The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, mul ...

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

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Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

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Long before the earliest written records, there are drawings that indicate a knowledge of mathematics and of measurement of time based on the stars. For example, paleontologists have discovered ochre rocks in a cave in South Africa adorned with scratched geometric patterns dating back more than 70,000 years [1]. Also prehistoric artifacts discovered in Africa and France, dated between 35000 BC and 20000 BC, indicate early attempts to quantify time Evidence exists that early counting involved women who kept records of their monthly biological ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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Long before the earliest written records, there are drawings that indicate a knowledge of mathematics and of measurement of time based on the stars. For example, paleontologists have discovered ochre rocks in a cave in South Africa adorned with scratched geometric patterns dating back more than 70,000 years [1]. Also prehistoric artifacts discovered in Africa and France, dated between 35000 BC and 20000 BC, indicate early attempts to quantify time Evidence exists that early counting involved women who kept records of their monthly biological ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Early written mathematics 2000 BC - 600 BC, History of mathematics - India 800 BC - 300 BC, History of mathematics - Greece and Hellenistic mathematics 400 BC - 200 BC, History of mathematics - Arab and Persian mathematics 650 - 1200, History of mathematics - Developing the concept of number through equations, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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In Europe at the dawn of the Renaissance, most of what is now called school mathematics -- addition, subtraction, multiplication, division, and geometry -- was known to educated people, though the notation was cumbersome: Roman numerals and words were used, but no symbols: no plus sign, no equal sign, no zero, and no use of x as an unknown. Almost all of the mathematics now taught in college had yet to be d ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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The Maya calendar utilized a base-20 number system which included the 'number' zero (also see Maya numerals). In China, Zu Chongzhi (5th century) of the Southern and Northern Dynasties was the first person to calculate the value of Pi to seven decimal places. ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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Greek mathematicians such as Eudoxus and Archimedes made informal use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In the 12th century the Indian mathematician Bhaskara gave an example of what would now be called a "differential coefficient" and the basic idea behind what is now known as Rolle's theorem. The 14th century Indian mathematician Madhava of Sangamagrama expressed various trigonometric functions as infinite series, and estimated the magnitude of th ...

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Mathematical analysis, Mathematical analysis - History, Mathematical analysis - Subdivisions

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Mathematical notation - Counting. It is believed that a mathematical notation was first developed at least 50,000 years ago in order to assist with counting. Early mathematical ideas for counting were represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient Sumer. The Census Quipu of the Andes and the Ishango Bone from Africa both used the tally mark method of accounting for numerical concepts. Mathemat ...

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Mathematical notation, Mathematical notation - Definition, Mathematical notation - Expressions, Mathematical notation - Precise semantic meaning, Mathematical notation - History, Mathematical notation - Counting, Mathematical notation - Geometry becomes analytic, Mathematical notation - Counting is mechanized, Mathematical notation - Computerized notation, Mathematical notation - Ideographic notation, Mathematical notation - Notes

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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

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Mathematics - History Mathematics - Inspiration pure and applied mathematics and aesthetics Mathematics - Notation language and rigor Mathematics - Is mathematics a science? Mathematics - Overview of fields of mathematics Mathematics - Major themes in mathematics
Mathematics - Quantity Mathematics - Structure Mathematics - Space Mathematics - Change Mathematics - Foundations and methods Mathematics - Discrete mathematics Mathematics - Applied mathematics Mathematics - Important theorems Mathematics - Important conjectures Mathematics - History and the world of mathematicians Mathematics - Mathematics and other fields
Mathematics - Common misconceptions

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Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. These topics are often studied in the context of real numbers, complex numbers, and their functions. However, they can also be defined and studied in any space of mathematical objects that is equipped with a definition of "nearness" (a topological space) or "distance" (a metric space). Mathematical analysis ...

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Mathematical analysis - History Mathematical analysis - Subdivisions

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In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

Including:

Sheaf mathematics - Introduction Sheaf mathematics - The formal definition
Sheaf mathematics - Definition of a presheaf Sheaf mathematics - The gluing axiom
Sheaf mathematics - Examples Sheaf mathematics - Morphisms of sheaves Sheaf mathematics - Stalks of a sheaf at a point and germs of functions Sheaf mathematics - The étale space of a sheaf Sheaf mathematics - Generalizations Sheaf mathematics - History

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Although the earliest Greek texts on mathematics were written after the Hellenistic period, many of these are considered to be copies of works written during, and some before, the Hellenistic period. Nevertheless, the dates of Greek mathematics are more certain than the dates of earlier mathematical writing, since a large number of chronologies exist that, overlapping, record events year by year up to the present day. Even so, many dates are uncertain; but the doubt is a matter of decades rather than centuries. Greek mathematics is th ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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The Islamic Caliphate (Islamic empire) established across the Middle East, North Africa, and in parts of India in the 8th century preserved and translated much of the Greek mathematics which was at that time largely forgotten in Europe. Indian mathematics, and the introduction of the Hindu-Arabic numerals, also had a major influence on Islamic mathematics. Like the Hindu mathematicians, Islamic mathematicians were especially interested in astronomy. The works of Brahmagupta were translated into Arabic circa 766. Al-Khwarizmi, the 9th ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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The oldest mathematics text discovered so far is the Moscow papyrus, which is an Egyptian Middle Kingdom papyrus dated circa 2050 BC - 1800 BC. Like many ancient mathematical texts, 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 pyramid: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square thi ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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In China, in 212 BC, the Emperor Qin Shi Huang (Shi Huang-ti) commanded that all books be burned. While this order was not universally obeyed, it means that little is known with certainty about ancient Chinese mathematics. Another problem is that the Chinese wrote on bamboo, a perishable medium. Dating from the Shang period (1500 BC - 1027 BC), the earliest extant Chinese mathematics consists of numbers scratched on tortoise shell. These numbers use a decimal system, so that the number 123 is written (from top to bottom) as the symbol ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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The earliest known mathematics in ancient India dates back to 3000 BC with the Indus Valley civilization. After its collapse in 1700 BC however, writing was absent in South Asia for a long period. There is considerable controversy regarding the dates when writing was re-developed in India and when the Brahmi script was developed.[5] Some scholars, such as Georg Bühler, date the Brahmi script as early as the 8th century BC, others from the Maurya dynasty in the 4th century BC. Some evidence dates it to 600 BC (see Brāhmī), while some schol ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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From 200 BC, mathematicians in India began studying mathematics for the sole purpose of science, starting with Jaina mathematicians between 200 BC and AD 400. They discovered transfinite numbers, set theory, logarithms, fundamental laws of indices, cubic equations, quartic equations, sequences and progressions, permutations and combinations, squaring and extracting square roots, and finite and infinite powers. Discoveries written in the Bakshali Manuscript include solutions of linear equations with upto five unknowns, the solution of ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, a ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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The 17th century saw an unprecedented explosion of mathematical and scientific ideas that not only fascinated philosophers but had industrial applications that began to make major changes in the way people lived. Copernicus, a Pole, had written that planets orbit the Sun. Galileo, an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe, a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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As we have seen, knowledge of the natural numbers, 1, 2, 3,..., as preserved in monolithic structures, is older then any surviving written text. The earliest civilizations, in Mesopotamia, Egypt, India, and China, knew arithmetic. One way to view the development of the various number systems of modern mathematics is to see new numbers invented to answer questions about arithmetic performed on older numbers. In prehistoric times, fractions answered the question: what number, when multiplied by 3, gives the answer 1. In India and China, ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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