Mathematics
Numbers, structures and proofs
Articles in this category are re-verified roughly every 365 days.
Mathematics
Bayes's theorem
Bayes's theorem is a rule for calculating how evidence changes the probability of a hypothesis. It combines the hypothesis's prior probability with the probability of observing the evidence if the hypothesis is true, while accounting for how often that evidence occurs overall. The rule underlies Bayesian inference and helps explain why test results must be interpreted alongside base rates.
Updated 19 Oct 2026
Mathematics
Big O notation
Big O notation describes an asymptotic upper bound on how a function, such as an algorithm’s running time or memory use, grows with input size. It formalizes statements about growth while setting aside constant factors and lower-order terms, allowing broad comparison of algorithmic resource requirements. Big O is a mathematical bound, not by itself a claim about an algorithm’s worst-case behavior or its exact running time.
Updated 30 Oct 2026
Mathematics
Boolean algebra
Boolean algebra is a mathematical structure whose operations model logical combination and complementation. In the two-valued interpretation, its elements are false and true (often written 0 and 1), with operations corresponding to NOT, AND and OR. It also has a set-theoretic interpretation and became important in the analysis and design of switching and digital circuits.
Updated 16 Oct 2026
Mathematics
Borsuk–Ulam Theorem
The Borsuk–Ulam theorem is a result in topology stating that every continuous map from an n-dimensional sphere to n-dimensional Euclidean space assigns the same value to at least one pair of antipodal points. It has notable consequences in geometry, including the ham sandwich theorem, and provides a classic example of how continuity and symmetry yield existence results.
Updated 10 Nov 2026
Mathematics
Brachistochrone
The brachistochrone is the curve of least travel time for an ideal particle released from rest at one point and sliding under gravity without friction to another, lower point. For the classical problem, the minimizing path is an arc of a cycloid—not a straight line. Posed as a mathematical challenge by Johann Bernoulli in 1696, it helped catalyze the calculus of variations.
Updated 26 Oct 2026
Mathematics
Brouwer fixed-point theorem
The Brouwer fixed-point theorem states that every continuous function from a closed ball in finite-dimensional Euclidean space to itself has at least one fixed point: a point that the function leaves unchanged. First proved by L. E. J. Brouwer in 1912, the result is a central theorem of topology and underlies existence arguments in areas such as economics. It guarantees existence, not necessarily a way to calculate the fixed point or its uniqueness.
Updated 26 Oct 2026
Mathematics
Calculus
Calculus is a branch of mathematics for describing change and accumulation. Differential calculus studies derivatives, which capture instantaneous rates of change; integral calculus studies totals built from continuously varying quantities. The fundamental theorem of calculus links the two: under suitable conditions, differentiation and integration are related inverse operations. Developed from earlier mathematical work and formalized independently by Isaac Newton and Gottfried Wilhelm Leibniz in the seventeenth century, calculus became a central tool across science and engineering.
Updated 13 Oct 2026
Mathematics
Cellular automaton
A cellular automaton is a mathematical model made of cells arranged on a grid, each holding a state that changes in discrete time steps according to a local rule. Repeated updates can produce large-scale patterns from simple interactions. Cellular automata are used to study computation, complex behaviour and dynamical processes; Conway’s Game of Life is a well-known example.
Updated 31 Oct 2026
Mathematics
Chaos theory
Chaos theory studies dynamical systems whose evolution follows precise rules but can be difficult to predict because small differences in starting conditions may grow over time. In mathematical models, chaotic behaviour is commonly associated with deterministic, nonlinear and aperiodic dynamics, though formal definitions vary. The field clarifies how unpredictability can arise without randomness and has applications in areas including meteorology, fluid dynamics, population biology and chemistry.
Updated 20 Oct 2026
Mathematics
Chinese remainder theorem
The Chinese remainder theorem describes when simultaneous congruences determine a unique solution modulo a product of moduli. If the moduli are pairwise coprime, every choice of remainders has exactly one solution modulo their product. The result connects modular arithmetic with number theory and computer science and has applications in efficient computation and cryptography.
Updated 30 Oct 2026
Mathematics
Euclidean algorithm
The Euclidean algorithm is a procedure for finding the greatest common divisor of two integers. It repeatedly replaces a pair of numbers with the smaller number and the remainder from division, stopping when that remainder is zero; the last nonzero remainder is the greatest common divisor. Its extended form also finds integer coefficients expressing that divisor as a linear combination of the original numbers.
