{ "ok": true, "world": "math", "count": 124, "terms": [ { "slug": "abel-ruffini-theorem", "term": "Abel–Ruffini theorem", "aka": [ "impossibility of the quintic" ], "category": "algebra", "short": "There is no general formula in radicals for solving polynomial equations of degree five or higher.", "definition": "The Abel–Ruffini theorem states that there is no general solution in radicals — using only the coefficients combined with addition, subtraction, multiplication, division, and root extraction — for polynomial equations of degree 5 or greater. Paolo Ruffini gave an incomplete proof and Niels Henrik Abel completed it; Galois theory later explained the result through the non-solvability of the symmetric group S₅. Specific high-degree equations may still be solvable, but no universal formula exists.", "example": "Degrees 1 through 4 each have a radical formula (the quadratic, cubic, and quartic formulas), but the quintic x⁵ − x − 1 = 0 has roots that cannot be written using radicals of its coefficients.", "related": [ "galois-theory", "polynomial", "quadratic-formula", "group" ], "source": "Dummit & Foote, Abstract Algebra" }, { "slug": "aleph-null", "term": "Aleph-Null (ℵ₀)", "aka": [ "aleph-naught", "countable infinity" ], "category": "constants-special-numbers", "short": "ℵ₀ is the cardinality of the natural numbers, the smallest infinite size.", "definition": "Aleph-null (ℵ₀) is the cardinal number measuring the size of the set of natural numbers, the smallest of the infinite cardinalities. A set has size ℵ₀ exactly when its elements can be put in one-to-one correspondence with the counting numbers, in which case it is called countably infinite. Cantor proved that the integers and even the rationals have size ℵ₀, but the real numbers form a strictly larger infinity, so not all infinities are equal.", "example": "The even numbers can be paired one-to-one with all natural numbers (n ↔ 2n), so there are exactly as many even numbers as natural numbers, namely ℵ₀; yet by Cantor's diagonal argument the real numbers cannot be listed this way and form a larger infinity.", "related": [ "transcendental-irrational-algebraic", "pi", "e" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "algebra", "term": "Algebra", "category": "disciplines-fields", "short": "The study of operations and the abstract structures — groups, rings, fields — that capture how symbols and quantities combine.", "definition": "Algebra began as the art of solving equations with unknowns and grew into the study of abstract structures defined by operations satisfying fixed rules. Modern (abstract) algebra organizes mathematics around groups, rings, and fields, and reveals shared patterns across seemingly different settings. It supplies the language of symmetry used throughout mathematics and physics.", "example": "The Abel–Ruffini theorem shows there is no general formula in radicals for solving the quintic; Galois theory explains exactly why, by linking solvability to the symmetry group of an equation's roots.", "related": [ "group", "ring", "field", "galois-theory", "linear-algebra" ], "source": "Dummit & Foote, Abstract Algebra" }, { "slug": "applied-mathematics", "term": "Applied Mathematics", "category": "disciplines-fields", "short": "The use of mathematical methods to model, analyze, and solve problems arising in science, engineering, and the world at large.", "definition": "Applied mathematics builds and studies models of real phenomena — fluids, populations, markets, signals — using tools from analysis, differential equations, probability, and numerical computation. It prizes results that predict or optimize behavior and is judged partly by how well its models match observation. The boundary with pure mathematics is porous: rigorous theorems and practical models constantly feed one another.", "example": "The Navier–Stokes equations model how fluids flow and are used to design aircraft and forecast weather, yet whether their solutions always remain smooth is a famous unsolved problem worth a Clay Millennium Prize.", "related": [ "pure-mathematics", "numerical-analysis", "differential-equations", "navier-stokes-equations" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "axiom-of-choice", "term": "Axiom of Choice", "aka": [ "AC" ], "category": "foundations-logic", "short": "The Axiom of Choice says that for any collection of nonempty sets, it is possible to select one element from each, even infinitely many at once.", "definition": "Formally, for any family of nonempty sets there exists a choice function assigning to each set one of its members. The axiom is independent of the other ZFC axioms: Kurt Goedel showed it is consistent with them, and Paul Cohen showed its negation is also consistent. It is equivalent to Zorn's Lemma and to the Well-Ordering Theorem.", "example": "Choice is needed to well-order the real numbers, and it yields counterintuitive results such as the Banach–Tarski decomposition of a solid ball into pieces reassembled into two balls of the same size.", "related": [ "zfc-axioms", "continuum-hypothesis", "cardinality" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "bayes-theorem", "term": "Bayes' Theorem", "aka": [ "Bayes' rule", "Bayes' law" ], "category": "probability-statistics", "short": "A rule for updating the probability of a hypothesis given new evidence.", "definition": "Bayes' theorem relates conditional probabilities by P(A | B) = P(B | A) · P(A) / P(B), provided P(B) > 0. The denominator can be expanded by the law of total probability as P(B) = Σ P(B | Aᵢ) · P(Aᵢ). It is the formal engine of belief updating: the posterior P(A | B) is proportional to the prior P(A) times the likelihood P(B | A).", "example": "A test is 99% sensitive and 99% specific for a disease present in 0.5% of people. If you test positive, Bayes gives P(disease | positive) = (0.99 · 0.005) / (0.99 · 0.005 + 0.01 · 0.995) ≈ 0.332 — only about a third, because the disease is rare.", "related": [ "kolmogorov-probability-axioms", "random-variable" ], "source": "Feller / Ross, probability texts" }, { "slug": "poisson-and-binomial-distributions", "term": "Binomial and Poisson Distributions", "aka": [ "binomial distribution", "Poisson distribution" ], "category": "probability-statistics", "short": "The binomial counts successes in n independent trials; the Poisson models rare events over a continuum.", "definition": "The binomial distribution gives P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ for k successes in n independent trials each with success probability p, with mean np and variance np(1 − p). The Poisson distribution gives P(X = k) = (λᵏ · e^(−λ)) / k! and has mean and variance both equal to λ. The Poisson arises as the limit of the binomial when n → ∞ and p → 0 with np = λ held fixed.", "example": "If a call center receives on average λ = 3 calls per minute, the probability of exactly 5 calls in a given minute is (3⁵ · e^(−3)) / 5! = (243 · 0.0498) / 120 ≈ 0.101.", "related": [ "random-variable", "expectation-and-variance", "central-limit-theorem" ], "source": "Feller / Ross, probability texts" }, { "slug": "binomial-coefficient", "term": "Binomial Coefficient", "aka": [ "n choose k", "combinations" ], "category": "discrete-combinatorics", "short": "The number of ways to choose k unordered items from a set of n, written C(n, k) = n! / (k!(n−k)!).", "definition": "The binomial coefficient C(n, k), read 'n choose k', counts the k-element subsets of an n-element set. It is given by C(n, k) = n! / (k!(n−k)!) for 0 ≤ k ≤ n. These numbers are the coefficients in the binomial theorem, (x + y)ⁿ = Σ C(n, k) xⁿ⁻ᵏ yᵏ, where the sum runs from k = 0 to n.", "example": "C(5, 2) = 5! / (2! · 3!) = 120 / (2 · 6) = 10 — there are exactly 10 ways to pick 2 items from 5. Likewise, (x + y)³ = x³ + 3x²y + 3xy² + y³, whose coefficients 1, 3, 3, 1 are C(3, 0) through C(3, 3).", "related": [ "pascals-triangle", "permutations-and-combinations", "generating-function" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "boolean-algebra", "term": "Boolean Algebra", "aka": [ "Boolean logic", "algebra of logic" ], "category": "foundations-logic", "short": "Boolean algebra is the algebra of truth values, using AND, OR, and NOT on the two elements 0 and 1.", "definition": "Introduced by George Boole in the mid-19th century, Boolean algebra operates on values {0, 1} (false, true) with operations conjunction (∧), disjunction (∨), and negation (¬). It satisfies laws such as distributivity, identity, and complementation, and De Morgan's laws relate the operations: ¬(a ∧ b) = ¬a ∨ ¬b and ¬(a ∨ b) = ¬a ∧ ¬b. It is the mathematical basis of digital logic and propositional reasoning.", "example": "De Morgan's law in action: 'not (raining and cold)' is logically equivalent to '(not raining) or (not cold)', i.e. ¬(a ∧ b) = ¬a ∨ ¬b.", "related": [ "godels-incompleteness-theorems", "p-versus-np" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "calculus", "term": "Calculus", "aka": [ "infinitesimal calculus" ], "category": "disciplines-fields", "short": "The mathematics of continuous change, built from two linked ideas: the derivative (rate of change) and the integral (accumulated total).", "definition": "Calculus studies how quantities vary and accumulate, using limits to make sense of instantaneous rates and infinitesimal sums. Developed independently by Newton and Leibniz in the late 17th century, it splits into differential calculus (slopes and rates) and integral calculus (areas and totals). It is the standard language for describing motion, growth, and any system that changes smoothly.", "example": "The fundamental theorem of calculus reveals that differentiation and integration are inverse operations: ∫ₐᵇ f′(x) dx = f(b) − f(a).", "related": [ "derivative", "integral", "fundamental-theorem-of-calculus", "epsilon-delta-limit", "differential-equations" ], "source": "Spivak, Calculus" }, { "slug": "cantor-diagonal-argument", "term": "Cantor's Diagonal Argument", "aka": [ "diagonalization", "Cantor diagonal" ], "category": "foundations-logic", "short": "Cantor's diagonal argument proves that the real numbers cannot be put in a one-to-one list, so infinity comes in different sizes.", "definition": "Georg Cantor showed in 1891 that no list of real numbers can be complete. Given any proposed enumeration of reals in [0,1], one constructs a new real by changing the n-th decimal digit of the n-th listed number, guaranteeing the new number differs from every entry. Hence the reals are uncountable: there is no surjection from ℕ onto ℝ.", "example": "If the list begins 0.1357..., 0.4826..., 0.9071..., flip the diagonal digits 1, 8, 7 to (say) 2, 9, 8, giving a number 0.298... that differs from every listed real in at least one position.", "related": [ "cardinality", "continuum-hypothesis", "godels-incompleteness-theorems", "zfc-axioms" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "cardinality", "term": "Cardinality and the Hierarchy of Infinities", "aka": [ "cardinal number", "aleph numbers" ], "category": "foundations-logic", "short": "Cardinality measures the size of a set, and Cantor proved that infinite sets come in a strictly increasing tower of sizes ℵ₀ < 2^ℵ₀ < ...", "definition": "Two sets have the same cardinality when a bijection exists between them. The smallest infinite cardinal is ℵ₀, the size of ℕ; sets of this size are called countable. Cantor's theorem states that for every set S, the power set 2^S is strictly larger than S, so |S| < |2^S| always. This generates an unending hierarchy of infinities.", "example": "The rationals ℚ are countable (|ℚ| = ℵ₀) even though they seem denser than the integers, but the reals are uncountable with |ℝ| = 2^ℵ₀ > ℵ₀.", "related": [ "cantor-diagonal-argument", "continuum-hypothesis", "zfc-axioms", "axiom-of-choice" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "category-theory", "term": "Category Theory", "aka": [ "theory of categories" ], "category": "disciplines-fields", "short": "A unifying framework that studies