theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. A lemma is a theorem that is demonstrated as an intermediate step in the proof of another, more basic theorem. A corollary is a theorem that follows as a direct consequence of another theorem or an axiom. There are many famous theorems in mathematics, often known by the name of their discoverer, e.g., the Pythagorean Theorem, concerning right triangles. One of the most famous problems of number theory was the proof of Fermat's Last Theorem (see Fermat, Pierre de); the theorem states that for an integer n greater than 2 the equation xⁿ+yⁿ=zⁿ admits no solutions where x, y, and z are also integers.

In mathematics or logic, a statement whose validity has been established or proved. It consists of a hypothesis and a conclusion, beginning with certain assumptions that are necessary and sufficient to establish a result. A system of theorems that build on and augment each other constitutes a theory. Within any theory, however, only statements that are essential, important, or of special interest are called theorems. Less important statements, usually stepping-stones in proofs of more important results, are called lemmas. A statement proved as a direct consequence of a theorem is a corollary of the theorem. Some theorems (and even lemmas and corollaries) are singled out and given h1s (e.g., Gödel's theorem, fundamental theorem of algebra, fundamental theorem of calculus, Pythagorean theorem).

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In mathematics, two theorems, one associated with differential calculus and one with integral calculus. The first proposes that any differentiable function defined on an interval has a mean value, at which a tangent line is parallel to the line connecting the endpoints of the function's graph on that interval. For example, if a car covers a mile from a dead stop in one minute, it must have been traveling exactly a mile a minute at some point along that mile. In integral calculus, the mean value of a function on an interval is, in essence, the arithmetic mean (see mean, median and mode) of its values over the interval. Because the number of values is infinite, a true arithmetic mean is not possible. The theorem shows how to find the mean value using a definite integral. Seealso Rolle's theorem.

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In statistics, any of several fundamental theorems in probability. Originally known as the law of errors, in its classic form it states that the sum of a set of independent random variables will approach a normal distribution regardless of the distribution of the individual variables themselves, given certain general conditions. Further, the mean (see mean, median, and mode) of the normal distribution will coincide with the (arithmetic) mean of the (statistical) means of each random variable.

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In algebra, a formula for expansion of the binomial (math.x + math.y) raised to any positive integer power. A simple case is the expansion of (math.x + math.y)², which is math.x² + 2math.xmath.y + math.y². In general, the expression (math.x + math.y)^math.n expands to the sum of (math.n + 1)terms in which the power of math.x decreases from math.n to 0 while the power of math.y increases from 0 to math.n in successive terms. The terms can be represented in factorial notation by the expression [math.n!/((math.n − math.r)!math.r!)]math.x^{math.n − math.r}math.y^math.r in which math.r takes on integer values from 0 to math.n.

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