Updated 21 Oct 2026
Mathematics
Fibonacci sequence
The Fibonacci sequence is a number sequence in which each term is the sum of the two preceding terms. Written as 0, 1, 1, 2, 3, 5, 8, …, it is defined by a simple recurrence whose ratios of consecutive terms approach the golden ratio. The sequence is named for the medieval Italian mathematician Fibonacci, who used a rabbit-counting puzzle to illustrate it in a widely read mathematical work.
Updated 13 Oct 2026
Mathematics
Four-color theorem
The four-color theorem states that any map drawn on a plane or sphere can have its regions colored with no more than four colors so that regions sharing a boundary line have different colors. Equivalently, every loopless planar graph has a proper vertex coloring with at most four colors. First posed in the nineteenth century, the theorem became famous for its computer-assisted proof.
Updated 20 Oct 2026
Mathematics
Fourier Transform
The Fourier transform is a mathematical integral transform that represents a suitable function as components indexed by frequency. It turns a signal or other function from its original domain into a frequency-domain representation; under appropriate conditions, an inverse transform recovers the original. Developed from Joseph Fourier’s work on heat conduction, the technique is now central to analysis, signal processing, imaging and many areas of physics and engineering.
Updated 19 Oct 2026
Mathematics
Fractals
Fractals are mathematical objects with intricate structure across multiple scales, often showing exact or statistical self-similarity. Fractal geometry uses iterative constructions and measures such as fractal dimension to study patterns that are difficult to describe with ordinary Euclidean geometry and to model some irregular forms in nature.
Updated 16 Oct 2026
Mathematics
Fundamental theorem of arithmetic
The fundamental theorem of arithmetic states that every integer greater than 1 is prime or can be expressed as a product of prime numbers in exactly one way, apart from the order of the factors. It underlies prime factorization, divisibility, greatest common divisors and least common multiples, and has connections to number theory and public-key cryptography.
Updated 30 Oct 2026
Mathematics
Game theory
Game theory is a branch of applied mathematics for analyzing choices whose outcomes depend on the choices of others. Models represent decision-makers, possible actions, information and payoffs. Their tools help study cooperation, competition and bargaining in fields such as economics, politics and biology. An equilibrium is a result under a model's assumptions, not automatically a fair or socially best outcome.
Updated 16 Oct 2026
Mathematics
Gödel's incompleteness theorems
Gödel's incompleteness theorems are results in mathematical logic about what can be proved inside formal systems that express arithmetic. In a modern form strengthened by J. Barkley Rosser, the first theorem says that any consistent, effectively axiomatized theory containing enough arithmetic leaves some sentence neither provable nor refutable within that theory. The second says that, under the theorem's conditions, such a theory cannot prove its own appropriately formalized consistency. These are limits on proof within a particular system, not claims that mathematics is inconsistent or that a statement can never be proved in a stronger system.
Updated 14 Oct 2026
Mathematics
Goldbach's conjecture
Goldbach's conjecture, also called the strong or binary Goldbach conjecture, states that every even integer greater than 2 can be written as the sum of two prime numbers. Despite its simple statement and extensive computational checks, no general proof is known. It is a central open problem in additive number theory, distinct from the weak (ternary) Goldbach conjecture, which has been proved.
Updated 30 Oct 2026
Mathematics
Graph theory
Graph theory is a branch of mathematics that studies structures made of vertices and edges, used to represent relationships between objects. By choosing what counts as an object and a connection, a graph can model questions such as whether a route exists, how to assign resources without conflicts, or how to connect a network efficiently. The field grew from Leonhard Euler's eighteenth-century analysis of the Königsberg bridge problem and now has applications in mathematics, computing and many other disciplines.
Updated 15 Oct 2026
Mathematics
Hindu-Arabic numeral system
The Hindu-Arabic numeral system is a decimal place-value system that uses ten digits, including zero, to represent numbers. Developed in India and transmitted through mathematical scholarship in the Islamic world, it became the basis of the numerals now commonly used in much of the world. Its positional notation makes it possible to express large numbers compactly and perform arithmetic with efficient written algorithms.
Updated 23 Oct 2026
Mathematics
Information theory
Information theory is a mathematical framework for measuring uncertainty and analyzing how information can be represented, stored and transmitted. Developed in its modern form by Claude Shannon, it uses concepts such as entropy and channel capacity to describe the limits of data compression and reliable communication. The theory addresses the statistical and engineering aspects of messages, not their meaning or value.