mathematical structures through the structure-preserving maps between them rather than their internal contents.", "definition": "Category theory describes mathematics in terms of objects and arrows (morphisms) that compose, focusing on how structures relate rather than what they are made of. Introduced by Samuel Eilenberg and Saunders Mac Lane in 1945, it provides a common language — functors, natural transformations, universal properties — that connects algebra, topology, and logic. It has become influential in theoretical computer science and the semantics of programming languages.", "example": "Many different constructions — the greatest common divisor of integers, the product of sets, the intersection of subspaces — turn out to be the same idea, a 'product,' once expressed by its universal property in category-theoretic terms.", "related": [ "algebra", "topology", "set-theory", "group", "mathematical-logic" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "complex-analysis-cauchy", "term": "Cauchy Integral Theorem", "aka": [ "Cauchy's theorem", "Cauchy–Goursat theorem" ], "category": "analysis-calculus", "short": "The integral of a holomorphic function around any closed loop in a simply connected region is exactly zero.", "definition": "Cauchy's integral theorem states that if f is holomorphic (complex-differentiable) on and inside a simple closed contour in a simply connected domain, then ∮ f(z) dz = 0. This single fact gives complex analysis its rigidity: holomorphic functions are automatically infinitely differentiable and equal to their own Taylor series. From it follows the Cauchy integral formula, which recovers a function's values inside a loop from its values on the boundary.", "example": "The Cauchy integral formula gives f(a) = (1/2πi) ∮ f(z)/(z − a) dz; for instance ∮ dz/z around the unit circle equals 2πi, encoding how the function wraps around the singularity at 0.", "related": [ "integral", "power-series", "taylor-series", "real-analysis" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "central-limit-theorem", "term": "Central Limit Theorem", "aka": [ "CLT" ], "category": "probability-statistics", "short": "The normalized sum of many independent, identically distributed variables converges to a normal distribution.", "definition": "If X₁, X₂, ... are independent and identically distributed with finite mean μ and finite variance σ², then the standardized sample mean converges in distribution to a standard normal: (X̄ₙ − μ) / (σ/√n) → N(0, 1) as n → ∞. Remarkably, this holds regardless of the shape of the original distribution. It explains why the Gaussian appears so often in nature and why averaging shrinks error like 1/√n.", "example": "Roll a fair die many times. A single roll is uniform, not bell-shaped, yet the average of, say, 1000 rolls is sharply normal-looking around 3.5 with standard deviation σ/√1000 ≈ 1.708/31.6 ≈ 0.054.", "related": [ "normal-distribution", "law-of-large-numbers", "expectation-and-variance" ], "source": "Feller / Ross, probability texts" }, { "slug": "combinatorics", "term": "Combinatorics", "aka": [ "combinatorial mathematics" ], "category": "disciplines-fields", "short": "The art of counting, arranging, and finding structure in finite collections of objects.", "definition": "Combinatorics answers how many ways something can be done, whether a given arrangement exists, and what unavoidable patterns must appear. It encompasses enumeration (permutations and combinations), graph theory, and existence results such as Ramsey theory. Despite often elementary statements, its problems can be deceptively hard and connect to algebra, probability, and computer science.", "example": "The number of ways to choose k items from n is the binomial coefficient C(n,k) = n! / (k!(n−k)!), the same numbers that form Pascal's triangle.", "related": [ "permutations-and-combinations", "binomial-coefficient", "graph-theory-paths-trees-coloring", "ramsey-theory", "generating-function" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "complex-number", "term": "Complex number", "category": "algebra", "short": "A number of the form a + bi, where i is a square root of −1, extending the reals into a two-dimensional plane.", "definition": "A complex number has the form a + bi, where a and b are real numbers and i satisfies i² = −1. They are added componentwise and multiplied using i² = −1, and they form a field ℂ. Geometrically each complex number is a point in the plane, with multiplication corresponding to scaling and rotation.", "example": "Multiplying (1 + i) by itself gives 1 + 2i + i² = 2i. Euler's identity e^(iπ) + 1 = 0 ties together the complex exponential, π, and the numbers 0 and 1 in a single equation.", "related": [ "field", "fundamental-theorem-of-algebra", "quadratic-formula", "polynomial" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "conic-sections", "term": "Conic Sections", "aka": [ "conics" ], "category": "geometry-topology", "short": "Conic sections are the curves — circle, ellipse, parabola, and hyperbola — obtained by intersecting a plane with a double cone.", "definition": "Depending on the angle at which a plane cuts a right circular double cone, the intersection is a circle, ellipse, parabola, or hyperbola. Algebraically they are exactly the curves described by a general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, classified by the discriminant B² − 4AC. Each also has a focus–directrix characterization governed by its eccentricity e.", "example": "A planet orbiting the Sun traces an ellipse with the Sun at one focus (Kepler's first law); a projectile under uniform gravity follows a parabola.", "related": [ "euclidean-geometry", "pythagorean-theorem", "polynomial", "non-euclidean-geometry" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "continuity", "term": "Continuity", "aka": [ "continuous function" ], "category": "analysis-calculus", "short": "A function is continuous at a point when its limit there equals its actual value, so the graph has no jumps or holes.", "definition": "A function f is continuous at a if lim(x→a) f(x) = f(a); it is continuous on an interval if this holds at every point. Continuity guarantees that small changes in input produce small changes in output. Continuous functions on closed bounded intervals are well behaved: they attain a maximum and minimum and satisfy the Intermediate Value Theorem.", "example": "The Intermediate Value Theorem says a function continuous on [a, b] takes every value between f(a) and f(b); since x³ − x − 1 is negative at x = 1 and positive at x = 2, it must have a root in between.", "related": [ "epsilon-delta-limit", "derivative", "real-analysis" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "cryptography", "term": "Cryptography (Mathematical)", "aka": [ "mathematical cryptography" ], "category": "disciplines-fields", "short": "The mathematics of secure communication, using number theory and algebra to encode messages so only intended parties can read them.", "definition": "Mathematical cryptography studies how to keep information confidential and authentic in the presence of adversaries, building on number theory, algebra, and computational complexity. Modern public-key systems rest on problems believed to be hard to reverse, such as factoring large integers or computing discrete logarithms. Security rests on the gap between the ease of encrypting and the apparent difficulty of breaking without the key.", "example": "The RSA cryptosystem (1977) lets anyone encrypt a message using a public key, while decryption requires factoring a large number into its two secret prime factors — a task with no known fast classical algorithm.", "related": [ "number-theory", "rsa-cryptosystem", "prime-number", "modular-arithmetic", "p-versus-np" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "derivative", "term": "Derivative", "aka": [ "differentiation", "instantaneous rate of change" ], "category": "analysis-calculus", "short": "The derivative is the instantaneous rate of change of a function, defined as the limit of average rates over shrinking intervals.", "definition": "The derivative of f at x is f′(x) = lim(h→0) [f(x + h) − f(x)] / h, the slope of the tangent line to the graph. It measures how fast f changes and underlies velocity, optimization, and the linear approximation of functions. Differentiability at a point implies continuity there, though continuity alone does not imply differentiability.", "example": "For f(x) = x², the difference quotient [(x + h)² − x²]/h = 2x + h, whose limit as h → 0 is f′(x) = 2x.", "related": [ "epsilon-delta-limit", "continuity", "integral", "fundamental-theorem-of-calculus", "taylor-series" ], "source": "Stewart, Calculus" }, { "slug": "determinant", "term": "Determinant", "aka": [ "det" ], "category": "algebra", "short": "A single number computed from a square matrix that measures how it scales volume and whether it is invertible.", "definition": "The determinant det(A) of a square matrix A is a scalar that records the signed factor by which A scales area, volume, or higher-dimensional content. A matrix is invertible exactly when its determinant is nonzero; a zero determinant signals that the columns are linearly dependent and the map collapses space into a lower dimension. Determinants are multiplicative, det(AB) = det(A) det(B), and a sign indicates whether orientation is preserved or flipped.", "example": "For [[a, b], [c, d]] the determinant is ad − bc; the matrix [[2, 0], [0, 3]] has determinant 6, meaning it stretches areas by a factor of 6.", "related": [ "matrix", "eigenvalues-and-eigenvectors", "linear-algebra", "vector" ], "source": "Strang, Introduction to Linear Algebra" }, { "slug": "differential-equations", "term": "Differential Equation", "aka": [ "ODE", "PDE" ], "category": "analysis-calculus", "short": "An equation relating a function to its own derivatives, describing how quantities change over space or time.", "definition": "A differential equation involves an unknown function and one or more of its derivatives; ordinary differential equations (ODEs) involve one variable, partial differential equations (PDEs) involve several. Solving one means finding the functions that satisfy it, often a whole family parameterized by initial conditions. They model nearly every dynamical process, from population growth to heat diffusion to planetary motion.", "example": "The equation y′ = ky has the general solution y = C·e^(kx); with k > 0 it describes exponential growth, the model behind compound interest and unchecked population increase.", "related": [ "derivative", "the-e-limit", "fourier-series", "integral" ], "source": "Stewart, Calculus" }, { "slug": "dirac-equation", "term": "Dirac Equation", "aka": [ "relativistic electron equation" ], "category": "mathematical-physics", "short": "The relativistic wave equation for spin-½ particles that combines quantum mechanics with special relativity.", "definition": "Formulated by Paul Dirac in 1928, the equation can be written (iℏγ^μ ∂μ − mc)ψ = 0, where the γ^μ are 4×4 matrices and ψ is a four-component spinor. It is first order in both space and time, consistent with E² = (pc)² + (mc²)², and it naturally accounts for the electron's spin. Its negative-energy solutions led Dirac to predict antimatter.", "example": "The Dirac equation implies the existence of the positron, the electron's antiparticle, which was discovered experimentally in 1932 — a prediction that emerged purely from the equation's mathematical structure.", "related": [ "schrodinger-equation", "mass-energy-equivalence", "hamiltonian-lagrangian" ], "source": "Griffiths, Introduction to Elementary Particles" }, { "slug": "discrete-mathematics", "term": "Discrete Mathematics", "category": "disciplines-fields", "short": "The mathematics of countable, separate objects — integers, graphs, logical statements — rather than continuous quantities.", "definition": "Discrete mathematics studies structures that take distinct, separated values: it includes