Literary Fund.[246]Anti-science attitudesWhile the scientific method is broadly accepted in the scientific community, some fractions of society reject certain scientific positions or are skeptical about science. Examples are the common notion that COVID-19 is not a major health threat to the US (held by 39% of Americans in August 2021)[247] or the belief that climate change is not a major threat to the US (also held by 40% of Americans, in late 2019 and early 2020).[248] Psychologists have pointed to four factors driving rejection of scientific results:[249]Scientific authorities are sometimes seen as inexpert, untrustworthy, or biased.Some marginalized social groups hold anti-science attitudes, in part because these groups have often been exploited in unethical experiments.[250]Messages from scientists may contradict deeply-held existing beliefs or morals.The delivery of a scientific message may not be appropriately targeted to a recipient's learning style.Anti-science attitudes seem to be often caused by fear of rejection in social groups. For instance, climate change is perceived as a threat by only 22% of Americans on the right side of the political spectrum, but by 85% on the left.[251] That is, if someone on the left would not consider climate change as a threat, this person may face contempt and be rejected in that social group. In fact, people may rather deny a scientifically accepted fact than lose or jeopardize their social status.[252]PoliticsPublic opinion on global warming in the United States by political party[253]Attitudes towards science are often determined by political opinions and goals. Government, business and advocacy groups have been known to use legal and economic pressure to influence scientific researchers. Many factors can act as facets of the politicization of science such as anti-intellectualism, perceived threats to religious beliefs, and fear for business interests.[254] Politicization of science is usually accomplished when scientific information is presented in a way that emphasizes the uncertainty associated with the scientific evidence.[255] Tactics such as shifting conversation, failing to acknowledge facts, and capitalizing on doubt of scientific consensus have been used to gain more attention for views that have been undermined by scientific evidence.[256] Examples of issues that have involved the politicization of science include the global warming controversy, health effects of pesticides, and health effects of tobacco.[256][257]History of mathematicsFrom Wikipedia, the free encyclopediaJump to navigationJump to searchA proof from Euclid's Elements (c. 300 BC), widely considered the most influential textbook of all time.[1]MathematicshideAreasNumber theoryGeometryAlgebraCalculus and analysisDiscrete mathematicsLogic and set theoryProbabilityStatistics and decision scienceshideRelationship with other fieldsPhysicsComputationBiologyLinguisticsEconomicsPhilosophyEducationPortalvteWe ask you, humbly: don't scroll away. Hi. Sorry to interrupt again, but this Sunday we humbly ask you to protect Wikipedia. This isn’t the first time we’ve asked recently, but only 2% of our readers give. Many think they’ll give later, but then forget. All we ask is $2.75, or what you can afford, to keep Wikipedia thriving. If Wikipedia has given you $2.75 worth of knowledge this year, take a minute to donate.Give $2.75 Give a different amount MAYBE LATER I ALREADY DONATEDCLOSE The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.The earliest mathematical texts available are from Mesopotamia and Egypt – Plimpton 322 (Babylonian c. 2000 – 1900 BC),[2] the Rhind Mathematical Papyrus (Egyptian c. 1800 BC)[3] and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".[4] Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.[5] Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers.[6][7] The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī.[8][9] Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.[10] Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century.Table of numeralsEuropean (descended from the West Arabic)0123456789Arabic-Indic٠١٢٣٤٥٦٧٨٩Eastern Arabic-Indic (Persian and Urdu)۰۱۲۳۴۵۶۷۸۹Devanagari (Hindi)०१२३४५६७८९Chinese〇一二三四五六七八九Tamil௧௨௩௪௫௬௭௮௯Contents1Prehistoric2Babylonian3Egyptian4Greek5Roman6Chinese7Indian8Islamic empires9Maya10Medieval European11Renaissance12Mathematics during the Scientific Revolution12.117th century12.218th century13Modern13.119th century13.220th century13.321st century14Future15See also16Notes17References18Further reading18.1General18.2Books on a specific period18.3Books on a specific topic19External links19.1Documentaries19.2Educational material19.3Bibliographies19.4Organizations19.5JournalsPrehistoric[edit]The origins of mathematical thought lie in the concepts of number, patterns in nature, magnitude, and form.[11] Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.[11]The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be more than 20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of sequences of prime numbers[12] or a six-month lunar calendar.[13] Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."[14] The Ishango bone, according to scholar Alexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.[15]Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.[16] All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.[17]Babylonian[edit]Main article: Babylonian mathematicsSee also: Plimpton 322Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (modern Iraq) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity.[18] The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC (Seleucid period).[19] It is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics.Geometry problem on a clay tablet belonging to a school for scribes; Susa, first half of the 2nd millennium BCEIn contrast to the sparsity of sources in Egyptian mathematics, knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.[20] Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.[21]The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around 2500 BC onward, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[22]The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.Babylonian mathematics were written using a sexagesimal (base-60) numeral system.[20] From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is likely the sexagesimal system was chosen because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.[20] Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the decimal system.[19] The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation.[19] The notational system of the Babylonians was the best of any civilization until the Renaissance,[23] and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.[23] The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.[19] By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.[19] This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.[19]Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, and their reciprocal pairs.[24] The tablets also include multiplication tables and methods for solving linear, quadratic equations and cubic equations, a remarkable achievement for the time.[25] Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem.[26] However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for proofs or logical principles.[21]Egyptian[edit]Main article: Egyptian mathematicsImage of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.[27] Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.[28]The most extensive Egyptian mathematical text is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000–1800 BC.[29] It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge,[30] including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6).[31] It also shows how to solve first order linear equations[32] as well as arithmetic and geometric series.[33]Another significant Egyptian mathematical text is the Moscow papyrus, also from the Middle Kingdom period, dated to c. 1890 BC.[34] It consists of what are today called word problems or story problems, which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum (truncated pyramid).Finally, the Berlin Papyrus 6619 (c. 1800 BC) shows that ancient Egyptians could solve a second-order algebraic equation.[35]Greek[edit]Main article: Greek mathematicsThe Pythagorean theorem. The Pythagoreans are generally credited with the first proof of the theorem.Greek mathematics refers to the mathematics written in the Greek language from the time of Thales of Miletus (~600 BC) to the closure of the Academy of Athens in 529 AD.[36] Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.[37]Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them.[38]Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.[39] Pythagoras established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".[40] It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the Pythagorean theorem,[41] though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers.[42][43] Although he was preceded by the Babylonians, Indians and the Chinese,[44] the Neopythagorean mathematician Nicomachus (60–120 AD) provided one of the earliest Greco-Roman multiplication tables, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the British Museum).[45] The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later Medieval name: the mensa Pythagorica.[46]Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others.[47] His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus, came.[48] Plato also discussed the foundations of mathematics,[49] clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions.[50] The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.[48]Eudoxus (408–c. 355 BC) developed the method of exhaustion, a precursor of modern integration[51] and a theory of ratios that avoided the problem of incommensurable magnitudes.[52] The former allowed the calculations of areas and volumes of curvilinear figures,[53] while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, Aristotle (384–c. 322 BC) contributed significantly to the development of mathematics by laying the foundations of logic.[54]One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[55]In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria.[56] It was there that Euclid (c. 300 BC) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time.[1] The Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.[57] The Elements was known to all educated people in the West up through the middle of the 20th century and its contents are still taught in geometry classes today.[58] In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry,[57] including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.[59]Archimedes used the method of exhaustion to approximate the value of pi.Archimedes (c. 287–212 BC) of Syracuse, widely considered the greatest mathematician of antiquity,[60] used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.[61] He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, 310/71 < π < 310/70.[62] He also studied the spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid),[61] and an ingenious method of exponentiation for expressing very large numbers.[63] While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles.[64] He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere.[65]Apollonius of Perga made significant advances in the study of conic sections.Apollonius of Perga (c. 262–190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.[66] He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").