Updated 14 Oct 2026
Mathematics
Intermediate value theorem
The intermediate value theorem states that a continuous real-valued function on a closed interval takes every value between its endpoint values. In particular, if its values at the endpoints have opposite signs, it must have a zero between them. The theorem guarantees existence, not the location or number of solutions, and underlies methods such as interval bisection for approximating roots.
Updated 30 Oct 2026
Mathematics
Kolmogorov complexity
Kolmogorov complexity, also called algorithmic complexity, measures the information content of an individual finite object by the length of its shortest effective description. For a binary string, it is defined relative to a universal computing machine as the length of the shortest program that outputs the string. The measure connects data compression, computation and randomness, but it is uncomputable in general.
Updated 31 Oct 2026
Mathematics
Linear algebra
Linear algebra is the branch of mathematics concerned with linear equations, vectors, matrices, vector spaces and linear transformations. It provides tools for representing systems compactly, understanding how transformations act on data and geometry, and solving both exact and approximate problems. Its methods are used throughout mathematics and in fields such as physics, engineering, economics and computing.
Updated 18 Oct 2026
Mathematics
Mandelbrot set
The Mandelbrot set is a set of complex parameters defined by repeatedly applying the rule zₙ₊₁ = zₙ² + c, starting at z₀ = 0. A parameter c belongs to the set when the resulting sequence remains bounded; parameters whose sequences escape without bound lie outside it. Computer images reveal an intricate boundary, connecting a compact mathematical definition to rich structures studied in complex dynamics and fractal geometry.
Updated 25 Oct 2026
Mathematics
Markov chain
A Markov chain is a stochastic process that moves among states according to transition probabilities, with the distribution of the next state depending only on the current state rather than the full history. In a time-homogeneous chain, these probabilities are represented by a transition matrix, and repeated matrix multiplication describes the process over multiple steps. Markov chains model evolving systems under uncertainty, but their predictions are conditional on the chosen states and transition model.
Updated 30 Oct 2026
Mathematics
Möbius strip
A Möbius strip is a surface made by giving a rectangular strip a half-twist and joining its ends. It has one connected boundary and is non-orientable: travelling around it reverses the orientation of a direction carried along the surface. First described independently by August Ferdinand Möbius and Johann Benedict Listing in 1858, it is a familiar model for exploring topology and the difference between a surface's appearance and its underlying properties.
Updated 21 Oct 2026
Mathematics
Non-Euclidean geometry
Non-Euclidean geometry is a family of geometrical systems that differ from Euclidean geometry, most notably in how they treat parallel lines. In hyperbolic geometry, multiple lines through a point can avoid a given line; in elliptic geometry, no such parallel lines exist. These systems emerged from nineteenth-century efforts to understand Euclid’s parallel postulate and became important both to mathematics and to later descriptions of curved space.
Updated 15 Oct 2026
Mathematics
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem gives a condition for reconstructing a band-limited, continuous-time signal from regularly spaced samples. For a baseband signal with no components above frequency B, sampling faster than 2B permits exact reconstruction in the ideal mathematical model. The theorem explains the sampling-rate requirements behind digital audio, image acquisition, communications, and measurement, as well as why undersampling can cause aliasing.
Updated 20 Oct 2026
Mathematics
P versus NP problem
The P versus NP problem asks whether every decision problem whose proposed solutions can be checked in polynomial time can also be solved in polynomial time. In standard terms, it asks whether the complexity classes P and NP are equal. A proof either way would have deep consequences for algorithms and complexity theory; despite its importance, the question remains open.
Updated 31 Oct 2026
Mathematics
P-value
A p-value is the probability, under a specified statistical model and null hypothesis, of obtaining a test statistic at least as extreme as the observed one. It describes how the data compare with that model; it is not the probability that the null hypothesis is true, nor a measure of an effect's size or practical importance. Its interpretation depends on the test, assumptions, and study design.
Updated 22 Oct 2026
Mathematics
Pascal's Triangle
Pascal's triangle is an array of binomial coefficients in which each interior number is the sum of the two entries above it. Its rows encode binomial expansions and combination counts, and related arrangements were studied in several mathematical traditions centuries before Pascal.
Updated 16 Oct 2026
Mathematics
Penrose tiling
A Penrose tiling is a family of non-periodic patterns that cover the plane using a small set of shapes and rules for how they fit. The familiar kite-and-dart version uses just two tile shapes, yet its matching rules prevent any translational repeat. Such patterns combine local order, including fivefold arrangements, with an overall structure that never repeats periodically; their self-similarity also makes them a useful mathematical model for understanding quasicrystal order.