combinatorics, graph theory, logic, set theory, and parts of number theory. Because computers operate on discrete data, the field is the mathematical backbone of computer science. Its questions often ask how to count, arrange, or connect things, and how efficiently a procedure can do so.", "example": "The pigeonhole principle — if 13 people are in a room, at least two share a birth month — is a one-line discrete fact that nonetheless forces surprising conclusions across the field.", "related": [ "combinatorics", "graph-theory-paths-trees-coloring", "mathematical-logic", "pigeonhole-principle" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "eigenvalues-and-eigenvectors", "term": "Eigenvalues and Eigenvectors", "aka": [ "eigenvalue", "eigenvector", "characteristic value" ], "category": "algebra", "short": "The special directions a matrix only stretches, and the factors by which it stretches them.", "definition": "For a square matrix A, a nonzero vector v is an eigenvector if A v = λ v for some scalar λ, the corresponding eigenvalue; the transformation merely scales v along its own direction without rotating it. Eigenvalues are the roots of the characteristic polynomial det(A − λI) = 0, and their eigenvectors reveal the intrinsic axes of a linear map. They underlie diagonalization, the long-term behavior of dynamical systems, and the principal modes of vibrating and quantum systems.", "example": "For [[2, 0], [0, 3]] the eigenvalues are 2 and 3, with eigenvectors along the x- and y-axes: the x-axis is stretched by 2 and the y-axis by 3.", "related": [ "matrix", "determinant", "linear-algebra", "vector", "vector-space" ], "source": "Strang, Introduction to Linear Algebra" }, { "slug": "einstein-field-equations", "term": "Einstein Field Equations", "aka": [ "EFE", "equations of general relativity" ], "category": "mathematical-physics", "short": "The equations of general relativity relating spacetime curvature to the energy and momentum it contains.", "definition": "The field equations are Gμν + Λgμν = (8πG/c⁴) Tμν, where Gμν is the Einstein tensor built from spacetime curvature, gμν is the metric, Λ is the cosmological constant, G is Newton's gravitational constant, and Tμν is the stress–energy tensor. The geometry of spacetime is determined by its matter and energy content; freely falling bodies follow geodesics of the resulting curved metric.", "example": "In the weak-field, slow-motion limit the equations reproduce Newton's law of gravitation, ∇²φ = 4πGρ, recovering classical gravity as an approximation.", "related": [ "mass-energy-equivalence", "noether-theorem", "hamiltonian-lagrangian" ], "source": "Misner, Thorne & Wheeler, Gravitation" }, { "slug": "equation", "term": "Equation", "aka": [ "equations" ], "category": "algebra", "short": "A statement asserting that two expressions are equal, true for some, all, or no values of its unknowns.", "definition": "An equation asserts the equality of two expressions joined by the sign =, and to solve it is to find the values of the unknowns that make the statement true. An identity holds for every value (as in (x + 1)² = x² + 2x + 1), while a conditional equation holds only for particular solutions. The same operation applied to both sides preserves equality, the principle that drives every algebraic solution method.", "example": "The linear equation 2x + 3 = 11 has the single solution x = 4, while the quadratic equation x² = 9 has two solutions, x = 3 and x = −3.", "related": [ "polynomial", "quadratic-formula", "function", "differential-equations", "fundamental-theorem-of-algebra" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "euclidean-algorithm", "term": "Euclidean Algorithm", "aka": [ "Euclid's algorithm", "gcd algorithm" ], "category": "number-theory", "short": "An ancient repeated-remainder procedure for computing the greatest common divisor of two integers.", "definition": "The greatest common divisor gcd(a, b) is the largest positive integer dividing both a and b. To compute it with a ≥ b > 0, replace the pair (a, b) by (b, a mod b) and repeat until the remainder is 0; the last nonzero remainder is the gcd, because gcd(a, b) = gcd(b, a mod b). Appearing in Euclid's Elements, it is one of the oldest algorithms still in routine use, and its extended form yields integers x and y satisfying Bézout's identity a·x + b·y = gcd(a, b).", "example": "gcd(48, 18): 48 mod 18 = 12, then 18 mod 12 = 6, then 12 mod 6 = 0, so the gcd is 6, and Bézout's identity gives 48·(−1) + 18·(3) = 6.", "related": [ "modular-arithmetic", "fundamental-theorem-of-arithmetic", "fermats-last-theorem" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "euclidean-geometry", "term": "Euclidean Geometry", "aka": [ "flat geometry" ], "category": "geometry-topology", "short": "The geometry of a flat plane or space built on Euclid's five postulates, including the parallel postulate.", "definition": "Euclidean geometry is the system codified in Euclid's Elements (c. 300 BCE), in which the parallel postulate guarantees that through a point not on a line there is exactly one parallel line. In this geometry the angles of every triangle sum to exactly 180° and the Pythagorean theorem holds. Relaxing the parallel postulate yields the consistent non-Euclidean geometries discovered in the 19th century.", "example": "In the Euclidean plane the interior angles of any triangle add to exactly 180°; on a sphere they add to more, signaling a different geometry.", "related": [ "non-euclidean-geometry", "pythagorean-theorem", "gaussian-curvature" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "euler-characteristic", "term": "Euler Characteristic", "aka": [ "Euler-Poincaré characteristic" ], "category": "geometry-topology", "short": "A topological number χ = V − E + F that stays the same no matter how a surface is divided up.", "definition": "For any polyhedron or triangulated surface, the Euler characteristic is χ = V − E + F (vertices minus edges plus faces), and it depends only on the surface's topology, not the particular subdivision. For any shape topologically equivalent to a sphere, χ = 2; for a torus χ = 0. The Euler characteristic is one of the simplest and most powerful topological invariants, distinguishing surfaces by counting handles.", "example": "A cube has V = 8, E = 12, F = 6, giving χ = 8 − 12 + 6 = 2, the same value as a sphere because a cube can be inflated into one.", "related": [ "topology", "moebius-strip", "gauss-bonnet-theorem", "manifold", "poincare-conjecture" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "euler-mascheroni-gamma", "term": "Euler-Mascheroni Constant (γ)", "aka": [ "gamma", "Euler's constant" ], "category": "constants-special-numbers", "short": "γ ≈ 0.57722 measures the limiting gap between the harmonic series and the natural logarithm.", "definition": "The Euler-Mascheroni constant γ ≈ 0.5772156649 is the limit of (1 + 1/2 + 1/3 + ⋯ + 1/n) − ln(n) as n grows without bound. It quantifies how far the slowly diverging harmonic series drifts above the natural logarithm. Remarkably, it remains unknown whether γ is even irrational, despite centuries of effort, making it one of the most famous open questions about a specific constant.", "example": "For n = 1,000,000 the harmonic sum exceeds ln(1,000,000) by roughly 0.5772, already matching γ to several digits; whether γ is rational or irrational has never been proved.", "related": [ "e", "pi", "transcendental-irrational-algebraic" ], "source": "NIST Digital Library of Mathematical Functions (DLMF, dlmf.nist.gov)" }, { "slug": "e", "term": "Euler's Number (e)", "aka": [ "Napier's constant", "base of the natural logarithm" ], "category": "constants-special-numbers", "short": "e ≈ 2.71828 is the base of the natural logarithm and the unique base for which the exponential function equals its own derivative.", "definition": "Euler's number e ≈ 2.71828182845 is the base of the natural logarithm. It is the limit of (1 + 1/n)ⁿ as n grows without bound, and equivalently the infinite sum 1 + 1/1! + 1/2! + 1/3! + ⋯. The function eˣ is the unique exponential whose derivative equals itself, which is why e governs continuous growth, decay, and compound interest. Like π, e is both irrational and transcendental.", "example": "Compounding $1 at 100% annual interest, split into n equal periods, yields (1 + 1/n)ⁿ dollars; as the compounding becomes continuous this approaches e ≈ $2.718.", "related": [ "pi", "imaginary-unit-i", "transcendental-irrational-algebraic", "euler-mascheroni-gamma" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "expectation-and-variance", "term": "Expectation and Variance", "aka": [ "mean and variance", "expected value" ], "category": "probability-statistics", "short": "The expectation is the probability-weighted average of a random variable; the variance measures its spread.", "definition": "The expectation (mean) is E[X] = Σ x·P(X = x) in the discrete case or ∫ x·f(x) dx in the continuous case. The variance is Var(X) = E[(X − E[X])²] = E[X²] − (E[X])², and its square root is the standard deviation. Expectation is linear — E[aX + bY] = a·E[X] + b·E[Y] always — whereas variance adds for independent variables: Var(X + Y) = Var(X) + Var(Y).", "example": "For a single fair die, E[X] = (1+2+3+4+5+6)/6 = 3.5 and Var(X) = E[X²] − 3.5² = 91/6 − 12.25 ≈ 2.917, so the standard deviation is about 1.708.", "related": [ "random-variable", "law-of-large-numbers", "central-limit-theorem" ], "source": "Feller / Ross, probability texts" }, { "slug": "exponentiation", "term": "Exponentiation", "aka": [ "powers", "raising to a power" ], "category": "algebra", "short": "Repeated multiplication of a base by itself, extended to any real or complex exponent.", "definition": "Exponentiation bⁿ means multiplying the base b by itself n times, and it obeys the laws bᵐ · bⁿ = bᵐ⁺ⁿ and (bᵐ)ⁿ = bᵐⁿ. These laws force the conventions b⁰ = 1, negative exponents as reciprocals, and fractional exponents as roots, so that b^(1/2) = √b. The definition extends continuously to all real and complex exponents through the exponential function, and its inverse is the logarithm.", "example": "2³ = 8 and 2⁻¹ = ½; the law of exponents gives 2³ · 2² = 2⁵ = 32.", "related": [ "logarithm", "polynomial", "function", "quadratic-formula" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "factorial", "term": "Factorial", "aka": [ "n!" ], "category": "discrete-combinatorics", "short": "The factorial n! is the product of all positive integers from 1 up to n, counting the number of ways to arrange n distinct objects in order.", "definition": "For a nonnegative integer n, the factorial n! is defined by n! = n × (n−1) × ⋯ × 2 × 1, with the convention 0! = 1. It satisfies the recurrence n! = n × (n−1)!, and it counts the permutations of n distinct objects. Factorials are the building blocks of binomial coefficients, since C(n, k) = n! / (k! (n−k)!), and they appear throughout the denominators of Taylor series.", "example": "5! = 5 × 4 × 3 × 2 × 1 = 120, so there are exactly 120 ways to order five distinct books on a shelf. Factorials grow astonishingly fast: 13! already exceeds six billion.", "related": [ "binomial-coefficient", "permutations-and-combinations", "recurrence-relation", "taylor-series" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "fermats-last-theorem", "term": "Fermat's Last Theorem", "category": "number-theory", "short": "No three positive integers satisfy aⁿ + bⁿ = cⁿ for any integer exponent n greater than 2.", "definition": "The theorem asserts that the equation aⁿ + bⁿ = cⁿ has no solution in positive integers a, b, c when n > 2. Fermat claimed a proof in a margin note around 1637, but it resisted proof for over 350 years. It was finally proved by Andrew Wiles, with a portion completed jointly with Richard Taylor, in 1994 via the modularity of elliptic curves.", "example": "For n = 2 there are infinitely many solutions, such as 3² + 4² = 5², but for n = 3 and beyond no positive-integer solution exists.", "related": [ "fermats-little-theorem", "euclidean-algorithm", "prime-number" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "fermats-little-theorem", "term": "Fermat's Little Theorem", "aka": [ "Fermat's theorem on congruences" ], "category": "number-theory", "short": "If p is prime and a is not divisible by p, then a^(p−1) ≡ 1 (mod p).", "definition": "Fermat's little theorem states that for a prime p and any integer a not divisible by p, a^(p−1) ≡ 1 (mod p); equivalently a^p ≡ a (mod p) for every integer a. It is a special case of Euler's theorem and provides the basis for fast primality tests and for the correctness of RSA decryption. The result is named for Pierre de Fermat, who stated it in 1640.", "example": "With p = 7 and a = 3: 3⁶ = 729 = 104·7 + 1, so 3⁶ ≡ 1 (mod 7).", "related": [ "modular-arithmetic", "rsa-cryptosystem", "prime-number", "fermats-last-theorem" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "fibonacci-sequence", "term": "Fibonacci Sequence", "aka": [ "Fibonacci numbers" ], "category": "discrete-combinatorics", "short": "The sequence 0, 1, 1, 2, 3, 5, 8, 13, … defined by Fₙ = Fₙ₋₁ + Fₙ₋₂, with ratios of successive terms approaching the golden ratio φ.", "definition": "The Fibonacci sequence is defined by the recurrence Fₙ = Fₙ₋₁ + Fₙ₋₂ with seed values F₀ = 0 and F₁ = 1. The ratio Fₙ₊₁ / Fₙ approaches the golden ratio φ = (1 + √5)/2 ≈ 1.618 as n grows. A closed form, Binet's formula, gives Fₙ = (φⁿ − ψⁿ)/√5 where ψ = (1 − √5)/2.", "example": "The terms grow as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. The number of ways to tile a 1×n strip with 1×1 squares and 1×2 dominoes is exactly Fₙ₊₁, a clean combinatorial appearance of the sequence.", "related": [ "recurrence-relation", "generating-function", "pascals-triangle" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "field", "term": "Field", "category": "algebra", "short": "A commutative ring in which every nonzero element has a multiplicative inverse, so you can add, subtract, multiply, and divide.", "definition": "A field is a set F with addition and multiplication such that (F, +) and (F∖{0}, ·) are both abelian groups and multiplication distributes over addition. Equivalently, it is a commutative ring with identity in which every nonzero element is invertible. Fields are the natural setting for solving linear equations and for defining vector spaces.", "example": "The rational numbers ℚ, the real numbers ℝ, and the complex numbers ℂ are all fields. So is the finite field of two elements {0, 1} with arithmetic done modulo 2; in it 1 + 1 = 0.", "related": [ "ring", "complex-number", "vector-space", "galois-theory" ], "source": "Dummit & Foote, Abstract Algebra" }, { "slug": "fourier-series", "term": "Fourier Series and Transform", "aka": [ "Fourier analysis", "harmonic analysis" ], "category": "analysis-calculus", "short": "Any well-behaved periodic function can be written as a sum of sines and cosines, and the Fourier transform extends this to nonperiodic signals.", "definition": "A Fourier series expresses a periodic function as Σ (aₙ cos(nx) + bₙ sin(nx)), with coefficients found by integrating the function against each harmonic. The Fourier transform generalizes this to nonperiodic functions, decomposing them into a continuum of frequencies. This frequency-domain view turns differentiation into multiplication and underlies signal processing, heat flow, and quantum mechanics.", "example": "A square wave of period 2π equals (4/π) Σ sin((2k−1)x)/(2k−1) over k ≥ 1, an infinite sum of odd harmonics that reconstructs the sharp edges from smooth sine waves.", "related": [ "integral", "power-series", "differential-equations" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "function", "term": "Function", "aka": [ "map", "mapping" ], "category": "analysis-calculus", "short": "A rule that assigns to every input from one set exactly one output in another set.", "definition": "A function f from a set A (the domain) to a set B (the codomain) assigns to each element x ∈ A a single, well-defined value f(x) ∈ B. The defining requirement is single-valuedness: one input never produces two different outputs, though different inputs may share an output. The set of values actually attained, {f(x) : x ∈ A}, is the range or image, and functions can be combined by composition, written (g ∘ f)(x) = g(f(x)).", "example": "The squaring rule f(x) = x² sends each real number to its square, so f(3) = 9 and f(−3) = 9; it is a valid function because each input yields exactly one output, even though two inputs can share one.", "related": [ "continuity", "derivative", "limit", "exponentiation", "logarithm" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "fundamental-theorem-of-algebra", "term": "Fundamental theorem of algebra", "category": "algebra", "short": "Every non-constant polynomial with complex coefficients has at least one complex root.", "definition": "The fundamental theorem of algebra states that every polynomial of degree n ≥ 1 with complex coefficients has at least one root in ℂ. By repeated factoring, it follows that such a polynomial has exactly n roots in ℂ counted with multiplicity, so ℂ is algebraically closed. Although called a theorem of algebra, all known proofs use analysis or topology rather than algebra alone.", "example": "The polynomial x² + 1 has no real roots, but over the complex numbers it factors as (x − i)(x + i), giving its two guaranteed roots. Every real polynomial likewise splits completely once complex roots are allowed.", "related": [ "complex-number", "polynomial", "quadratic-formula", "field" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "fundamental-theorem-of-arithmetic", "term": "Fundamental Theorem of Arithmetic", "aka": [ "unique factorization theorem" ], "category": "number-theory", "short": "Every integer greater than 1 factors uniquely into primes, apart from the order of the factors.", "definition": "The theorem states that each integer n > 1 can be written as a product of primes, and this representation is unique up to the order in which the factors are written. Existence follows by induction, and uniqueness follows from Euclid's lemma that if a prime divides a product it divides one of the factors. This is why the primes are called the multiplicative building blocks of the integers.", "example": "The number 360 factors as 2³ · 3² · 5, and no other multiset of primes multiplies to 360.", "related": [ "prime-number", "euclidean-algorithm", "prime-number-theorem" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "fundamental-theorem-of-calculus", "term": "Fundamental Theorem of Calculus", "aka": [ "FTC" ], "category": "analysis-calculus", "short": "Differentiation and integration are inverse operations, linking the area under a curve to an antiderivative of the function.", "definition": "The first part states that if F(x) = ∫ₐˣ f(t) dt for continuous f, then F′(x) = f(x). The second part states that ∫ₐᵇ f(x) dx = F(b) − F(a) whenever F is any antiderivative of f. Together they reduce the hard problem of computing areas (limits of sums) to the comparatively easy problem of finding antiderivatives.", "example": "∫₀^π sin(x) dx = [−cos(x)]₀^π = (−cos π) − (−cos 0) = 1 − (−1) = 2, evaluated instantly via an antiderivative rather than summing infinitely many rectangles.", "related": [ "derivative", "integral", "continuity" ], "source": "Spivak, Calculus" }, { "slug": "galois-theory", "term": "Galois theory", "category": "algebra", "short": "The theory connecting the solvability of polynomial equations to the symmetry group of their roots.", "definition": "Galois theory, developed by Évariste Galois, establishes a correspondence between field extensions and groups of symmetries (the Galois group) that permute the roots of a polynomial while preserving algebraic relations. A polynomial is solvable by radicals exactly when its Galois group is a solvable group. This insight explains why some equations admit formulas in radicals and others do not.", "example": "The general quintic x⁵ + ax⁴ + ⋯ has Galois group S₅, which is not solvable, so no general radical formula exists for degree 5. By contrast, quadratics have a small abelian Galois group, matching the existence of the quadratic formula.", "related": [ "group", "field", "abel-ruffini-theorem", "polynomial" ], "source": "Dummit & Foote, Abstract Algebra" }, { "slug": "game-theory", "term": "Game Theory", "category": "disciplines-fields", "short": "The mathematical study of strategic decision-making, where each player's best move depends on what the others choose.", "definition": "Game theory models situations in which rational agents interact, analyzing strategies, payoffs, and the outcomes that result. Founded by John von Neumann and Oskar Morgenstern in 1944, it formalizes cooperation, competition, and bargaining. It is widely applied in economics, evolutionary biology, political science, and the design of auctions and markets.", "example": "A Nash equilibrium, named for John Nash, is a set of strategies in which no player can do better by changing alone; the Prisoner's Dilemma shows such an equilibrium can leave everyone worse off than cooperation would.", "related": [ "applied-mathematics", "probability-theory", "expectation-and-variance", "combinatorics" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "gauss-bonnet-theorem", "term": "Gauss-Bonnet Theorem", "aka": [ "Gauss–Bonnet formula" ], "category": "geometry-topology", "short": "On a closed surface, total curvature equals 2π times the Euler characteristic, tying geometry to topology.", "definition": "The Gauss-Bonnet theorem states that for a closed (compact, boundaryless) surface, the integral of the Gaussian curvature K over the whole surface equals 2π·χ, where χ is the Euler characteristic. The left side is geometric and can change locally as the surface bends; the right side is topological and fixed. The theorem is a landmark bridge, forcing the total bending of a surface to be determined entirely by how many handles it has.", "example": "For any sphere, ∫K dA = 2π·χ = 2π·2 = 4π, regardless of how the sphere is dented; for a torus the total curvature is 2π·0 = 0, so its positive and negative curvature exactly cancel.", "related": [ "gaussian-curvature", "euler-characteristic", "manifold", "topology" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "gaussian-curvature", "term": "Gaussian Curvature", "aka": [ "intrinsic curvature" ], "category": "geometry-topology", "short": "A number measuring how a surface bends at a point, which an inhabitant could detect without ever leaving the surface.", "definition": "The Gaussian curvature K at a point is the product of the two principal curvatures of the surface there; it is positive at dome-like points, negative at saddle points, and zero on flat or cylindrical ones. Gauss's Theorema Egregium ('remarkable theorem') states that K is intrinsic — preserved by any bending that does not stretch the surface — so it can be measured purely from distances within the surface. This is why a flat map of the Earth must distort, since a sphere (K > 0) and a plane (K = 0) cannot be matched without stretching.", "example": "A cylinder has Gaussian curvature zero because it can be unrolled flat without stretching, whereas a sphere of radius r has constant positive curvature K = 1/r² and cannot be flattened.", "related": [ "gauss-bonnet-theorem", "manifold", "non-euclidean-geometry", "euclidean-geometry" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "generating-function", "term": "Generating Function", "aka": [ "ordinary generating function" ], "category": "discrete-combinatorics", "short": "A formal power series Σ aₙ xⁿ that encodes a sequence (aₙ) as the coefficients of x, turning combinatorial problems into algebra.", "definition": "The ordinary generating function of a sequence a₀, a₁, a₂, … is the formal power series A(x) = Σ aₙ xⁿ summed over n ≥ 0. Operations on sequences (shifting, convolution, summation) become algebraic operations on the series, so recurrences turn into