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

In number theory, the prime number theorem ( PNT ) describes the asymptotic distribution of the prime numbers.

In non-Cartesian ( non-square ) or curved coordinate systems, the Pythagorean theorem holds only on infinitesimal length scales and must be augmented with a more general metric tensor g < sub > μν </ sub >, which can vary from place to place and which describes the local geometry in the particular coordinate system.

In complex analysis, a branch of mathematics, the Casorati – Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities.

The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number µ during this convergence.

This conjecture can be justified ( but not proven ) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem.

Building on Hartley's foundation, Shannon's noisy channel coding theorem ( 1948 ) describes the maximum possible efficiency of error-correcting methods versus levels of noise interference and data corruption.

In probability theory, the law of large numbers ( LLN ) is a theorem that describes the result of performing the same experiment a large number of times.

The Cartan – Dieudonné theorem describes the structure of the orthogonal group for a non-singular form.

A perfect matching describes a way of simultaneously satisfying all job-seekers and filling all jobs ; the marriage theorem provides a characterization of the bipartite graphs which allow perfect matchings.

Ptolemy describes this theorem in the Almagest XII. 1.

* This describes the original proof of the theorem ( Atiyah and Singer never published their original proof themselves, but only improved versions of it.

In probability theory, the Girsanov theorem ( named after Igor Vladimirovich Girsanov ) describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure.

The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which describes the probability that an underlying instrument ( such as a share price or interest rate ) will take a particular value or values to the risk-neutral measure which is a very useful tool for pricing derivatives on the underlying.

In law and economics, the Coase theorem ( pronounced / ˈkoʊs /), attributed to Nobel Prize laureate Ronald Coase, describes the economic efficiency of an economic allocation or outcome in the presence of externalities.

Angular velocity is a vector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which it is rotating ( the existence of this instantaneous axis is guaranteed by the Euler's rotation theorem ).

In mathematics, the Nagell – Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.

Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field Q of rational numbers.

In fact a thorough use of Pontryagin duality here shows that the whole Kronecker theorem describes the closure of < P > as the intersection of the kernels of the χ with

: The contour is chosen so that the contour follows the part of the complex plane that describes the real-valued integral, and also encloses singularities of the integrand so application of the Cauchy integral formula or residue theorem is possible

He also describes Smith's theorem that " the division of labour is limited by the extent of the market " as the " core of a theory of the functions of firm and industry " and a " fundamental principle of economic organisation.

In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.

In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.

Eilenberg's theorem, often referred to as the variety theorem describes a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups.

Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.

Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.

* The Ham sandwich theorem: For any compact sets A < sub > 1 </ sub >,..., A < sub > n </ sub > in ℝ < sup > n </ sup > we can always find a hyperplane dividing each of them into two subsets of equal measure.

The binomial theorem also holds for two commuting elements of a Banach algebra.

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus.

* ( Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation ).

As a consequence of the Nyquist – Shannon sampling theorem, any spatial waveform that can be displayed must consist of at least two pixels, which is proportional to image resolution.

( These two observations combine as real and imaginary parts in Cauchy's integral theorem.

which can be obtained by two consecutive applications of Pythagoras ' theorem.

It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free ( and hence has only one prime factor of two ) will have at least one odd prime factor, and thus results in an even dividing an odd, a contradiction.

The convolution theorem for the discrete-time Fourier transform indicates that a convolution of two infinite sequences can be obtained as the inverse transform of the product of the individual transforms.

It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization ( which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

Image: Pythagorean. svg | Pythagoras ' theorem: The sum of the areas of the two squares on the legs ( a and b ) of a right triangle equals the area of the square on the hypotenuse ( c ).

The celebrated Pythagorean theorem ( book I, proposition 47 ) states that in any right triangle, the area of the square whose side is the hypotenuse ( the side opposite the right angle ) is equal to the sum of the areas of the squares whose sides are the two legs ( the two sides that meet at a right angle ).

Animation illustrating Pythagorean theorem | Pythagoras ' rule for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.

In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

The intuitive statement of the four color theorem, i. e. ' that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color ', needs to be interpreted appropriately to be correct.

In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, " every planar graph is four-colorable " (; ).

In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.