[67] His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton.[68] While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.[69]Around the same time, Eratosthenes of Cyrene (c. 276–194 BC) devised the Sieve of Eratosthenes for finding prime numbers.[70] The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline.[71] Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers.[71] Hipparchus of Nicaea (c. 190–120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle.[72] Heron of Alexandria (c. 10–70 AD) is credited with Heron's formula for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots.[73] Menelaus of Alexandria (c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem.[74] The most complete and influential trigonometric work of antiquity is the Almagest of Ptolemy (c. AD 90–168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years.[75] Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.[76]Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics.[77] During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis".[78] The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations.[79] The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares).[80] Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation.[79]The Hagia Sophia was designed by mathematicians Anthemius of Tralles and Isidore of Miletus.Among the last great Greek mathematicians is Pappus of Alexandria (4th century AD). He is known for his hexagon theorem and centroid theorem, as well as the Pappus configuration and Pappus graph. His Collection is a major source of knowledge on Greek mathematics as most of it has survived.[81] Pappus is considered the last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work.The first woman mathematician recorded by history was Hypatia of Alexandria (AD 350–415). She succeeded her father (Theon of Alexandria) as Librarian at the Great Library[citation needed] and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed.[82] Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as Proclus, Simplicius and Eutocius.[83] Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic Academy of Athens by the emperor Justinian in 529 AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletus, the architects of the Hagia Sophia.[84] Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and the centers of mathematical innovation were to be found elsewhere by this time.[85]Roman[edit]Further information: Roman abacus and Roman numeralsEquipment used by an ancient Roman land surveyor (gromatici), found at the site of Aquincum, modern Budapest, HungaryAlthough ethnic Greek mathematicians continued under the rule of the late Roman Republic and subsequent Roman Empire, there were no noteworthy native Latin mathematicians in comparison.[86][87] Ancient Romans such as Cicero (106–43 BC), an influential Roman statesman who studied mathematics in Greece, believed that Roman surveyors and calculators were far more interested in applied mathematics than the theoretical mathematics and geometry that were prized by the Greeks.[88] It is unclear if the Romans first derived their numerical system directly from the Greek precedent or from Etruscan numerals used by the Etruscan civilization centered in what is now Tuscany, central Italy.[89]Using calculation, Romans were adept at both instigating and detecting financial fraud, as well as managing taxes for the treasury.[90] Siculus Flaccus, one of the Roman gromatici (i.e. land surveyor), wrote the Categories of Fields, which aided Roman surveyors in measuring the surface areas of allotted lands and territories.[91] Aside from managing trade and taxes, the Romans also regularly applied mathematics to solve problems in engineering, including the erection of architecture such as bridges, road-building, and preparation for military campaigns.[92] Arts and crafts such as Roman mosaics, inspired by previous Greek designs, created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each tessera tile, the opus tessellatum pieces on average measuring eight millimeters square and the finer opus vermiculatum pieces having an average surface of four millimeters square.[93][94]The creation of the Roman calendar also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during the Roman Kingdom and included 356 days plus a leap year every other year.[95] In contrast, the lunar calendar of the Republican era contained 355 days, roughly ten-and-one-fourth days shorter than the solar year, a discrepancy that was solved by adding an extra month into the calendar after the 23rd of February.[96] This calendar was supplanted by the Julian calendar, a solar calendar organized by Julius Caesar (100–44 BC) and devised by Sosigenes of Alexandria to include a leap day every four years in a 365-day cycle.[97] This calendar, which contained an error of 11 minutes and 14 seconds, was later corrected by the Gregorian calendar organized by Pope Gregory XIII (r. 1572–1585), virtually the same solar calendar used in modern times as the international standard calendar.[98]At roughly the same time, the Han Chinese and the Romans both invented the wheeled odometer device for measuring distances traveled, the Roman model first described by the Roman civil engineer and architect Vitruvius (c. 80 BC – c. 15 BC).[99] The device was used at least until the reign of emperor Commodus (r. 177 – 192 AD), but its design seems to have been lost until experiments were made during the 15th century in Western Europe.[100] Perhaps relying on similar gear-work and technology found in the Antikythera mechanism, the odometer of Vitruvius featured chariot wheels measuring 4 feet (1.2 m) in diameter turning four-hundred times in one Roman mile (roughly 4590 ft/1400 m). With each revolution, a pin-and-axle device engaged a 400-tooth cogwheel that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.[101]Chinese[edit]Main article: Chinese mathematicsFurther information: Book on Numbers and ComputationThe Tsinghua Bamboo Slips, containing the world's earliest decimal multiplication table, dated 305 BC during the Warring States periodAn analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of the world, leading scholars to assume an entirely independent development.[102] The oldest extant mathematical text from China is the Zhoubi Suanjing, variously dated to between 1200 BC and 100 BC, though a date of about 300 BC during the Warring States Period appears reasonable.[103] However, the Tsinghua Bamboo Slips, containing the earliest known decimal multiplication table (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.[44]Counting rod numeralsOf particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.[104] Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.[105] Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the suan pan, or Chinese abacus. The date of the invention of the suan pan is not certain, but the earliest written mention dates from AD 190, in Xu Yue's Supplementary Notes on the Art of Figures.The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.[106] It also defined the concepts of circumference, diameter, radius, and volume.[107]The Nine Chapters on the Mathematical Art, one of the earliest surviving mathematical texts from China (2nd century AD).In 212 BC, the Emperor Qin Shi Huang commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the book burning of 212 BC, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art, the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying, and includes material on right triangles.[103] It created mathematical proof for the Pythagorean theorem,[108] and a mathematical formula for Gaussian elimination.[109] The treatise also provides values of π,[103] which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 AD) provided a figure of 3.1457 and subsequently Zhang Heng (78–139) approximated pi as 3.1724,[110] as well as 3.162 by taking the square root of 10.[111][112] Liu Hui commented on the Nine Chapters in the 3rd century AD and gave a value of π accurate to 5 decimal places (i.e. 3.14159).[113][114] Though more of a matter of computational stamina than theoretical insight, in the 5th century AD Zu Chongzhi computed the value of π to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years.[113][115] He also established a method which would later be called Cavalieri's principle to find the volume of a sphere.[116]The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty (960–1279), with the development of Chinese algebra. The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method.[113] The Precious Mirror also contains a diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.[117] The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (AD 1238–1298).[117]Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.[117]Japanese mathematics, Korean mathematics, and Vietnamese mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to the Confucian-based East Asian cultural sphere.[118] Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's Ming dynasty (1368–1644).[119] For instance, although Vietnamese mathematical treatises were written in either Chinese or the native Vietnamese Chữ Nôm script, all of them followed the Chinese format of presenting a collection of problems with algorithms for solving them, followed by numerical answers.[120] Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of mathematicians and astronomers, whereas in Japan it was more prevalent in the realm of private schools.[121]Indian[edit]Main article: Indian mathematicsSee also: History of the Hindu–Arabic numeral systemThe numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.Indian numerals in stone and copper inscriptions[122]Ancient Brahmi numerals in a part of IndiaThe earliest civilization on the Indian subcontinent is the Indus Valley civilization (mature phase: 2600 to 1900 BC) that flourished in the Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.[123]The oldest extant mathematical records from India are the Sulba Sutras (dated variously between the 8th century BC and the 2nd century AD),[124] appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.[125] As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual.[124] The Sulba Sutras give methods for constructing a circle with approximately the same area as a given square, which imply several different approximations of the value of π.[126][127][a] In addition, they compute the square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the Pythagorean theorem.[127] All of these results are present in Babylonian mathematics, indicating Mesopotamian influence.[124] It is not known to what extent the Sulba Sutras influenced later Indian mathematicians. As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity.[124]Pāṇini (c. 5th century BC) formulated the rules for Sanskrit grammar.[128] His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion.[129] Pingala (roughly 3rd–1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral system.[130][131] His discussion of the combinatorics of meters corresponds to an elementary version of the binomial theorem. Pingala's work also contains the basic ideas of Fibonacci numbers (called mātrāmeru).[132]The next significant mathematical documents from India after the Sulba Sutras are the Siddhantas, astronomical treatises from the 4th and 5th centuries AD (Gupta period) showing strong Hellenistic influence.