equations one can solve for A(x) in closed form. The coefficients are then recovered by series expansion.", "example": "The generating function for the Fibonacci numbers is Σ Fₙ xⁿ = x / (1 − x − x²). The simplest example, 1/(1 − x) = 1 + x + x² + x³ + ⋯, is the generating function for the all-ones sequence.", "related": [ "recurrence-relation", "fibonacci-sequence", "binomial-coefficient" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "geometry", "term": "Geometry", "category": "disciplines-fields", "short": "The study of shape, size, distance, and space, from the points and lines of antiquity to curved and higher-dimensional spaces.", "definition": "Geometry investigates the properties of figures and the spaces they inhabit — lengths, angles, areas, curvature, and how these change under transformation. Euclid systematized plane and solid geometry from axioms around 300 BCE, and the 19th-century discovery of non-Euclidean geometries showed his parallel postulate was a choice, not a necessity. Modern geometry extends to manifolds and curved spaces that describe the physical universe.", "example": "The Pythagorean theorem, a² + b² = c² for a right triangle, is one of the oldest known mathematical results and still anchors how distance is measured.", "related": [ "euclidean-geometry", "non-euclidean-geometry", "pythagorean-theorem", "manifold", "topology" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "godels-incompleteness-theorems", "term": "Goedel's Incompleteness Theorems", "aka": [ "incompleteness", "Goedel incompleteness", "Gödel's theorems" ], "category": "foundations-logic", "short": "Goedel proved that any consistent formal system rich enough to express arithmetic contains true statements it cannot prove, and cannot prove its own consistency.", "definition": "In 1931 Kurt Goedel established two results. The First Incompleteness Theorem: any consistent, effectively axiomatized system capable of expressing basic arithmetic is incomplete — there are statements neither provable nor refutable within it. The Second: such a system cannot prove its own consistency. The proof encodes statements as numbers (Goedel numbering) to build a sentence that asserts its own unprovability.", "example": "The self-referential sentence essentially says 'This statement is not provable in the system.' If the system could prove it, the system would be inconsistent; so if consistent, the statement is true yet unprovable.", "related": [ "russells-paradox", "cantor-diagonal-argument", "p-versus-np" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "goldbach-conjecture", "term": "Goldbach's Conjecture and Twin Primes", "aka": [ "Goldbach conjecture", "twin prime conjecture" ], "category": "number-theory", "short": "Two famous unproved claims: every even number above 2 is a sum of two primes, and there are infinitely many primes that differ by 2.", "definition": "Goldbach's conjecture, from a 1742 letter by Christian Goldbach to Leonhard Euler, asserts that every even integer greater than 2 is the sum of two primes; it is verified by computer to enormous bounds but unproved in general. The twin prime conjecture asserts there are infinitely many pairs (p, p+2) that are both prime, such as (11, 13). A landmark step came in 2013 when Yitang Zhang proved infinitely many prime pairs differ by at most a fixed bound, later reduced by collaborative work to a gap of 246.", "example": "Goldbach: 100 = 3 + 97 = 47 + 53. Twin primes: the pairs (3, 5), (5, 7), (11, 13), and (17, 19) all differ by exactly 2.", "related": [ "prime-number", "prime-number-theorem", "riemann-hypothesis" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "golden-ratio-phi", "term": "Golden Ratio (φ)", "aka": [ "phi", "divine proportion", "golden section" ], "category": "constants-special-numbers", "short": "φ = (1 + √5)/2 ≈ 1.61803 is the ratio in which the whole relates to the larger part as the larger part relates to the smaller.", "definition": "The golden ratio φ = (1 + √5)/2 ≈ 1.61803398875 is the positive solution of x² = x + 1, equivalently the value for which a + b is to a as a is to b. It is irrational and equals its own reciprocal plus one, so 1/φ = φ − 1. Ratios of consecutive Fibonacci numbers converge to φ as the numbers grow.", "example": "Fibonacci ratios approach φ: 5/3 ≈ 1.667, 8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615, closing in on 1.61803.", "related": [ "sqrt2", "pi", "transcendental-irrational-algebraic" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "graph-theory-paths-trees-coloring", "term": "Graph Theory: Paths, Trees, and Coloring", "aka": [ "Eulerian trail", "Hamiltonian path", "spanning tree", "four color theorem", "Königsberg bridges" ], "category": "discrete-combinatorics", "short": "The study of vertices joined by edges, covering Euler trails (zero or two odd-degree vertices), Hamiltonian paths, trees (connected, acyclic, with exactly n−1 edges on n vertices), and the four-color theorem for planar maps.", "definition": "A graph is a set of vertices joined by edges. An Eulerian trail uses every edge once; Euler proved a connected graph has one iff it has zero or two odd-degree vertices, resolving the Seven Bridges of Königsberg in 1736. A Hamiltonian path visits every vertex once, and a tree is a connected graph with no cycles — equivalently, a connected graph on n vertices with exactly n−1 edges. The four color theorem, proved by Appel and Haken in 1976, states that every planar graph can be properly colored with four colors.", "example": "Königsberg's four landmasses all had odd degree, so no walk crosses each bridge exactly once. Cayley's formula gives the count of labeled trees on n vertices as nⁿ⁻², so on 4 vertices there are 4² = 16 distinct labeled trees.", "related": [ "ramsey-theory", "pigeonhole-principle", "recurrence-relation" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "group", "term": "Group", "aka": [ "abstract group" ], "category": "algebra", "short": "A set with one associative operation that has an identity element and inverses for every element.", "definition": "A group is a set G together with a binary operation · satisfying four axioms: closure, associativity (a·b)·c = a·(b·c), an identity element e with e·a = a·e = a, and an inverse a⁻¹ for each a with a·a⁻¹ = e. When the operation also commutes (a·b = b·a) the group is called abelian. Groups formalize the notion of symmetry and are the foundational object of abstract algebra.", "example": "The integers ℤ under addition form an abelian group: 0 is the identity and the inverse of n is −n. The set of rotations of a square (0°, 90°, 180°, 270°) under composition forms a 4-element cyclic group.", "related": [ "ring", "field", "galois-theory", "abel-ruffini-theorem" ], "source": "Dummit & Foote, Abstract Algebra" }, { "slug": "hypothesis-testing", "term": "Hypothesis Testing", "aka": [ "statistical hypothesis test", "significance testing" ], "category": "probability-statistics", "short": "Hypothesis testing is a procedure for deciding, from sample data, whether observed evidence is strong enough to reject a default claim about a population.", "definition": "One formulates a null hypothesis H₀ (a default claim, often of no effect) against an alternative hypothesis H₁, then computes a test statistic from the data. The p-value is the probability, assuming H₀ is true, of observing data at least as extreme as what was seen; if it falls below a chosen significance level α, H₀ is rejected. The framework trades off two errors: rejecting a true H₀ (Type I) and failing to reject a false H₀ (Type II).", "example": "To test whether a coin is fair, H₀ is p = 0.5; observing 90 heads in 100 tosses yields a tiny p-value, leading to rejection of fairness at the conventional α = 0.05 level.", "related": [ "regression", "normal-distribution", "expectation-and-variance", "central-limit-theorem" ], "source": "Ross, A First Course in Probability" }, { "slug": "imaginary-unit-i", "term": "Imaginary Unit (i)", "aka": [ "√(−1)", "the imaginary number" ], "category": "constants-special-numbers", "short": "i is defined by i² = −1, the building block of the complex numbers.", "definition": "The imaginary unit i is defined by the property i² = −1, supplying a square root for negative numbers that the real line lacks. Adjoining i to the reals produces the complex numbers a + bi, which form an algebraically closed field: every nonconstant polynomial with complex coefficients has a complex root. The powers of i cycle with period four: i, −1, −i, 1, then repeat.", "example": "Euler's identity ties together five fundamental constants: e^(iπ) + 1 = 0, combining e, i, π, 1, and 0 in a single equation.", "related": [ "e", "pi", "transcendental-irrational-algebraic" ], "source": "Dummit & Foote, Abstract Algebra" }, { "slug": "infinity", "term": "Infinity", "aka": [ "∞" ], "category": "foundations-logic", "short": "Infinity refers to that which is unbounded or endless; in set theory it is made precise through infinite sets and their cardinalities.", "definition": "Informally infinity denotes a quantity without bound, and the symbol ∞ is used in limits and improper integrals to indicate unbounded growth rather than an actual number. Cantor made the infinite rigorous by defining a set as infinite when it can be put in one-to-one correspondence with a proper subset of itself, and by showing infinities come in different sizes measured by cardinal numbers. The smallest infinite cardinal is ℵ₀, the size of the natural numbers.", "example": "Cantor's diagonal argument proves the real numbers cannot be listed in a sequence, so they are strictly more numerous than the integers — there are infinities larger than ℵ₀.", "related": [ "cardinality", "aleph-null", "cantor-diagonal-argument", "continuum-hypothesis", "set" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "integral", "term": "Integral", "aka": [ "definite integral", "Riemann integral" ], "category": "analysis-calculus", "short": "The definite integral is the limit of sums of thin rectangles, giving the signed area under a curve.", "definition": "The Riemann integral ∫ₐᵇ f(x) dx is defined as the limit of Riemann sums Σ f(xᵢ*) Δxᵢ as the partition of [a, b] is refined. When the limit exists independent of the sample points, f is integrable on [a, b]. Geometrically it measures signed area; physically it accumulates a quantity from its rate of change.", "example": "∫₀¹ x² dx = 1/3, found as the limit of Riemann sums or via the antiderivative x³/3 evaluated from 0 to 1.", "related": [ "derivative", "fundamental-theorem-of-calculus", "epsilon-delta-limit", "fourier-series" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "kolmogorov-probability-axioms", "term": "Kolmogorov Probability Axioms", "aka": [ "axioms of probability", "probability axioms" ], "category": "probability-statistics", "short": "The three axioms that any probability measure P on a sample space must satisfy.", "definition": "For a sample space Ω with events drawn from a σ-algebra, a probability measure P assigns to each event a number subject to three rules: (1) non-negativity, P(A) ≥ 0; (2) normalization, P(Ω) = 1; and (3) countable additivity, P(A₁ ∪ A₂ ∪ ...) = Σ P(Aᵢ) for pairwise disjoint events. From these alone the entire calculus of probability follows. Andrey Kolmogorov gave this axiomatic foundation in 1933.", "example": "From the axioms one immediately derives P(Aᶜ) = 1 − P(A): since A and Aᶜ are disjoint and their union is Ω, P(A) + P(Aᶜ) = P(Ω) = 1.", "related": [ "bayes-theorem", "random-variable", "expectation-and-variance" ], "source": "Feller / Ross, probability texts" }, { "slug": "hamiltonian-lagrangian", "term": "Lagrangian and Hamiltonian Mechanics", "aka": [ "principle of least action", "analytical mechanics" ], "category": "mathematical-physics", "short": "Two reformulations of classical mechanics built on energy functions rather than forces.", "definition": "Lagrangian mechanics uses L = T − V, the kinetic minus the potential energy, and derives