Updated 25 Oct 2026
Mathematics
Poincaré conjecture
The Poincaré conjecture is a theorem in topology stating that every closed, simply connected three-dimensional manifold is homeomorphic to the 3-sphere. Formulated by Henri Poincaré in 1904, it was proved by Grigori Perelman through work on Ricci flow announced in 2002–03.
Updated 20 Oct 2026
Mathematics
Prime numbers
A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. Primes are the multiplicative building blocks of the positive integers: every integer greater than 1 has a unique factorization into primes, apart from the order of the factors. Studied since antiquity, primes are infinite in number and follow a large-scale distribution described by the prime number theorem. They also play a role in some public-key cryptographic systems, including RSA.
Updated 13 Oct 2026
Mathematics
Prime-counting function
The prime-counting function, written π(x), gives the number of prime numbers less than or equal to a real number x. Its values record how primes are distributed among the integers. The prime number theorem says that, for large x, π(x) is asymptotic to x divided by the natural logarithm of x, providing a simple estimate of how prime numbers become less dense as numbers grow.
Updated 22 Oct 2026
Mathematics
Probability theory
Probability theory is the mathematical study of uncertain outcomes. It assigns probabilities to events in a model of possible results and provides rules for reasoning about combinations of events, conditional information, and random variables. Used throughout statistics and the sciences, it quantifies uncertainty under stated assumptions; it does not guarantee what will happen in any one trial.
Updated 12 Oct 2026
Mathematics
Pythagorean theorem
The Pythagorean theorem is a central result of Euclidean geometry: in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse, written a^2 + b^2 = c^2. It provides a way to calculate unknown side lengths, underlies the distance formula in coordinate geometry, and has many geometric proofs. Although named for the ancient Greek mathematician Pythagoras, records show that the relationship was known in earlier civilizations.
Updated 12 Oct 2026
Mathematics
Riemann hypothesis
The Riemann hypothesis is a major open conjecture in number theory about the nontrivial zeros of the Riemann zeta function. It states that every such zero has real part one-half. The zeta function is closely connected to the distribution of prime numbers, so the conjecture would sharpen understanding of how primes are distributed; despite extensive numerical checks, no general proof is known.
Updated 30 Oct 2026
Mathematics
Rolle's theorem
Rolle's theorem guarantees that a differentiable curve with equal values at the ends of an interval has at least one interior point with a horizontal tangent, provided it is continuous on the closed interval and differentiable on its interior. It is a special case of the mean value theorem and a foundational result in differential calculus.
Updated 30 Oct 2026
Mathematics
Set Theory
Set theory is the branch of mathematics that studies sets—collections whose members are treated as mathematical objects—and the relationships between them. It provides a precise language for describing structures such as numbers, functions and relations, and serves as a widely used foundation for much of modern mathematics. Its study of infinite sets revealed that different infinite collections can have different cardinalities. Modern axiomatic systems, especially Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC), specify how sets may be formed and avoid contradictions associated with unrestricted collection-making.
Updated 17 Oct 2026
Mathematics
Standard deviation
Standard deviation is a measure of how dispersed numerical values are around their arithmetic mean. Defined as the square root of variance, it expresses spread in the same units as the data. The formula differs according to whether the values describe a whole population or a sample used to estimate one. Because squared deviations give extra weight to distant observations, standard deviation is sensitive to outliers and is most informative when interpreted alongside the data's distribution and other measures of spread.
Updated 18 Oct 2026
Mathematics
Topology
Topology is a branch of mathematics concerned with properties of spaces that remain unchanged under continuous deformation. Its geometric intuition is often described as stretching or bending without tearing, but modern topology also studies abstract spaces defined by collections of open sets. Concepts such as continuity, connectedness, and holes help mathematicians compare spaces and investigate problems in geometry, analysis, and other fields.
Updated 14 Oct 2026
Mathematics
Turing machine
A Turing machine is an abstract mathematical model of step-by-step computation, defined by a finite set of states and rules for reading, writing and moving across a tape of symbols. Its universal form can simulate other Turing machines, while results such as the undecidability of the halting problem show that some questions have no general algorithmic solution.
Updated 19 Oct 2026
Mathematics
Tusi couple
The Tusi couple is a geometric theorem that produces straight-line oscillation from two uniform circular motions. Devised by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi, it became a tool for constructing models in medieval astronomy.
Updated 9 Nov 2026