[133] They are significant in that they contain the first instance of trigonometric relations based on the half-chord, as is the case in modern trigonometry, rather than the full chord, as was the case in Ptolemaic trigonometry.[134] Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya".[134]Explanation of the sine rule in YuktibhāṣāAround 500 AD, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.[135] Though about half of the entries are wrong, it is in the Aryabhatiya that the decimal place-value system first appears. Several centuries later, the Muslim mathematician Abu Rayhan Biruni described the Aryabhatiya as a "mix of common pebbles and costly crystals".[136]In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in Brahma-sphuta-siddhanta, he lucidly explained the use of zero as both a placeholder and decimal digit, and explained the Hindu–Arabic numeral system.[137] It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, all of which evolved from the Brahmi numerals. Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, Halayudha's commentary on Pingala's work contains a study of the Fibonacci sequence and Pascal's triangle, and describes the formation of a matrix.[citation needed]In the 12th century, Bhāskara II[138] lived in southern India and wrote extensively on all then known branches of mathematics. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, derivatives, the mean value theorem and the derivative of the sine function. To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics.[139]In the 14th century, Madhava of Sangamagrama, the founder of the Kerala School of Mathematics, found the Madhava–Leibniz series and obtained from it a transformed series, whose first 21 terms he used to compute the value of π as 3.14159265359. Madhava also found the Madhava-Gregory series to determine the arctangent, the Madhava-Newton power series to determine sine and cosine and the Taylor approximation for sine and cosine functions.[140] In the 16th century, Jyesthadeva consolidated many of the Kerala School's developments and theorems in the Yukti-bhāṣā.[141] [142] It has been argued that the advances of the Kerala school, which laid the foundations of the calculus, were transmitted to Europe in the 16th century[143] via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time and, as a result, directly influenced later European developments in analysis and calculus.[144] However, other scholars argue that the Kerala School did not formulate a systematic theory of differentiation and integration, and that there is not any direct evidence of their results being transmitted outside Kerala.[145][146][147][148]Islamic empires[edit]Main article: Mathematics in medieval IslamSee also: History of the Hindu–Arabic numeral systemPage from The Compendious Book on Calculation by Completion and Balancing by Muhammad ibn Mūsā al-Khwārizmī (c. AD 820)The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in Arabic, most of them were not written by Arabs, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. Persians contributed to the world of Mathematics alongside Arabs.In the 9th century, the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote an important book on the Hindu–Arabic numerals and one on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[149] and he was the first to teach algebra in an elementary form and for its own sake.[150] He also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as al-jabr.[151] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[152]In Egypt, Abu Kamil extended algebra to the set of irrational numbers, accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions.[153] His works formed an important foundation for the development of algebra and influenced later mathematicians, such as al-Karaji and Fibonacci.Further developments in algebra were made by Al-Karaji in his treatise al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.[154] The historian of mathematics, F. Woepcke,[155] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Also in the 10th century, Abul Wafa translated the works of Diophantus into Arabic. Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.[156]In the late 11th century, Omar Khayyam wrote Discussions of the Difficulties in Euclid, a book about what he perceived as flaws in Euclid's Elements, especially the parallel postulate. He was also the first to find the general geometric solution to cubic equations. He was also very influential in calendar reform.[157]In the 13th century, Nasir al-Din Tusi (Nasireddin) made advances in spherical trigonometry. He also wrote influential work on Euclid's parallel postulate. In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner.Other achievements of Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine, al-Kindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam and the development of an algebraic notation by al-Qalasādī.[158]During the time of the Ottoman Empire and Safavid Empire from the 15th century, the development of Islamic mathematics became stagnant.Maya[edit]The Maya numerals for numbers 1 through 19, written in the Maya scriptIn the Pre-Columbian Americas, the Maya civilization that flourished in Mexico and Central America during the 1st millennium AD developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, Egyptian, and Asian mathematics.[159] Maya numerals utilized a base of twenty, the vigesimal system, instead of a base of ten that forms the basis of the decimal system used by most modern cultures.[159] The Maya used mathematics to create the Maya calendar as well as to predict astronomical phenomena in their native Maya astronomy.[159] While the concept of zero had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.[159]Medieval European[edit]Further information: Category:Medieval European mathematics, List of medieval European scientists, and European science in the Middle AgesSee also: Latin translations of the 12th centuryMedieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato's Timaeus and the biblical passage (in the Book of Wisdom) that God had ordered all things in measure, and number, and weight.[160]Boethius provided a place for mathematics in the curriculum in the 6th century when he coined the term quadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica, a free translation from the Greek of Nicomachus's Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Elements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.[161][162]In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester, and the complete text of Euclid's Elements, translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona.[163][164] These and other new sources sparked a renewal of mathematics.Leonardo of Pisa, now known as Fibonacci, serendipitously learned about the Hindu–Arabic numerals on a trip to what is now Béjaïa, Algeria with his merchant father. (Europe was still using Roman numerals.) There, he observed a system of arithmetic (specifically algorism) which due to the positional notation of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. Leonardo wrote Liber Abaci in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as the Fibonacci sequence (known to Indian mathematicians for hundreds of years before that) which was used as an unremarkable example within the text.The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.[165] One important contribution was development of mathematics of local motion.Thomas Bradwardine proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R).[166] Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.[167]Nicole Oresme (1323–1382), shown in this contemporary illuminated manuscript with an armillary sphere in the foreground, was the first to offer a mathematical proof for the divergence of the harmonic series.[168]One of the 14th-century Oxford Calculators, William Heytesbury, lacking differential calculus and the concept of limits, proposed to measure instantaneous speed "by the path that would be described by [a body] if... it were moved uniformly at the same degree of speed with which it is moved in that given instant".[169]Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".[170]Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.[171] In a later mathematical commentary on Euclid's Elements, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.[172]Renaissance[edit]Further information: Mathematics and artDuring the Renaissance, the development of mathematics and of accounting were intertwined.[173] While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in Flanders and Germany) or abacus schools (known as abbaco in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing bookkeeping operations, but for complex bartering operations or the calculation of compound interest, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.Piero della Francesca (c. 1415–1492) wrote books on solid geometry and linear perspective, including De Prospectiva Pingendi (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and De quinque corporibus regularibus (On the Five Regular Solids).[174][175][176]Portrait of Luca Pacioli, a painting traditionally attributed to Jacopo de' Barbari, 1495, (Museo di Capodimonte).Luca Pacioli's Summa de Arithmetica, Geometria, Proportioni et Proportionalità (Italian: "Review of Arithmetic, Geometry, Ratio and Proportion") was first printed and published in Venice in 1494. It included a 27-page treatise on bookkeeping, "Particularis de Computis et Scripturis" (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the mathematical puzzles it contained, and to aid the education of their sons.[177] In Summa Arithmetica, Pacioli introduced symbols for plus and minus for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. Summa Arithmetica was also the first known book printed in Italy to contain algebra. Pacioli obtained many of his ideas from Piero Della Francesca whom he plagiarized.In Italy, during the first half of the 16th century, Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations. Gerolamo Cardano published them in his 1545 book Ars Magna, together with a solution for the quartic equations, discovered by his student Lodovico Ferrari. In 1572 Rafael Bombelli published his L'Algebra in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations.Simon Stevin's book De Thiende ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of decimal notation, which influenced all later work on the real number system.Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.[178]During the Renaissance the desire of artists to represent the natural world realistically, together with the rediscovered philosophy of the Greeks, led artists to study mathematics. They were also the engineers and architects of that time, and so had need of mathematics in any case. The art of painting in perspective, and the developments in geometry that involved, were studied intensely.[179]Mathematics during the Scientific Revolution[edit]17th century[edit]Gottfried Wilhelm Leibniz.The 17th century saw an unprecedented increase of mathematical and scientific ideas across Europe. Galileo observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. By his position as Brahe's assistant, Johannes Kepler was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention of logarithms by John Napier and Jost Bürgi. Kepler succeeded in formulating mathematical laws of planetary motion.