motion from the Euler–Lagrange equations d/dt(∂L/∂q̇) − ∂L/∂q = 0, which extremize the action ∫L dt. Hamiltonian mechanics instead uses H = T + V expressed in coordinates q and momenta p, with the equations q̇ = ∂H/∂p and ṗ = −∂H/∂q. The two formulations are equivalent and underpin both quantum mechanics and statistical mechanics.", "example": "For a free particle the action is minimized by straight-line motion at constant speed, recovering Newton's first law from the principle of least action.", "related": [ "noether-theorem", "schrodinger-equation", "einstein-field-equations" ], "source": "Goldstein, Classical Mechanics" }, { "slug": "law-of-large-numbers", "term": "Law of Large Numbers", "aka": [ "LLN" ], "category": "probability-statistics", "short": "As a sample grows, its average converges to the true expected value.", "definition": "For independent, identically distributed variables with finite mean μ, the sample mean X̄ₙ = (X₁ + ... + Xₙ)/n converges to μ as n → ∞. The weak law gives convergence in probability, while the strong law gives almost-sure convergence. It justifies the frequentist interpretation that long-run relative frequency approximates probability.", "example": "Flip a fair coin: the fraction of heads need not be near 1/2 after 10 flips, but after 100,000 flips it sits very close to 0.5. Crucially, the law concerns the average, not the count — the absolute gap between heads and tails can grow even as the proportion converges.", "related": [ "central-limit-theorem", "expectation-and-variance" ], "source": "Feller / Ross, probability texts" }, { "slug": "limit", "term": "Limit", "aka": [ "limiting value" ], "category": "analysis-calculus", "short": "The value a function or sequence approaches as its input gets arbitrarily close to a target.", "definition": "The limit of f(x) as x approaches a, written lim(x→a) f(x) = L, captures the value the outputs cluster around as the inputs draw near a, regardless of whether f(a) itself is defined. The notion is made precise by the epsilon–delta definition: for every ε > 0 there is a δ > 0 so that 0 < |x − a| < δ forces |f(x) − L| < ε. Limits are the foundation on which continuity, the derivative, and the integral are all built.", "example": "Although (sin x)/x is undefined at x = 0, its limit there is 1: as x shrinks toward 0, the ratio approaches 1 ever more closely, written lim(x→0) (sin x)/x = 1.", "related": [ "continuity", "derivative", "function", "logarithm" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "epsilon-delta-limit", "term": "Limit (Epsilon-Delta Definition)", "aka": [ "ε-δ definition", "rigorous limit", "limiting value" ], "category": "analysis-calculus", "short": "A limit is the value a function approaches as its input nears a point, made precise by the rule that every tolerance ε > 0 can be met by some closeness δ > 0.", "definition": "We say lim(x→a) f(x) = L if for every ε > 0 there exists δ > 0 such that whenever 0 < |x − a| < δ, we have |f(x) − L| < ε. This replaces the vague idea of a function 'approaching' a value with a precise challenge-and-response game, and it makes sense of rates and sums that would otherwise divide by zero. It is the logical foundation on which continuity, derivatives, and integrals are built.", "example": "For f(x) = 2x at a = 3, choosing δ = ε/2 guarantees |2x − 6| = 2|x − 3| < ε whenever |x − 3| < δ; separately, lim(x→0) sin(x)/x = 1 despite the form 0/0.", "related": [ "continuity", "real-analysis", "derivative" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "linear-algebra", "term": "Linear algebra", "category": "algebra", "short": "The study of vectors, vector spaces, and linear maps, using matrices, determinants, and eigenvalues as its core tools.", "definition": "Linear algebra studies linear maps between vector spaces and the systems of linear equations they represent. A linear map is encoded by a matrix, a rectangular array of numbers, and matrix multiplication corresponds to composing maps (it is associative but generally not commutative). The determinant of a square matrix is a single number whose magnitude is the volume-scaling factor and whose vanishing signals non-invertibility, while an eigenvector is a nonzero vector v with Av = λv, scaled by its eigenvalue λ without changing direction.", "example": "The 2×2 matrix [[2, 0], [0, 3]] has determinant ad − bc = 6, scaling areas by 6; the vector (1, 0) is an eigenvector with eigenvalue 2 and (0, 1) an eigenvector with eigenvalue 3. The eigenvalues are precisely the roots of the characteristic polynomial det(A − λI) = 0.", "related": [ "vector-space", "field", "polynomial", "complex-number" ], "source": "Strang, Introduction to Linear Algebra" }, { "slug": "logarithm", "term": "Logarithm", "aka": [ "log", "natural log", "ln" ], "category": "analysis-calculus", "short": "The inverse of exponentiation: the exponent to which a fixed base must be raised to produce a given number.", "definition": "The logarithm to base b of a positive number x, written log_b(x), is the exponent y for which bʸ = x; thus logarithm and exponentiation are inverse operations. Its defining property turns products into sums, log_b(xy) = log_b(x) + log_b(y), which once made multiplication of large numbers tractable. The natural logarithm ln(x) uses base e and equals the area ∫ from 1 to x of dt/t.", "example": "Because 10³ = 1000, we have log₁₀(1000) = 3; likewise ln(e) = 1 since e¹ = e.", "related": [ "exponentiation", "function", "integral", "power-series" ], "source": "NIST Digital Library of Mathematical Functions (DLMF, dlmf.nist.gov)" }, { "slug": "manifold", "term": "Manifold", "aka": [ "smooth manifold" ], "category": "geometry-topology", "short": "A space that looks like ordinary flat Euclidean space when you zoom in close enough to any point.", "definition": "A manifold of dimension n is a topological space in which every point has a neighborhood that looks like (is homeomorphic to) an open region of n-dimensional Euclidean space. The Earth's surface is a 2-dimensional manifold: it is curved globally yet locally flat, which is why local maps work. Adding a smooth structure lets calculus be done on manifolds, providing the setting for differential geometry and general relativity.", "example": "A sphere is a 2-manifold: any small patch can be flattened onto a map without tearing, even though the whole sphere cannot be flattened without distortion.", "related": [ "topology", "gaussian-curvature", "non-euclidean-geometry", "euler-characteristic", "poincare-conjecture" ], "source": "Lee, Introduction to Smooth Manifolds" }, { "slug": "markov-chain", "term": "Markov Chain", "aka": [ "discrete-time Markov chain" ], "category": "probability-statistics", "short": "A sequence of random states in which the next state depends only on the current state, not the full history.", "definition": "A Markov chain is a stochastic process satisfying the Markov property: P(Xₙ₊₁ = j | Xₙ = i, Xₙ₋₁, ...) = P(Xₙ₊₁ = j | Xₙ = i). Its dynamics are captured by a transition matrix P whose entry Pᵢⱼ is the probability of moving from state i to state j; each row sums to 1. Many chains converge to a stationary distribution π satisfying π = πP, independent of the starting state.", "example": "Model weather with two states {Sunny, Rainy} and transition matrix rows [0.9, 0.1] and [0.5, 0.5]. Solving π = πP gives the long-run stationary distribution π ≈ (0.833, 0.167) — sunny about 5/6 of the time.", "related": [ "bayes-theorem", "random-variable", "poisson-and-binomial-distributions" ], "source": "Feller / Ross, probability texts" }, { "slug": "mass-energy-equivalence", "term": "Mass-Energy Equivalence", "aka": [ "E equals mc squared", "rest energy relation" ], "category": "mathematical-physics", "short": "The principle that the rest energy of a body equals its mass times the square of the speed of light, E = mc².", "definition": "For a body at rest, energy and mass are proportional: E = mc², where c is the speed of light in vacuum. The full relativistic relation including momentum p is E² = (mc²)² + (pc)², which reduces to E = mc² when p = 0 and to E = pc for a massless particle such as a photon. Because c² is enormous, a tiny mass corresponds to an immense energy.", "example": "One kilogram of mass corresponds to E = (1 kg)(c²) ≈ 9 × 10¹⁶ joules, since c ≈ 3 × 10⁸ m/s and c² ≈ 9 × 10¹⁶ m²/s².", "related": [ "einstein-field-equations", "dirac-equation", "schrodinger-equation" ], "source": "Griffiths, Introduction to Elementary Particles" }, { "slug": "mathematical-analysis", "term": "Mathematical Analysis", "aka": [ "analysis" ], "category": "disciplines-fields", "short": "The rigorous foundation of calculus, studying limits, continuity, and convergence with full logical precision.", "definition": "Analysis grew out of the need to put calculus on firm logical ground, replacing intuitions about the infinitely small with precise definitions of limit, continuity, and convergence. Real analysis treats functions of real numbers using the epsilon–delta definition of a limit; complex analysis studies functions of complex variables and reveals striking rigidity. The field underlies differential equations, probability, and much of applied mathematics.", "example": "The epsilon–delta definition makes 'the limit of f(x) as x approaches a is L' exact: for every ε greater than 0 there is a δ greater than 0 so that 0 less than |x − a| less than δ forces |f(x) − L| less than ε.", "related": [ "real-analysis", "continuity", "epsilon-delta-limit", "complex-analysis-cauchy", "power-series" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "mathematical-logic", "term": "Mathematical Logic", "aka": [ "symbolic logic", "formal logic" ], "category": "disciplines-fields", "short": "The formal study of reasoning, proof, and the limits of what mathematical systems can establish about themselves.", "definition": "Mathematical logic makes the rules of valid inference themselves an object of mathematical study, covering proof theory, model theory, computability, and set theory. It asks what can be proved, what is true, and whether the two always coincide. Its discoveries draw sharp boundaries around the power of any formal system.", "example": "Gödel's incompleteness theorems (1931) show that any consistent formal system rich enough for arithmetic contains true statements it cannot prove, and cannot prove its own consistency.", "related": [ "godels-incompleteness-theorems", "set-theory", "boolean-algebra", "russells-paradox", "p-versus-np" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "matrix", "term": "Matrix", "aka": [ "matrices" ], "category": "algebra", "short": "A rectangular array of numbers that encodes a linear transformation or a system of linear equations.", "definition": "A matrix is a rectangular grid of entries arranged in m rows and n columns, written as an m × n matrix. Matrices add entrywise and multiply by a rule that composes the linear maps they represent, so multiplication is associative but generally not commutative. A square matrix may have an inverse, and the identity matrix I leaves every vector unchanged under multiplication.", "example": "The 2 × 2 matrix [[0, −1], [1, 0]] rotates every vector in the plane 90° counterclockwise; applying it to the vector (1, 0) yields (0, 1).", "related": [ "determinant", "eigenvalues-and-eigenvectors", "vector", "linear-algebra", "vector-space" ], "source": "Strang, Introduction to Linear Algebra" }, { "slug": "maxwell-equations", "term": "Maxwell's Equations", "aka": [ "equations of electromagnetism" ], "category": "mathematical-physics", "short": "The four field equations that unify electricity, magnetism, and light into a single classical theory of the electromagnetic field.", "definition": "In differential form (SI units) the four equations are Gauss's law ∇·E = ρ/ε₀, the absence of magnetic monopoles ∇·B = 0, Faraday's law ∇×E = −∂B/∂t, and the Ampère–Maxwell law ∇×B = μ₀J + μ₀ε₀ ∂E/∂t. Here E and B are the electric and magnetic fields, ρ is charge