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Mathematics is the study of numbers, quantity, space, pattern, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

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Georg Ferdinand Ludwig Philipp Cantor (December 3, 1845, St. Petersburg, Russia – January 6, 1918, Halle, Germany) was a German mathematician who is best known as the creator of set theory. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities." He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.

Cantor's work encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré, and later from Hermann Weyl and L.E.J. Brouwer. Ludwig Wittgenstein raised philosophical objections. Nowadays, the vast majority of mathematicians who are neither constructivists nor finitists accept Cantor's work on transfinite sets and arithmetic, recognizing it as a major paradigm shift.

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animation of the act of "unrolling" a circle's circumference, illustrating the ratio pi (π)
Credit: John Reid

Pi, represented by the Greek letter π, is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space (i.e., on a flat plane); it is also the ratio of a circle's area to the square of its radius. (These facts are reflected in the familiar formulas from geometry, C = π d and A = π r2.) In this animation, the circle has a diameter of 1 unit, giving it a circumference of π. The rolling shows that the distance a point on the circle moves linearly in one complete revolution is equal to π. Pi is an irrational number and so cannot be expressed as the ratio of two integers; as a result, the decimal expansion of π is nonterminating and nonrepeating. To 50 decimal places, π  3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510, a value of sufficient precision to allow the calculation of the volume of a sphere the size of the orbit of Neptune around the Sun (assuming an exact value for this radius) to within 1 cubic angstrom. According to the Lindemann–Weierstrass theorem, first proved in 1882, π is also a transcendental (or non-algebraic) number, meaning it is not the root of any non-zero polynomial with rational coefficients. (This implies that it cannot be expressed using any closed-form algebraic expression—and also that solving the ancient problem of squaring the circle using a compass and straightedge construction is impossible). Perhaps the simplest non-algebraic closed-form expression for π is 4 arctan 1, based on the inverse tangent function (a transcendental function). There are also many infinite series and some infinite products that converge to π or to a simple function of it, like 2/π; one of these is the infinite series representation of the inverse-tangent expression just mentioned. Such iterative approaches to approximating π first appeared in 15th-century India and were later rediscovered (perhaps not independently) in 17th- and 18th-century Europe (along with several continued fractions representations). Although these methods often suffer from an impractically slow convergence rate, one modern infinite series that converges to 1/π very quickly is given by the Chudnovsky algorithm, first published in 1989; each term of this series gives an astonishing 14 additional decimal places of accuracy. In addition to geometry and trigonometry, π appears in many other areas of mathematics, including number theory, calculus, and probability.

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