[180] The analytic geometry developed by René Descartes (1596–1650) allowed those orbits to be plotted on a graph, in Cartesian coordinates.Building on earlier work by many predecessors, Isaac Newton discovered the laws of physics explaining Kepler's Laws, and brought together the concepts now known as calculus. Independently, Gottfried Wilhelm Leibniz, developed calculus and much of the calculus notation still in use today. Science and mathematics had become an international endeavor, which would soon spread over the entire world.[181]In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling. Pascal, with his wager, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of utility theory in the 18th–19th century.18th century[edit]Leonhard Euler by Emanuel Handmann.The most influential mathematician of the 18th century was arguably Leonhard Euler (1707–1783). His contributions range from founding the study of graph theory with the Seven Bridges of Königsberg problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol i, and he popularized the use of the Greek letter {\displaystyle \pi } to stand for the ratio of a circle's circumference to its diameter. He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.Other important European mathematicians of the 18th century included Joseph Louis Lagrange, who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics.Modern[edit]This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2021) (Learn how and when to remove this template message)19th century[edit]Carl Friedrich Gauss.Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.Behavior of lines with a common perpendicular in each of the three types of geometryThis century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalizes the ideas of curves and surfaces.The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra. The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1. Boolean algebra is the starting point of mathematical logic and has important applications in electrical engineering and computer science. Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the calculus in a more rigorous fashion.Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem). Other 19th-century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks. On the other hand, the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers.Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry.In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of mathematical logic in the hands of Peano, L.E.J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics.The 19th century saw the founding of a number of national mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Circolo Matematico di Palermo in 1884, the Edinburgh Mathematical Society in 1883, and the American Mathematical Society in 1888. The first international, special-interest society, the Quaternion Society, was formed in 1899, in the context of a vector controversy.In 1897, Kurt Hensel introduced p-adic numbers.20th century[edit]The 20th century saw mathematics become a major profession. Every year, thousands of new Ph.D.s in mathematics were awarded, and jobs were available in both teaching and industry. An effort to catalogue the areas and applications of mathematics was undertaken in Klein's encyclopedia.In a 1900 speech to the International Congress of Mathematicians, David Hilbert set out a list of 23 unsolved problems in mathematics. These problems, spanning many areas of mathematics, formed a central focus for much of 20th-century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.A map illustrating the Four Color TheoremNotable historical conjectures were finally proven. In 1976, Wolfgang Haken and Kenneth Appel proved the four color theorem, controversial at the time for the use of a computer to do so. Andrew Wiles, building on the work of others, proved Fermat's Last Theorem in 1995. Paul Cohen and Kurt Gödel proved that the continuum hypothesis is independent of (could neither be proved nor disproved from) the standard axioms of set theory. In 1998 Thomas Callister Hales proved the Kepler conjecture.Mathematical collaborations of unprecedented size and scope took place. An example is the classification of finite simple groups (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages. A group of French mathematicians, including Jean Dieudonné and André Weil, publishing under the pseudonym "Nicolas Bourbaki", attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.[182]Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star, with relativistic precession of apsidesDifferential geometry came into its own when Albert Einstein used it in general relativity. Entirely new areas of mathematics such as mathematical logic, topology, and John von Neumann's game theory changed the kinds of questions that could be answered by mathematical methods. All kinds of structures were abstracted using axioms and given names like metric spaces, topological spaces etc. As mathematicians do, the concept of an abstract structure was itself abstracted and led to category theory. Grothendieck and Serre recast algebraic geometry using sheaf theory. Large advances were made in the qualitative study of dynamical systems that Poincaré had begun in the 1890s. Measure theory was developed in the late 19th and early 20th centuries. Applications of measures include the Lebesgue integral, Kolmogorov's axiomatisation of probability theory, and ergodic theory. Knot theory greatly expanded. Quantum mechanics led to the development of functional analysis. Other new areas include Laurent Schwartz's distribution theory, fixed point theory, singularity theory and René Thom's catastrophe theory, model theory, and Mandelbrot's fractals. Lie theory with its Lie groups and Lie algebras became one of the major areas of study.Non-standard analysis, introduced by Abraham Robinson, rehabilitated the infinitesimal approach to calculus, which had fallen into disrepute in favour of the theory of limits, by extending the field of real numbers to the Hyperreal numbers which include infinitesimal and infinite quantities. An even larger number system, the surreal numbers were discovered by John Horton Conway in connection with combinatorial games.The development and continual improvement of computers, at first mechanical analog machines and then digital electronic machines, allowed industry to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with this: Alan Turing's computability theory; complexity theory; Derrick Henry Lehmer's use of ENIAC to further number theory and the Lucas-Lehmer test; Rózsa Péter's recursive function theory; Claude Shannon's information theory; signal processing; data analysis; optimization and other areas of operations research. In the preceding centuries much mathematical focus was on calculus and continuous functions, but the rise of computing and communication networks led to an increasing importance of discrete concepts and the expansion of combinatorics including graph theory. The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too time-consuming to deal with by pencil and paper calculations, leading to areas such as numerical analysis and symbolic computation. Some of the most important methods and algorithms of the 20th century are: the simplex algorithm, the fast Fourier transform, error-correcting codes, the Kalman filter from control theory and the RSA algorithm of public-key cryptography.At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved the truth or falsity of all statements formulated about the natural numbers plus either addition or multiplication (but not both), was decidable, i.e. could be determined by some algorithm. In 1931, Kurt Gödel found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as Peano arithmetic, was in fact incompletable. (Peano arithmetic is adequate for a good deal of number theory, including the notion of prime number.) A consequence of Gödel's two incompleteness theorems is that in any mathematical system that includes Peano arithmetic (including all of analysis and geometry), truth necessarily outruns proof, i.e. there are true statements that cannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic, and David Hilbert's dream of making all of mathematics complete and consistent needed to be reformulated.The absolute value of the Gamma function on the complex plane.One of the more colorful figures in 20th-century mathematics was Srinivasa Aiyangar Ramanujan (1887–1920), an Indian autodidact who conjectured or proved over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. He also made major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.Paul Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. Mathematicians have a game equivalent to the Kevin Bacon Game, which leads to the Erdős number of a mathematician. This describes the "collaborative distance" between a person and Erdős, as measured by joint authorship of mathematical papers.Emmy Noether has been described by many as the most important woman in the history of mathematics.[183] She studied the theories of rings, fields, and algebras.As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century there were hundreds of specialized areas in mathematics and the Mathematics Subject Classification was dozens of pages long.[184] More and more mathematical journals were published and, by the end of the century, the development of the World Wide Web led to online publishing.21st century[edit]See also: List of unsolved problems in mathematics § Problems solved since 1995In 2000, the Clay Mathematics Institute announced the seven Millennium Prize Problems, and in 2003 the Poincaré conjecture was solved by Grigori Perelman (who declined to accept an award, as he was critical of the mathematics establishment).Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched. There is an increasing drive toward open access publishing, first popularized by arXiv.Future[edit]Main article: Future of mathematicsThere are many observable trends in mathematics, the most notable being that the subject is growing ever larger, computers are ever more important and powerful, the application of mathematics to bioinformatics is rapidly expanding, and the volume of data being produced by science and industry, facilitated by computers, is expanding exponentially.[citation needed]Mathematics, or math, is often defined as the study of quantity, magnitude, and relations of numbers or symbols. It embraces the subjects of arithmetic, geometry, algebra, calculus, probability, statistics, and many other special areas of research.​There are two major divisions of mathematics: pure and applied. Pure mathematics investigates the subject solely for its theoretical interest. Applied mathematics develops tools and techniques for solving specific problems of business and engineering or for highly theoretical applications in the sciences.​Mathematics is pervasive throughout modern life. Baking a cake or building a house involves the use of numbers, geometry, measures, and space. The design of precision instruments, the development of new technologies, and advanced computers all use more technical mathematics. (See also mathematics at a glance.)​History​Mathematics first arose from the practical need to measure time and to count. Thus, the history of mathematics begins with the origins of numbers and recognition of the dimensions and properties of space and time. The earliest evidence of primitive forms of counting occurs in notched bones and scored pieces of wood and stone. Early uses of geometry are revealed in patterns found on ancient cave walls and pottery.​Ancient Periods​As civilizations arose in Asia and the Near East, the field of mathematics evolved. Both sophisticated number systems and basic knowledge of arithmetic, geometry, and algebra began to develop.