density, and J is current density. Together they predict electromagnetic waves traveling at speed c = 1/√(μ₀ε₀).", "example": "In a vacuum with no charges or currents, the equations combine to give the wave equation ∇²E = (1/c²) ∂²E/∂t², so light is an electromagnetic wave with c = 1/√(μ₀ε₀).", "related": [ "wave-equation", "mass-energy-equivalence", "noether-theorem" ], "source": "Griffiths, Introduction to Electrodynamics" }, { "slug": "mersenne-prime", "term": "Mersenne Prime", "category": "number-theory", "short": "A prime of the form 2ᵖ − 1, tied to the perfect numbers by the Euclid–Euler theorem.", "definition": "A Mersenne number has the form Mₙ = 2ⁿ − 1, and when it is prime it is called a Mersenne prime; a necessary condition is that the exponent n be prime, though that is not sufficient. A perfect number equals the sum of its proper divisors, and the Euclid–Euler theorem says the even perfect numbers are exactly those of the form 2^(p−1)·(2ᵖ − 1) with 2ᵖ − 1 a Mersenne prime. The largest known primes are typically Mersenne primes, found via the efficient Lucas–Lehmer test.", "example": "2⁵ − 1 = 31 is a Mersenne prime, yielding the perfect number 2⁴·31 = 496; but 2¹¹ − 1 = 2047 = 23·89 is composite, so a prime exponent does not guarantee a Mersenne prime.", "related": [ "prime-number", "fundamental-theorem-of-arithmetic" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "moebius-strip", "term": "Möbius Strip", "aka": [ "Mobius band", "Möbius band" ], "category": "geometry-topology", "short": "A surface with only one side and one edge, made by giving a strip a half-twist before joining its ends.", "definition": "A Möbius strip is formed by taking a rectangular strip, giving one end a half-twist, and gluing it to the other end. The result is a non-orientable surface: there is no consistent way to choose an 'up' direction across the whole surface, and it has a single side and a single boundary edge. It was described independently by Möbius and Listing around 1858 and is a standard first example of non-orientability.", "example": "Cutting a Möbius strip along its center line does not produce two loops as intuition suggests, but instead yields a single longer loop with a full twist.", "related": [ "topology", "euler-characteristic", "manifold" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "modular-arithmetic", "term": "Modular Arithmetic", "aka": [ "clock arithmetic", "congruences" ], "category": "number-theory", "short": "Arithmetic of integers where numbers wrap around upon reaching a fixed modulus.", "definition": "Two integers a and b are congruent modulo n, written a ≡ b (mod n), when n divides a − b. Congruence is compatible with addition, subtraction, and multiplication, so one can compute within the finite set {0, 1, …, n−1}. Introduced systematically by Gauss in his 1801 Disquisitiones Arithmeticae, it underlies divisibility tests, calendar calculations, and modern cryptography.", "example": "On a 12-hour clock, 9 + 5 = 14 ≡ 2 (mod 12), so five hours after 9 o'clock is 2 o'clock.", "related": [ "fermats-little-theorem", "rsa-cryptosystem", "euclidean-algorithm" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "navier-stokes-equations", "term": "Navier–Stokes Equations", "aka": [ "equations of viscous fluid flow" ], "category": "mathematical-physics", "short": "The nonlinear PDEs governing the motion of a viscous fluid, by applying Newton's second law to a fluid element.", "definition": "For an incompressible Newtonian fluid, the equations are ρ(∂u/∂t + u·∇u) = −∇p + μ∇²u + f together with the incompressibility condition ∇·u = 0. Here u is the velocity field, p the pressure, ρ the density, μ the viscosity, and f an external force. The convective term u·∇u makes the system nonlinear, which is the source of turbulence.", "example": "Whether smooth solutions in three dimensions always remain smooth (no finite-time blow-up) is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems, still unsolved.", "related": [ "wave-equation", "schrodinger-equation" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "noether-theorem", "term": "Noether's Theorem", "aka": [ "symmetry and conservation laws" ], "category": "mathematical-physics", "short": "The theorem that every continuous symmetry of a physical system's action corresponds to a conserved quantity.", "definition": "Proved by Emmy Noether in 1918, the theorem states that for a system described by a Lagrangian, each continuous symmetry of the action yields a conservation law. Invariance under time translation gives conservation of energy, invariance under spatial translation gives conservation of momentum, and rotational invariance gives conservation of angular momentum. It is one of the deepest organizing principles of theoretical physics.", "example": "Because the laws of physics do not change if you shift your clock's zero point, energy is conserved — symmetry under time translation directly implies the conservation of energy.", "related": [ "hamiltonian-lagrangian", "maxwell-equations", "einstein-field-equations" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "non-euclidean-geometry", "term": "Non-Euclidean Geometry", "aka": [ "hyperbolic geometry", "spherical geometry" ], "category": "geometry-topology", "short": "Consistent geometries that reject Euclid's parallel postulate, in which triangle angles sum to less than or more than 180°.", "definition": "Non-Euclidean geometries arise by replacing the parallel postulate. In hyperbolic geometry (developed by Lobachevsky and Bolyai) there are infinitely many parallels through a point and triangle angles sum to less than 180°; in spherical or elliptic geometry there are none and the angles sum to more than 180°. These geometries are internally consistent, showing the parallel postulate cannot be derived from Euclid's other axioms.", "example": "On the surface of a sphere a triangle with three right angles can be drawn — from the north pole down to two points on the equator a quarter-turn apart — giving an angle sum of 270°.", "related": [ "euclidean-geometry", "gaussian-curvature", "manifold", "gauss-bonnet-theorem" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "normal-distribution", "term": "Normal (Gaussian) Distribution", "aka": [ "Gaussian distribution", "bell curve" ], "category": "probability-statistics", "short": "The bell-shaped continuous distribution with density determined by a mean μ and variance σ².", "definition": "The normal distribution N(μ, σ²) has probability density f(x) = (1 / (σ·√(2π))) · e^(−(x − μ)² / (2σ²)). It is symmetric about μ, and the total area under the curve is 1, reflecting the Gaussian integral ∫ e^(−x²) dx = √π over the whole real line. The standard normal N(0, 1) is obtained by the substitution z = (x − μ)/σ.", "example": "The empirical 68–95–99.7 rule: about 68% of a normal population lies within ±1σ of the mean, about 95% within ±2σ, and about 99.7% within ±3σ.", "related": [ "central-limit-theorem", "random-variable", "expectation-and-variance" ], "source": "NIST Digital Library of Mathematical Functions (DLMF, dlmf.nist.gov)" }, { "slug": "number-theory", "term": "Number Theory", "aka": [ "higher arithmetic" ], "category": "disciplines-fields", "short": "The study of the integers and their deep properties, above all the primes and how numbers divide one another.", "definition": "Number theory investigates whole numbers: divisibility, prime factorization, modular arithmetic, and Diophantine equations whose solutions must themselves be integers. Carl Friedrich Gauss called it the queen of mathematics for its blend of elementary statements and profound difficulty. Many of its questions are simple to state yet have resisted proof for centuries.", "example": "Fermat's Last Theorem — that aⁿ + bⁿ = cⁿ has no positive-integer solutions for n greater than 2 — was conjectured around 1637 and finally proved by Andrew Wiles in 1994.", "related": [ "prime-number", "fundamental-theorem-of-arithmetic", "fermats-last-theorem", "riemann-hypothesis", "modular-arithmetic" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "numerical-analysis", "term": "Numerical Analysis", "category": "disciplines-fields", "short": "The study of algorithms that compute approximate but reliable answers to mathematical problems that have no exact closed-form solution.", "definition": "Numerical analysis designs and analyzes methods for approximating solutions — to equations, integrals, differential equations, and linear systems — using finite arithmetic on computers. Central concerns are accuracy, convergence rate, stability, and how rounding errors accumulate. It is the engine room of scientific computing, turning mathematical models into runnable simulations.", "example": "Newton's method finds a root of f by iterating xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ), often roughly doubling the number of correct digits each step when it converges.", "related": [ "applied-mathematics", "differential-equations", "linear-algebra", "taylor-series", "mathematical-analysis" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "p-versus-np", "term": "P versus NP", "aka": [ "P vs NP", "P = NP problem" ], "category": "foundations-logic", "short": "P versus NP asks whether every problem whose solution can be checked quickly can also be solved quickly.", "definition": "P is the class of decision problems solvable by a deterministic algorithm in polynomial time; NP is the class whose 'yes' answers can be verified in polynomial time given a certificate. The open question is whether P = NP. It is one of the seven Clay Millennium Prize Problems, each carrying a one-million-dollar award, and is widely conjectured to be false (P ≠ NP).", "example": "Verifying that a given assignment satisfies a Boolean formula is fast (in NP), but finding a satisfying assignment from scratch (the SAT problem) has no known fast algorithm — and SAT is NP-complete, so a fast solver for it would settle P = NP.", "related": [ "boolean-algebra", "godels-incompleteness-theorems" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "pascals-triangle", "term": "Pascal's Triangle", "aka": [ "arithmetical triangle" ], "category": "discrete-combinatorics", "short": "A triangular array in which each entry is the sum of the two entries above it, with row n giving the binomial coefficients C(n, 0) through C(n, n).", "definition": "Pascal's triangle arranges the binomial coefficients so that the entry in row n, position k equals C(n, k). Every interior entry is the sum of the two entries diagonally above it, expressing Pascal's rule C(n, k) = C(n−1, k−1) + C(n−1, k). The rows begin 1; 1 1; 1 2 1; 1 3 3 1; and so on.", "example": "The sum of the entries in row n equals 2ⁿ, since Σ C(n, k) = 2ⁿ counts all subsets of an n-set. For row 4: 1 + 4 + 6 + 4 + 1 = 16 = 2⁴. The shallow diagonals of the triangle sum to the Fibonacci numbers.", "related": [ "binomial-coefficient", "permutations-and-combinations", "fibonacci-sequence" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "permutations-and-combinations", "term": "Permutations and Combinations", "aka": [ "counting arrangements and selections" ], "category": "discrete-combinatorics", "short": "Permutations count ordered arrangements, P(n, k) = n!/(n−k)!, while combinations count unordered selections, C(n, k) = n!/(k!(n−k)!).", "definition": "A permutation is an ordered arrangement; the number of ways to arrange k of n distinct objects is P(n, k) = n! / (n−k)!. A combination is an unordered selection; the number of ways to choose k of n objects is C(n, k) = n! / (k!(n−k)!). The two are linked by P(n, k) = k! · C(n, k), since each combination can be ordered in k! ways.", "example": "From 4 people, the number of ordered seatings of 2 is P(4, 2) = 4!