Flashcards: Articles containing proofs

Rational trigonometry is a proposed reformulation of metrical planar and solid geometries (which includes trigonometry) by Canadian mathematician Norman J.

In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no algebraic solution—that is, solution in radicals—to the general polynomial equations of degree five or higher with arbitrary coefficients.

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and integer n it holds that where i is the imaginary unit (i2 = −1).

This property is the key in the proof of the fundamental theorem of arithmetic.

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that, every number after the first two is the sum of the two preceding ones: Often, especially in modern usage, the sequence is extended by one more initial term: .

In mathematics, Fermat's theorem (also known as Interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function derivative is zero in that point).

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors.

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.

In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem (such as a differential equation) to a discrete problem.

In combinatorics (combinatorial mathematics), the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as The principle is more clearly seen in the case of three sets, which for the sets A, B and C is given by Generalizing the results of these examples gives the principle of inclusion–exclusion.

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought.

In mathematics, the mean value theorem states, roughly, that given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

Multiplication (often denoted by the cross symbol "×", by a point "·", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division.

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary.

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients.

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean, and it informally measures how far a set of (random) numbers are spread out from their mean.

In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if f : M → N is a continuous function between two metric spaces, and M is compact, then f is uniformly continuous.

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, …, n, for some positive integer n.

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side.

Variants of Nim have been played since ancient times.

Rouché's theorem, named after Eugène Rouché, states that if the complex-valued functions f and g are holomorphic inside and on some closed contour K, with |g(z)| < |f(z)| on K, then f and f + g have the same number of zeros inside K, where each zero is counted as many times as its multiplicity.

In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter".

In number theory, Zolotarev's lemma states that the Legendre symbol for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation: where ε denotes the signature of a permutation and πa is the permutation of the nonzero residue classes mod p induced by multiplication by a.

In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces.

In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions.

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.

The square root of 2, written in mathematics as √2 or 2 1⁄2, is the positive algebraic number that, when multiplied by itself, gives the number 2.

In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form then for x in (x0, x1) the derivative of this integral is thus expressible as provided that f and its partial derivative fx are both continuous over a region in the form [x0, x1] × [y0, y1].

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for a short exact sequence are equivalent.

In the mathematical discipline of graph theory the Tutte theorem, named after William Thomas Tutte, is a characterization of graphs with perfect matchings.

In mathematics, Pascal's rule is a combinatorial identity about binomial coefficients.

In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths , , , and area , the following inequality holds: .

In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series.

A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum where a0 is a nonnegative integer, and a1, a2, .

In mathematics, a Gödel numbering for sequences provides us an effective way to represent each finite sequence of natural numbers as a single natural number.

In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single (n − 1)-dimensional hyperplane.

Goursat's lemma, named after the French mathematician Édouard Goursat, is an algebraic theorem about subgroups of the direct product of two groups.

The ratio estimator is a statistical parameter and is defined to be the ratio of means of two variates.