​Egypt and Mesopotamia​© sculpies/© sculpies/​The earliest continuous records of mathematical activity that have survived in written form are from the 2nd millennium bc. The Egyptian pyramids reveal evidence of a fundamental knowledge of surveying and geometry as early as 2900 bc. Written testimony of what the Egyptians knew, however, is known from documents drawn up about 1,000 years later.​Two of the best-known sources for our current knowledge of ancient Egyptian mathematics are the Rhind papyrus and the Moscow papyrus. These present many different kinds of practical mathematical problems, including applications to surveying, salary distributions, calculations of the areas of simple geometric surfaces and volumes such as the truncated pyramid, and simple solutions for first- and second-degree equations.​Egyptian arithmetic, based on counting in groups of ten, was relatively simple. Base-10 systems, the most widespread throughout the world, probably arose for biological reasons. The fingers of both hands facilitated natural counting in groups of ten. Numbers are sometimes called digits from the Latin word for finger. In the Egyptians’ base-10 arithmetic, hieroglyphs stood for individual units and groups of tens, hundreds, and thousands. Higher powers of ten made it possible to count numbers into the millions. Unlike our familiar number system, which is both decimal and positional (23 is not the same as 32), the Egyptians’ arithmetic was not positional but additive.​Unlike the Egyptians, the Babylonians of ancient Mesopotamia developed flexible techniques for dealing with fractions. They also succeeded in developing a more sophisticated base-10 arithmetic that was positional, and they kept mathematical records on clay tablets. The most remarkable feature of Babylonian arithmetic was its use of a sexagesimal (base 60) place-valued system in addition to a decimal system. Thus the Babylonians counted in groups of sixty as well as ten. Babylonian mathematics is still used to tell time—an hour consists of 60 minutes, and each minute is divided into 60 seconds—and circles are measured in divisions of 360 degrees.​The Babylonians apparently adopted their base-60 number system for economic reasons. Their principal units of weight and money were the mina, consisting of 60 shekels, and the talent, consisting of 60 mina. This sexagesimal arithmetic was used in commerce and astronomy. Surviving tablets also show the Babylonians’ facility in computing compound interest, squares, and square roots.​Because their base-60 system was especially flexible for computation and handling fractions, the Babylonians were particularly strong in algebra and number theory. Tablets survive giving solutions to first-, second-, and some third-degree equations. Despite rudimentary knowledge of geometry, the Babylonians knew many cases of the Pythagorean theorem for right triangles. They also knew accurate area formulas for triangles and trapezoids. Since they used a crude approximation of three for the value of pi, they achieved only rough estimates for the areas of circles.​Greece and Rome​The Greeks were the first to develop a truly mathematical spirit. They were interested not only in the applications of mathematics but in its philosophical significance, which was especially appreciated by Plato.​The Greeks developed the idea of using mathematical formulas to prove the validity of a proposition. Some Greeks, such as Aristotle, engaged in the theoretical study of logic, the analysis of correct reasoning. No previous mathematics had dealt with abstract entities or the idea of a mathematical proof.​©©​Encyclopædia Britannica, Inc.Encyclopædia Britannica, Inc.​Pythagoras provided one of the first proofs in mathematics and discovered incommensurable magnitudes, or irrational numbers. The Pythagorean theorem relates the sides of a right triangle with their corresponding squares. The discovery of irrational magnitudes had another consequence for the Greeks: since the lengths of diagonals of squares could not be expressed by rational numbers of the form a/b, the Greek number system was inadequate for describing them. Due to the incompleteness of their number system, the Greeks developed geometry at the expense of algebra. The only systematic contribution to algebra was made much later in antiquity by Diophantus. Called the father of algebra, he devised symbols to represent operations, unknown quantities, and frequently occurring constants.​​Ancient knowledge of the sciences was often wrong and wholly unsatisfactory by modern standards. However, the mathematics of Euclid, Apollonius of Perga, and Archimedes—the three greatest mathematicians of antiquity—remains as valid today as it was more than 2,000 years ago. Euclid’s Elements of Geometry used logic and deductive reasoning to set up axioms, postulates, and a collection of theorems related to plane and solid geometry, as well as a theory of proportions used to resolve the difficulty of irrational numbers. Despite its flaws, the Elements remains a historic example of how to establish universally agreed-upon knowledge by following a rigorous course of deductive logic. Apollonius, best known for his work on conic sections, coined the terms parabola, hyperbola, and ellipse. Another great figure was Ptolemy, who contributed to the development of trigonometry and mathematical astronomy.​Roman mathematicians, in contrast to the Greeks, are renowned for being very practical. The Roman mind did not favor the abstract side of mathematics, which had so delighted the Greeks. The Romans cared instead for the usefulness of mathematics in measuring and counting. As the fortunes of the Roman Empire declined, a rising interest in mathematics developed elsewhere, in India and among Arab scholars.​The Middle Ages​MukerjeeMukerjee​Indian mathematicians were especially skilled in arithmetic, methods of calculation, algebra, and trigonometry. Aryabhata calculated pi to a very accurate value of 3.1416, and Brahmagupta and Bhaskara II advanced the study of indeterminate equations. Because Indian mathematicians were not concerned with such theoretical problems as irrational numbers, they were able to make great strides in algebra. Their decimal place-valued number system, including zero, was especially suited for easy calculation. Indian mathematicians, however, lacked interest in a sense of proof. Most of their results were presented simply as useful techniques for given situations, especially in astronomical or astrological computations.​© Juulijs/Fotolia© Juulijs/Fotolia​One of the greatest scientific minds of Islam was al-Khwarizmi, who introduced the name (al-jabr) that became known as algebra. Consequently, the numbers familiar to most people are still referred to as Arabic numerals. Arab mathematicians also translated and commented on Ptolemy’s astronomy before it was brought to the attention of Europeans. Islamic scholars not only translated the works of Euclid, Archimedes, Apollonius, and Ptolemy into Arabic but advanced beyond what the Greek mathematicians had done to provide new results of their own.​By the end of the 8th century the influence of Islam had extended as far west as Spain. It was there, primarily, that Arabic, Jewish, and Western scholars eventually translated Greek and Islamic manuscripts into Latin. By the 13th century, original mathematical work by European authors had begun to appear.​Renaissance Period​Most of the early mathematical activity of the Renaissance was centered in Italy, where the mathematician Luca Pacioli wrote a standard text on arithmetic, algebra, and geometry that served to introduce the subject to students for generations. The solution of the cubic equation instigated great rivalries and priority claims between Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano. Among the advances in algebra made during the 16th century, the use of letters of the alphabet to denote constants, variables, and unknowns in equations is notable. This symbolic algebra later proved to be the key to advances in geometry, algebra, and the infinitesimal calculus.​17th Century​Courtesy of the Musée de la Ville de Narbonne, FranceCourtesy of the Musée de la Ville de Narbonne, France​Mathematics received considerable stimulus in the 17th century from astronomical problems. The astronomer Johannes Kepler, for example, who discovered the elliptical shape of the planetary orbits, was especially interested in the problem of determining areas bounded by curved figures. Kepler and other mathematicians used infinitesimal methods of one sort or another to find a general solution for the problem of areas. In connection with such questions, the French mathematician Pierre de Fermat investigated properties of maxima and minima. He also discovered a method of determining tangents to curves, a problem closely related to the almost simultaneous development of the differential and integral calculus by Isaac Newton and Gottfried Wilhelm Leibniz later in the century.​Statens Museum for Kunst (National Gallery of Denmark); (Public domai...Statens Museum for Kunst (National Gallery of Denmark); (Public domain)​Of equal importance to the invention of the calculus was the independent discovery of analytic geometry by Fermat and René Descartes. Of the two, Descartes used a better notation and devised superior techniques. Above all, he showed how the solution of simultaneous equations was facilitated through the application of analytic geometry. Many geometric problems could be translated directly into equivalent algebraic terms for solution.​Developed in the 17th century, projective geometry involves, in part, the analysis of conic sections in terms of their projections. Its value was not fully appreciated until the 19th century. The study of probability as related to games of chance had also begun.​Courtesy of the Collection Haags Gemeentemuseum, The HagueCourtesy of the Collection Haags Gemeentemuseum, The Hague​The greatest achievement of the century was the discovery of methods that applied mathematics to the study of motion. An example is Galileo’s analysis of the parabolic path of projectiles, published in 1638. At the same time, the Dutch mathematician Christiaan Huygens was publishing works on the analysis of conic sections and special curves. He also presented theorems related to the paths of quickest descent of falling objects.​© North Wind Picture Archives© North Wind Picture Archives​The unsurpassed master of the application of mathematics to problems of physics was Isaac Newton, who used analytic geometry, infinite series, and calculus to make numerous mathematical discoveries. Newton also developed his method of fluxions and fluents—the differential and integral calculus. He showed that the two methods—derivatives and integrals—were inversely related to one another. Newton and Leibniz were studying similar problems of physics and mathematics at the same time. Having made his own discovery of the calculus in 1674, Leibniz published a rather obscure version of his methods in 1684, some years before Newton published a full version of his own methods. The sequence of mathematical developments that flows out of the discovery of the calculus is called analysis.​Although the new calculus was an immediate success, its methods were sharply criticized because infinitesimals were sometimes treated as if they were finite and, at other times, as if they were zero. Doubts about the foundations of the calculus were unresolved until the 19th century.