 # Mathematics Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day. History Evolution Etymology Definitions of mathematics

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List of theorems This is a list of theorems, by Wikipedia page. See also Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields. 0–9 Metric space The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Polynomial The graph of a polynomial function of degree 3 Etymology According to the Oxford English Dictionary, polynomial succeeded the term binomial, and was made simply by replacing the Latin root bi- with the Greek poly-, which comes from the Greek word for many.

Fractals Representing geometry in SecondLife Some comments on the suitability of SecondLife as a way of presenting fractal geometry Fractal forms found by using Google Earth Readers are encouraged to contribute their own findings. POVRay Fractal Raytracing Contest Property (philosophy) Daniel Dennett distinguishes between lovely properties (such as loveliness itself), which, although they require an observer to be recognised, exist latently in perceivable objects; and suspect properties which have no existence at all until attributed by an observer (such as being a suspect in a murder enquiry) Property dualism: the exemplification of two kinds of property by one kind of substance Property dualism describes a category of positions in the philosophy of mind which hold that, although the world is constituted of just one kind of substance—the physical kind—there exist two distinct kinds of properties: physical properties and mental properties. In other words, it is the view that non-physical, mental properties (such as beliefs, desires and emotions) inhere in some physical substances (namely brains).

Theorem Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from the hypotheses. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. Boundary (topology) A set (in light blue) and its boundary (in dark blue). A connected component of the boundary of S is called a boundary component of S. If the set consists of discrete points only, then the set has only a boundary and no interior. There are several common (and equivalent) definitions to the boundary of a subset S of a topological space X: These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure.

Fundamental theorem of algebra The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. History Peter Rothe, in his book Arithmetica Philosophica (published in 1608), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L'invention nouvelle en l'Algèbre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds “unless the equation is incomplete”, by which he meant that no coefficient is equal to 0.

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