​18th Century​©©​The discovery of analytic geometry and invention of the calculus made possible the application of mathematics to a wide range of problems in the 18th century. The Bernoullis, a Swiss family of mathematicians, were pioneers in the application of the calculus to physics. However, they were not the only ones to advance the calculus in the 18th century. Mathematicians in France and England also tried to extend the range of the work of Newton and Leibniz.​Kean Collection/Hulton Archive/Getty ImagesKean Collection/Hulton Archive/Getty Images​The greatest development of mathematics in the 18th century took place on the Continent, where monarchs such as Louis XIV, Frederick the Great, and the Empress Catherine the Great of Russia provided generous support for science, including mathematics. The most prolific 18th-century mathematician was Leonhard Euler of Switzerland. He published hundreds of research papers, and his major books dealt with both the differential and integral infinitesimal calculus as well as with algebra, geometry, mechanics, and the calculus of variations.​Joseph-Louis Lagrange contributed to mechanics, foundations of the calculus, the calculus of variations, probability theory, and the theories of numbers and equations. While analysis was being developed by some French mathematicians, others were turning to geometry and probability theory. The French astronomer Pierre-Simon Laplace succeeded in applying probability theory and analysis to the Newtonian theory of celestial mechanics. He was thereby able to establish the dynamic stability of the solar system.​19th Century​The 19th century witnessed tremendous change in mathematics with increased specialization and new theories of algebra and number theory. The entire scope of mathematics was enriched by the discovery of controversial areas of study such as non-Euclidean geometries and transfinite set theory. Non-Euclidean geometries, in showing that consistent geometries could be developed for which Euclid’s parallel postulate did not hold, raised significant questions pertaining to the foundation of mathematics.​© Nicku/© Nicku/​In Germany, Carl Friedrich Gauss discovered the law of quadratic reciprocity, proved the fundamental theorem of algebra, and developed the theory of complex numbers. The Norwegian mathematician Niels Henrik Abel also made great strides during the 19th century, particularly with his theory of integrals of algebraic functions and a theorem that led to the Abelian functions, later advanced by Karl Gustav Jacobi.​David BenbennickDavid Benbennick​The German mathematician Karl Weierstrass brought new levels of rigor to analysis by reducing its elements to arithmetic principles and by using power series as a foundation for the theory of complex functions. August Möbius, also from Germany, worked in the area of analytic geometry and was a pioneer in topology. He discovered the Möbius strip, a topological space obtained by twisting one end of a rectangular strip and pasting it to the other.​Mathematicians in England slowly began to take an interest in advances made on the Continent during the previous century. The Analytic Society was formed in 1812 to promote the new notation and ideas of the calculus commonly used by the French. A form of noncommutative algebra called quaternions was discovered by William Rowan Hamilton, and other mathematical forms were applied to the theory of electromagnetism.​H. Roger-ViolletH. Roger-Viollet​In the United States indigenous groups of mathematicians were beginning to form, particularly in the areas of linear associative algebra and logic. In France mathematicians made significant contributions to work in geometry and analysis, especially analysis of elliptic functions. Other advances were made in complex analysis, modular functions, number theory, and invariant theory. Augustin-Louis Cauchy advanced nearly every branch of mathematics, but especially real and complex analysis. Henri Poincaré made significant contributions to mathematical physics, automorphic functions, differential equations, topology, probability theory, and the foundations of mathematics. Italian mathematics in the 19th century tended to stress geometry and analysis.​Two related areas of mathematics established in the 19th century proved to be of major significance in the 20th century: set theory and mathematical logic. These were closely related to questions concerning the foundations of mathematics and the continuum of real numbers as investigated by Richard Dedekind and Georg Cantor. It was Cantor who created set theory and the theory of transfinite numbers.​Modern Times​Modern mathematics is highly specialized and abstract. The advance of set theory and discoveries involving infinite sets, transfinite numbers, and purely logical paradoxes caused much concern as to the foundations of mathematics. In addition to purely theoretical developments, devices such as high-speed computers influenced both the content and the teaching of mathematics. Among the areas of mathematical research that were developed since the 20th century are abstract algebra, non-Euclidean geometry, abstract analysis, mathematical logic, and the foundations of mathematics.​Modern abstract algebra includes the study of groups, rings, algebras, lattices, and a host of other subjects developed from a formal, abstract point of view. This approach formed the cornerstone of the work of a group of mathematicians called Bourbaki. Bourbaki used abstract algebra in an axiomatic framework to develop virtually all branches of higher mathematics, including set theory, algebra, and general topology.​The significance of non-Euclidean geometry was realized early in the 20th century when the geometry was applied in mathematical physics. It has come to play an essential role in the theory of relativity and has also raised controversial philosophical questions about the nature of mathematics and its foundations.​Another area of mathematics, abstract analysis, has produced theories of the derivatives and integrals in abstract and infinite-dimensional spaces. There are many areas of special interest in the field of abstract analysis, including functional analysis, harmonic analysis, families of functions, integral equations, divergent and asymptotic series, summability, and the study of functions of a complex variable. Analysis has advanced with the introduction of nonstandard analysis. By developing infinitesimals this theory provides an alternative to the traditional approach of using limits in the calculus.​The most notable development in the area of logic began in the 20th century with the work of two English logicians and philosophers, Bertrand Russell and Alfred North Whitehead. The object of their three-volume publication, Principia Mathematica (1910–13), was to show that mathematics can be deduced from a very small number of logical principles. In the 1930s questions about the logical consistency and completeness of axiomatic systems helped to spark interest in mathematical logic and concern for the foundations of mathematics. Since the 1940s mathematical logic has become increasingly specialized.​The foundations of mathematics have many “schools.” At the beginning of the 20th century, David Hilbert was determined to preserve the powerful methods of transfinite set theory and the use of the infinite in mathematics, despite apparent paradoxes and numerous objections. He believed it was possible to find finite means of establishing the truth of mathematical propositions, even when the infinite was involved. To this end Hilbert devoted considerable effort to developing a metamathematical theory of proofs. His program was virtually abandoned in the 1930s when Kurt Gödel demonstrated that for any general axiomatic system there are always theorems that cannot be proved or disproved.​Hilbert’s followers, known as formalists, view mathematics in terms of abstract structures. The axioms are developed as arbitrary rules. When applied to the unspecified elements of the theory, they can be used to establish the validity of theorems. Mathematical “truth” is thus reduced to the question of logical self-consistency. Those opposed to the formalist view, called intuitionists, believe that the basic truths of mathematics present themselves as fundamental intuitions of thought. The oldest philosophy of mathematics is usually ascribed to Plato. Platonism asserts the existence of eternal truths, independent of the human mind. In this philosophy the truths of mathematics arise from an abstract, ideal reality.​Subdivisions of Mathematics​Throughout history mathematics has become increasingly complex and diversified. At the same time, however, it has become increasingly general and abstract. Among the major subdivisions of modern mathematics are the following:​Arithmetic​Arithmetic comes from the word arithmos, meaning “number” in Greek. It is the study of the nature and properties of numbers. It includes study of the algorithms of calculation with numbers, namely the basic operations of addition, subtraction, multiplication, and division, as well as the taking of powers and roots. Arithmetic is often applied in the calculation of fractions, ratios, percentages, and proportions.​Algebra​Algebra has often been described as “arithmetic with letters.” Unlike arithmetic, which deals with specific numbers, algebra introduces variables that greatly extend the generality and scope of arithmetic. The algebra taught in high schools involves techniques for solving relatively simple equations.​Modern algebra, or abstract algebra, is a more general branch of mathematics that analyzes algebraic axioms and operations with arbitrary sets of symbols. Special areas of abstract algebra include the study of groups, rings, fields, the algebra of matrices, and a large variety of nonassociative and noncommutative algebras. Special algebras of sets and vectors and Boolean algebras arise in the study of logic. Algebra is used in the calculation of compound interest, in the solution of distance-rate-time problems, or in any situation in the sciences where the determination of unknown quantities from a body of known data is required.​Geometry​The word geometry is derived from the Greek meaning “earth measurement.” Although geometry originated for practical purposes in ancient Egypt and Babylonia, the Greeks investigated it in a more systematic and general way.​In the 19th century, Euclidean geometry’s status as the primary geometry was challenged by the discovery of non-Euclidean geometries. These inspired a new approach to the subject by presenting theorems in terms of axioms applied to properties assigned to undefined elements called points and lines. This led to many new geometries, including elliptical, hyperbolic, and parabolic geometries. Modern abstract geometry deals with very general questions of space, shape, size, and other properties of figures. Projective geometry, for example, is an abstract geometry concerned with the geometric properties that remain invariant under the projection of figures onto other figures, as in the case of mathematical perspective.​A very useful approach to geometry is found in topology, the study of the properties of a geometric figure that remain the same when a figure is subjected to continuous transformation without loss of identity of any of its parts. Differential geometry is the study of geometry in terms of infinitesimals.​Analytic Geometry and Trigonometry​Analytic geometry combines the generality of algebra with the precision of geometry. It is sometimes called Cartesian geometry, after Descartes, who was the first to exploit the methods of algebra in geometry. Analytic geometry addresses geometric problems from an algebraic point of view by associating any curve with variables by means of a coordinate system. For example, in a two-dimensional coordinate system, any point on a curve can be associated with a pair of points (a,b). General properties of such curves can then be studied in terms of their algebraic properties.​Trigonometry is the study of triangles, angles, and their relations. It also involves the study of trigonometric functions. There are six trigonometric ratios associated with an angle: sine, cosine, tangent, cotangent, secant, and cosecant. These are especially useful in determining unknown angles or the sides of triangles based upon known trigonometric ratios. In antiquity, trigonometry was used with considerable success by surveyors and astronomers.​Calculus​The calculus discovered in the 17th century by Newton and Leibniz used infinitesimal quantities to determine tangents to curves and to facilitate calculation of lengths and areas of curved figures. These operations were found to be inversely related. Newton called them “fluxions” and “fluents,” corresponding to what are now termed derivatives and integrals. Leibniz called them “differences” and “sums.”​In the 19th century, in response to questions about its rigorous foundations, the calculus was developed in terms of a theory of limits. Analysis—differential and integral calculus—was subsequently approached even more rigorously by those who sought to establish its results by strictly arithmetic means. This required an exact definition of the continuity of the real numbers. Others extended the power of analysis with very general theories of measure.​Analysis gives primary emphasis to functions, convergence of sequences, series, continuity, differentiability, and questions about the completeness of the real numbers. Introductory courses in calculus generally include study of logarithms, exponential functions, trigonometric functions, and transcendental functions.​Complex Analysis​Complex analysis extends the methods of analysis from real to complex variables. Complex numbers first arose to permit general solutions to algebraic equations. They take the form a + bi, where a and b are real numbers. The variable a is called the real part of the number; b, the imaginary part of the number; and i represents the complex, or “imaginary,” number signified by the square root of –1. Because complex numbers have two independent components, a and b, they are especially useful in applications whenever two variables must be treated simultaneously. For example, complex analysis has proven particularly valuable in applications to fluid dynamics, where both pressure and velocity vary from point to point. Complex numbers were made more acceptable to many in the 19th century when they were given a geometric interpretation.​Number Theory​It has been said that any unsolved mathematical problem that is over a century old and is still considered interesting belongs to number theory. This branch of mathematics involves the study of the properties of numbers and the structure of different number systems. It is concerned with integers, or whole numbers. Many problems in number theory deal with prime numbers. These are integers larger than 1 that have only themselves and 1 as factors.​Questions about highest common factors, least common multiples, decompositions into primes, and the representation of natural numbers in certain forms as well as their divisibility are all the province of number theory. Computers have recently been applied to the solution of certain number-theory problems.​Probability Theory and Statistics​The branch of mathematics concerned with the analysis of random phenomena is called probability theory. The entire set of possible outcomes of a random event is called the sample space. Each outcome in this space is assigned a probability, a number indicating the likelihood that the particular event will arise in a single instance. An example of a random experiment is the tossing of a coin. The sample space consists of the two outcomes, heads or tails, and the probability assigned to each is one half.​Statistics applies probability theory to real cases and involves the analysis of empirical data. The word statistics reflects the original application of mathematical methods to data collected for purposes of the state. Such studies led to general techniques for analyzing data and computing various values, drawing correlations, using methods of sampling, counting, estimating, and ranking data according to certain criteria.​Set Theory​Created in the 19th century by the German mathematician Georg Cantor, set theory was originally meant to provide techniques for the mathematical analysis of the infinite. Set theory deals with the properties of well-defined collections of objects. Sets may be finite or infinite. A finite set has a definite number of members; such a set might consist of all the integers from 1 to 1,000. An infinite set has an endless number of members. For example, all of the positive integers compose an infinite set.​Cantor developed a theory of infinite numbers and transfinite arithmetic to go along with them. His Continuum Hypothesis conjectures that the set of all real numbers is the second smallest infinite set. The smallest infinite set is composed of the integers or any set equivalent to it.​Early in the 20th century certain contradictions of set theory concerning infinite sets, transfinite numbers, and purely logical paradoxes brought about attempts to axiomatize set theory in hopes of eliminating such difficulties. When Kurt Gödel showed that, for any axiomatic system, propositions could be devised that were neither true nor false, it seemed that the traditional certainty of mathematics had been suddenly lost.​In the 1960s Paul Cohen succeeded in showing the independence of the Continuum Hypothesis, namely that it could be neither proved nor disproved within a given axiomatization of set theory. This meant that it was possible to contemplate non-Cantorian set theories in which the Continuum Hypothesis might be negated, much as non-Euclidean geometries treat geometry without assuming the necessary validity of Euclid’s parallel postulate.​Logic​Logic is the study of the way in which valid conclusions may be drawn from given premises. It was first treated systematically by Aristotle and later developed in terms of an algebra of logic. Symbolic logic arose from traditional logic by using symbols to stand for propositions and relations between them. Modern logicians use algebraic and formal methods to study the relations between logical propositions. This has led to model theory and model logic.​Additional Reading​Asimov, Isaac. Realm of Numbers (Fawcett, 1981). Boyer, C.B. A History of Mathematics (Princeton, 1985). Courant, R. and Robbins, H. What Is Mathematics? (Oxford, 1978). Dauben, J.W., ed. The History of Mathematics from Antiquity to the Present (Garland, 1985). Hershey, R.L. How to Think with Numbers (Kaufmann, 1982). Hoffman, Paul. Archimedes‘ Revenge. The Joys and Perils of Mathematics (Norton, 1988). James, E. and Barkin, C. What Do You Mean by “Average”? Means, Medians, and Modes (Lothrop, 1978). Kline, Morris. Mathematical Thought from Ancient to Modern Times (Oxford, 1972). Struik, D.J. A Concise History of Mathematics, 3rd rev. ed. (Dover, 1967). Introduction​More than 5,000 years ago an Egyptian ruler recorded, perhaps with a bit of exaggeration, the capture of 120,000 prisoners, 400,000 oxen, and 1,422,000 goats. This event was inscribed on a ceremonial mace which now is in a museum in Oxford, England.​The ancient Egyptians developed the art of counting to a high degree, but their system of numeration was very crude. For example, the number 1,000 was symbolized by a picture of a lotus flower, and the number 2,000 was symbolized by a picture of two lotus flowers growing out of a bush. Although these symbols, called hieroglyphics, permitted the Egyptians to write large numbers, the numeration system was clumsy and awkward to work with. The number 999, for instance, required 27 individual marks.​In our system of numeration, we use ten symbols called digits—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—and combinations of these symbols. Our system of numeration is called the decimal, or base-ten, system. There is little doubt that our ten fingers influenced the development of a numeration system based on ten digits. (See also mathematics.)​Other numeration systems were developed in early cultures and societies. Two of the most common were the base-five system, related to the number of fingers on one hand, and the base-twenty system, related to the number of fingers and toes.​In some languages the word for five is the same as the word for hand, and the word for ten is the same as the word for two hands. In our own language the word digit is a synonym for the word finger—that is, ten digits, ten fingers.​Still another early system of numeration was a base-sixty system developed by the Mesopotamians and used for centuries. These ancient people divided the year into 360 days (6 × 60); today we still divide the hour into 60 minutes and the minute into 60 seconds. Numeration systems of current interest include a binary, or base-two, system used in electronic computers and a base-twelve, or duodecimal, system.​It is worthwhile for everyone to become familiar with the principles of the base-twelve system of numeration and with those of base-two, base-five, or other systems. Working with other bases gives one a deeper insight into the decimal system that most people in the United States have used since childhood. Before considering other systems, however, let us investigate our own familiar decimal system.​The Decimal System​The ten digits of our numeration system are used to name the numbers of dots shown in these frames:​​​If we were not familiar with our system of numeration, a reasonable method for naming the number of dots in frames showing 10, 11, and 12 dots might be somewhat as follows:​​​A system of numeration based on such a procedure as this is called an additive system. The numeral 995 would represent the sum 9 + 9 + 5. However, in our system of numeration, we use a positional, or place-value, system invented by ancient Hindu mathematicians. The positional system has many advantages over a simple additive system.​In a positional system of numeration, the value assigned to a digit depends upon its position in the numeral. For example, in our decimal positional system the 1 in the numeral 103 refers to one group of 100; the 0 to zero groups of 10; and the 3 to three 1s.​​​The symbol 102 is a convenient shorthand notation for the product of 10 × 10. The symbol 2 is called an exponent, and it tells you that 10 is to be multiplied by itself. In the same notation,​10 × 10 × 10 = 103 and 10 × 10 × 10 × 10 = 104.​Using this shorthand notation, the numeral 1,572 can be expressed:​1,572 = [1 × 103] + [5 × 102] + [7 × 10] + [2 × 1].​Numbers such as 1, 6, 230, and 1,572 are called whole numbers. The ten digits may be used to represent not only whole numbers but also numbers less than one and numbers which are sums of whole numbers and numbers less than one. These numbers can be represented either as fractions or as decimals. The value assigned to each digit is again determined by its position in the numeral. For example:​​​​Base-Five Numeration System​Now, consider a frame with 17 dots, thinking of it in terms of one group of 10 dots and 7 more dots:​​​​But we may also think of it as three groups of 5 dots and 2 more dots:​​​​The numeral 32five is a shorthand notation for [3 × (5)] + [2 × (1)]. The subscript “five” tells you that the numeral is expressed in the base-five system of numeration. In our discussion a numeral without a subscript is considered to be base-ten. Thus,​17 = [1 × (10)] + [7 × (1)]​ = [3 × (5)] + [2 × (1)] = 32five.​Similarly, the numeral 214five represents​​​​In the base-five system of numeration, just five digits are used: 0, 1, 2, 3, and 4.​An easy way to recognize the significance and meaning of positional, or place, value and the relationship between base-ten and base-five is to think about the ways in which we express amounts of money. In the base-ten system you may think about pennies (units), dimes (tens), and dollars (hundreds), depending upon the position of the digit in the numeral. Similarly, in the base-five system you may think about pennies (units), nickels (fives), and quarters (twenty-fives).​For example, 431five (cents) may be exchanged for​4(quarters) + 3(nickels) + 1(cent).​In turn, this may be exchanged for​1(dollar) + 1(dime) + 6 (cents),​which may be exchanged for 116 (cents). Thus, 431five =116ten.​The Base-Two, or Binary, System​Encyclopædia Britannica, Inc.Encyclopædia Britannica, Inc.​Consider a frame with seven dots. Instead of thinking in terms of groups of ten or groups of five, we may think in terms of groups of two:​​​​The numeral 111two is a shorthand notation for​[1 × (2 × 2)] + [1 × (2)] + [1 × (1)].​The subscript “two” tells you that the numeral is expressed in the base-two system of numeration. In this system only two digits, 0 and 1, are used.​Let us see how one would do arithmetic problems in the binary system.​​​​Here are some exercises in binary arithmetic:​​​​​​​The simplicity of binary arithmetic makes the binary system well suited for use in electronic computers. The digit 1 may correspond to a light turned on, and the digit 0 to a light turned off. For example, the number 10110two may correspond to this arrangement of lights:​​​​The Base-Twelve, or Duodecimal, System​Consider a frame with 30 dots. We may consider groups of 12:​​​​The numeral 26twelve is a shorthand notation for​[2 × (12)] + [6 × (1)].​The subscript “twelve” tells you that the numeral is expressed in the base-twelve, or duodecimal, system of numeration. This system requires 12 symbols, so we shall use the 10 digits and the letters X and E.​​​Thus,​​​​When working with base-twelve it may be helpful to think about units, dozens, and gross. For example, 431twelve (units) is the same as​4 (gross) + 3 (dozen) + 1 (unit).​In turn, this is the same as​[4 × (12 × 12)] (units) + [3 × (12)] (units)​+ [1 × (1)] (units),​which is the same as 613 units. Thus, 431twelve = 613.​You may also think about inches and feet. For example, XEtwelve (inches) is the same as​X (feet) + E (inches), or 10 (feet) + 11 (inches).​This is the same as 131 (inches). Thus, XEtwelve = 131.​Following are some base-twelve addition problems with related examples shown alongside.​​​​Converting from One Base to Another​You already know how to convert base-two, base-five, and base-twelve numerals into base-ten numerals. Now, using the idea of place value, you can easily convert a numeral of any given base to a base-ten numeral. For example:​37eight = [3 × (8)] + [7 × (1)] = 24 + 7 = 31​146seven = 01 × (7 × 7)] + [4 × (7)] + [6 × (1)]​ = 49+28+ 6 = 83​It may require a little more thought to convert a base-ten numeral into a numeral of another base. You can do this by memorizing some rules and following them blindly. But it is more important to be able to make the conversion in a natural way by applying the basic idea of place value. The following examples show how you can apply the principles of place value to convert from base-ten to another base.​Example 1. Suppose that you wish to convert the base-ten numeral 19 into a base-two numeral. First notice that​​​​Example 2. Convert the base-ten numeral 67 into a base-five numeral. Notice that​​​and 67 = [2 × (5 × 5)] + [3 × (5)] +[2 × (1)] = 232five.​Example 3. This is a second method of converting the base-ten numeral 67 into a base-five numeral. Recall that​​​​Example 4. Convert the base-ten numeral 587 into a base-twelve numeral. Notice that​​​​Just for fun, here are some conversions for you to verify:​(a) 365twelve = 509​(b) 11010110two = 214​(c) 11010110two = 1324five​(d) 424seven1324five​Hindu-Arabic and Roman Numeral Systems​It was stated previously that the ancient Hindus are credited with discovering the decimal system of numeration we use today. This system was translated into Arabic prior to its introduction into Europe by traveling merchants around the 13th century. Hence it is also known as the Hindu-Arabic system.​Adoption of the Hindu-Arabic system met resistance due to the widespread use of the Roman numeral system during this period. Gradually, however, the superior Hindu-Arabic system was learned by the Europeans, and eventually it replaced the Roman system (see Roman numeral).​The Roman numerals are still sometimes used. Some examples of items on or in which Roman numerals still appear include clock faces and books, for numbering introductory pages and chapters.​The Roman system, like others that are not based on the principle of position, does not provide an efficient and easy method of computation. Here are some examples of computations using the Roman system. Equivalent computations using the Hindu-Arabic system are alongside.​​​​​​​The Real Numbers​A numeration system provides the numerals by which we may name numbers. Now, let us use our decimal system of numeration to study the set of real numbers.​When we say that a man is 6 feet tall, weighs 180 pounds, and is 30 years old, we are using numbers as measures of magnitudes. Numbers which are used in this way may be referred to as numbers-of-arithmetic (see arithmetic).​Numbers which are used as measures of directed change are called real numbers. For example, if we are speaking about temperature, a 5-degree increase in temperature may be measured by the real number +5 (say “positive five”), while a 5-degree decrease in temperature may be measured by the real number –5 (say “negative five”). The set of real numbers consists of all the positive numbers, all the negative numbers, and zero. The number-of-arithmetic 0 corresponds to the real number 0.​Corresponding to each nonzero number-of-arithmetic—that is, to each number-of-arithmetic other than 0—there are exactly two real numbers. For example, corresponding to the number-of-arithmetic 5 there are exactly two real numbers: +5 and –5. The number-of-arithmetic 5 is called the arithmetic value of the real numbers +5 and –5.​The sum of a pair of real numbers is a measure of the resultant of a pair of directed changes. For example, if the temperature increases by 5 degrees and later decreases by 3 degrees, the resultant change would be an increase of 2 degrees. The sum +5 + –3 is the same as +2. For short,​+5 + –3 = +2.​A decrease of 5 degrees followed by an increase of 3 degrees would result in a decrease of 2 degrees:​–5 + +3 = –2.​A decrease of 5 degrees followed by another decrease of 3 degrees would result in a decrease of 8 degrees:​–5 + –3 = –8.​Thinking of real numbers as measures of directed change also motivates rules for multiplying the real numbers. Let us suppose, for example, that the temperature on January 1 is 0 degrees. If the temperature then increases 5 degrees per day, the resultant change in temperature 3 days later is an increase of 15 degrees:​+5 × +3 = +15.​Rule: The product of a positive number multiplied by a positive number is a positive number.​If the temperature decreases 5 degrees per day, the resultant change in temperature 3 days later is a decrease of 15 degrees:​–5 × +3 = –15.​Rule: The product of a negative number multiplied by a positive number is a negative number.​Suppose that the temperature had been decreasing 5 degrees per day for the past 3 days. Then 3 days earlier the temperature was 15 degrees higher:​–5 × –3 = +15.​Rule: The product of a negative number multiplied by a negative number is a positive number.​Notice that operations on real numbers correspond in a natural way with operations on their arithmetic values. The arithmetic value of a positive real number is often written as an abbreviation for the real number.​​​The set of real numbers may be pictured as the set of points on a line, and we speak of the set of real numbers as the number line. A point on the line is chosen to represent the number 0, and other points are chosen for the numbers +1, +2, +3, +4, and so on.​The set of positive integers is called a subset of the set of real numbers and consists of all the positive numbers. The set of negative integers is also a subset of the set of real numbers and consists of all the negative numbers: –1, –2, –3, –4, and so on.​​​​The drawing of the number line may be extended to the right and to the left as necessary.​The set of integers consists of all the positive integers, all the negative integers, and zero.​When one integer is divided by another nonzero integer, the quotient is called a rational number. Rational numbers may be represented as fractions.​For example, the fraction +1/+2 represents the quotient of +1 divided by +2. The symbol above the line is called the numerator, and the symbol below the line is called the denominator.​Rational numbers may also be represented as decimals. A fraction can be converted by division to the decimal representation as follows:​​​​You may have noticed that the rational numbers include both positive and negative integers. For example, –4 is a rational number because –4/+1 = –4.​In the real number system, every rational number corresponds to a point on the number line. For example, the point corresponding to +(7/3) may be found by dividing the segment between 0 and +1 into thirds and then constructing the segment 7 times as long as the segment from 0 to +(1/3).​​​​However, not every point that can be shown on a real-number line designates a rational number. The ancient Greek geometers were the first to discover that there are some real numbers which are not rational. They showed that if we construct a square measuring 1 unit by 1 unit, the length of the diagonal (denoted by d in the drawing) is not a rational number.​​​The Pythagorean Theorem, named after the Greek geometer Pythagoras, states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. By this theorem,​d2 = 12 + 12 = 1 + 1 = 2. So, d = √2.​The real numbers which are not rational are called irrational numbers. Other irrational numbers are √3, √5, and π. (Say “square root of three,” “square root of five,” and “pi.” Pi is the circumference of a circle whose diameter is 1 unit in length.)​The decimal of a rational number obtained by dividing an integer by a nonzero integer either terminates or is an infinite decimal which repeats a certain pattern. For example:​1/5= .2 terminates; but 2/7= .285714285714285714 . . . repeats.​An irrational number cannot be obtained by dividing an integer by a nonzero integer. The decimal representation of an irrational number does not terminate, nor does it repeat a certain pattern. In the case of π:​π = 3.141592653589793238462643383279502884