/2! = 12, while the number of unordered pairs is C(4, 2) = 6. Note 12 = 2! · 6, matching P(n, k) = k! · C(n, k).", "related": [ "binomial-coefficient", "pascals-triangle", "pigeonhole-principle" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "pi", "term": "Pi (π)", "aka": [ "the circle constant", "Archimedes' constant" ], "category": "constants-special-numbers", "short": "π is the ratio of a circle's circumference to its diameter, approximately 3.14159.", "definition": "Pi (π) is the constant ratio of any circle's circumference to its diameter, with value π ≈ 3.14159265. It is irrational, so its decimal expansion neither terminates nor repeats, and it is also transcendental, meaning it is not a root of any nonzero polynomial with integer coefficients. Its transcendence, proved by Lindemann in 1882, settles the ancient problem of squaring the circle as impossible with compass and straightedge.", "example": "For a circle of diameter 1, the circumference is exactly π. Pi also appears far from circles: the sum 1 + 1/4 + 1/9 + 1/16 + ⋯ of reciprocal squares equals π²/6.", "related": [ "e", "tau", "transcendental-irrational-algebraic", "euler-mascheroni-gamma" ], "source": "NIST Digital Library of Mathematical Functions (DLMF, dlmf.nist.gov)" }, { "slug": "pigeonhole-principle", "term": "Pigeonhole Principle", "aka": [ "Dirichlet box principle", "Dirichlet's drawer principle" ], "category": "discrete-combinatorics", "short": "If n items are placed into m containers and n > m, then at least one container holds more than one item.", "definition": "The pigeonhole principle states that if more than m objects are distributed among m boxes, some box must contain at least two objects. The generalized form says that distributing n objects among m boxes forces some box to hold at least ⌈n/m⌉ objects. Despite its simplicity, it underlies many nonconstructive existence proofs in combinatorics and number theory.", "example": "Among any 13 people, at least two share a birth month, since there are only 12 months. More strikingly, of any n+1 distinct integers chosen from {1, …, 2n}, some pair must have one dividing the other.", "related": [ "permutations-and-combinations", "ramsey-theory", "graph-theory-paths-trees-coloring" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "poincare-conjecture", "term": "Poincaré Conjecture", "aka": [ "Poincaré-Perelman theorem" ], "category": "geometry-topology", "short": "Every simply connected closed 3-dimensional manifold is topologically a 3-sphere — now a proven theorem.", "definition": "Posed by Henri Poincaré in 1904, the conjecture asserts that any closed 3-manifold in which every loop can be continuously shrunk to a point (simply connected) must be homeomorphic to the 3-dimensional sphere. It is the 3-dimensional analogue of the fact that the ordinary sphere is the only simply connected closed surface. Grigori Perelman proved it in 2002–2003 using Richard Hamilton's Ricci flow, settling the only Clay Millennium Prize Problem solved to date.", "example": "Intuitively, on an ordinary 2-sphere any rubber band can be slid to a point, but on a torus a band around the hole cannot; the conjecture says that in three dimensions this 'no snags' property pins the shape down to the 3-sphere.", "related": [ "manifold", "topology", "euler-characteristic" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "polynomial", "term": "Polynomial", "category": "algebra", "short": "An expression built from a variable using only addition, subtraction, and multiplication, such as 3x² − 2x + 1.", "definition": "A polynomial in one variable is an expression of the form aₙxⁿ + ⋯ + a₁x + a₀, where the coefficients lie in a ring or field and n, the degree, is a nonnegative integer. Polynomials can be added, subtracted, and multiplied to form a ring, and divided with remainder much like integers. Their roots are the central objects of much of algebra.", "example": "The polynomial x² − 5x + 6 factors as (x − 2)(x − 3), so its roots are 2 and 3. A polynomial of degree n has at most n real roots, and exactly n complex roots counted with multiplicity.", "related": [ "ring", "fundamental-theorem-of-algebra", "quadratic-formula", "complex-number" ], "source": "Dummit & Foote, Abstract Algebra" }, { "slug": "power-series", "term": "Power Series", "aka": [ "analytic expansion" ], "category": "analysis-calculus", "short": "A power series is an infinite polynomial Σ cₙ(x − a)ⁿ that converges inside a radius around its center a.", "definition": "A power series Σ cₙ(x − a)ⁿ converges for |x − a| < R and diverges for |x − a| > R, where R is the radius of convergence determined by the coefficients. Inside that interval the series defines a smooth function that can be differentiated and integrated term by term. Functions equal to their power series near a point are called analytic.", "example": "The series Σ xⁿ (n from 0) equals 1/(1 − x) for |x| < 1; at x = 1/2 it sums to 2, but it diverges at x = 1.", "related": [ "taylor-series", "complex-analysis-cauchy", "fourier-series" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "prime-number", "term": "Prime Number", "aka": [ "prime" ], "category": "number-theory", "short": "A natural number greater than 1 whose only positive divisors are 1 and itself.", "definition": "A prime is an integer p > 1 with no positive divisors other than 1 and p; an integer greater than 1 that is not prime is called composite. Euclid proved that there are infinitely many primes. The number 1 is excluded by convention so that factorization into primes is unique.", "example": "The primes below 20 are 2, 3, 5, 7, 11, 13, 17, and 19; note that 2 is the only even prime, since every larger even number is divisible by 2.", "related": [ "fundamental-theorem-of-arithmetic", "prime-number-theorem", "mersenne-prime", "goldbach-conjecture" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "prime-number-theorem", "term": "Prime Number Theorem", "aka": [ "PNT" ], "category": "number-theory", "short": "The number of primes up to x is asymptotically x / ln x.", "definition": "Letting π(x) count the primes not exceeding x, the prime number theorem asserts that π(x) ~ x / ln x, meaning the ratio of the two sides tends to 1 as x → ∞. Equivalently, the average gap between primes near x is about ln x. It was proved independently in 1896 by Hadamard and de la Vallée Poussin using properties of the Riemann zeta function.", "example": "For x = 1,000,000 the true count π(x) = 78,498, while x / ln x ≈ 72,382, a relative error of roughly 8% that shrinks as x grows.", "related": [ "prime-number", "riemann-hypothesis", "fundamental-theorem-of-arithmetic" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "probability-theory", "term": "Probability Theory", "category": "disciplines-fields", "short": "The mathematical study of randomness and uncertainty, assigning numbers between 0 and 1 to how likely events are.", "definition": "Probability theory builds a rigorous calculus of chance, modeling random outcomes and the events built from them. Kolmogorov's axioms (1933) placed the field on a measure-theoretic foundation, defining probability as a normalized measure on a sample space. Its limit theorems explain why averages of many random quantities behave predictably even when individual outcomes do not.", "example": "The central limit theorem says that the sum of many independent random variables tends toward a normal (bell-curve) distribution, which is why the bell curve appears so widely in nature and data.", "related": [ "kolmogorov-probability-axioms", "random-variable", "central-limit-theorem", "law-of-large-numbers", "normal-distribution" ], "source": "Ross, A First Course in Probability" }, { "slug": "pure-mathematics", "term": "Pure Mathematics", "aka": [ "theoretical mathematics" ], "category": "disciplines-fields", "short": "The study of mathematical structures for their own sake, pursued for internal coherence and beauty rather than immediate application.", "definition": "Pure mathematics develops and proves theorems about abstract objects — numbers, groups, spaces, sets, functions — guided by rigor and curiosity rather than a target application. Its branches include number theory, algebra, topology, and analysis. A recurring lesson is that ideas invented as pure abstraction often become essential tools generations later.", "example": "Number theory was long called the purest, most useless branch of mathematics; today its results on prime numbers underpin the RSA cryptosystem that secures internet traffic.", "related": [ "applied-mathematics", "number-theory", "algebra", "topology" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "pythagorean-theorem", "term": "Pythagorean Theorem", "aka": [ "Pythagoras' theorem" ], "category": "geometry-topology", "short": "In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.", "definition": "For a right triangle with legs of length a and b and hypotenuse c, the relation a² + b² = c² holds. It is equivalent to the fact that distance in the Euclidean plane is measured by √(Δx² + Δy²), so the theorem is really a statement about the flat geometry of the plane. The converse is also true: if a² + b² = c² for a triangle's side lengths, the triangle has a right angle opposite c.", "example": "A 3-4-5 triangle satisfies 3² + 4² = 9 + 16 = 25 = 5², the smallest right triangle with whole-number sides.", "related": [ "euclidean-geometry", "non-euclidean-geometry", "gaussian-curvature" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "quadratic-formula", "term": "Quadratic formula", "category": "algebra", "short": "The closed-form solution x = (−b ± √(b² − 4ac)) / (2a) for any quadratic equation ax² + bx + c = 0.", "definition": "For a quadratic equation ax² + bx + c = 0 with a ≠ 0, the two roots are given by x = (−b ± √(b² − 4ac)) / (2a). The quantity under the root, Δ = b² − 4ac, is the discriminant: it is positive for two distinct real roots, zero for one repeated real root, and negative for a complex-conjugate pair. The formula follows from completing the square.", "example": "For x² − 5x + 6 = 0 we have a = 1, b = −5, c = 6, so x = (5 ± √(25 − 24)) / 2 = (5 ± 1) / 2, giving x = 3 and x = 2. For x² + 1 = 0 the discriminant is −4, yielding the complex roots ±i.", "related": [ "polynomial", "complex-number", "fundamental-theorem-of-algebra", "abel-ruffini-theorem" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "ramsey-theory", "term": "Ramsey Theory", "aka": [ "Ramsey's theorem", "party problem" ], "category": "discrete-combinatorics", "short": "The principle that within any sufficiently large structure, complete disorder is impossible — some ordered substructure must always appear.", "definition": "Ramsey theory studies conditions under which order must emerge in large structures. Ramsey's theorem guarantees that for any r and s, a sufficiently large complete graph whose edges are 2-colored contains a monochromatic complete subgraph; the threshold is the Ramsey number R(r, s). These numbers grow rapidly and are notoriously hard to compute — even R(5, 5) is unknown, with only bounds established.", "example": "The Ramsey number R(3, 3) = 6: among any 6 people, there are always 3 who all mutually know each other or 3 who are all mutual strangers, yet with 5 people this can fail. This is the classic 'party problem'.", "related": [ "pigeonhole-principle", "graph-theory-paths-trees-coloring", "binomial-coefficient" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "random-variable", "term": "Random Variable", "aka": [ "stochastic variable" ], "category": "probability-statistics", "short": "A measurable function mapping outcomes of a random experiment to real numbers.", "definition": "A random variable X is a function X: Ω → ℝ from the sample space to the reals such that the preimage of every interval is an event. It is described by its distribution: a discrete random variable has a probability mass function, while a continuous one has a probability density function f with P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. The cumulative distribution function F(x) = P(X ≤ x) characterizes either type.", "example": "Let X be the number of heads in two fair coin flips. Then X takes values {0, 1, 2} with probabilities {1/4, 1/2, 1/4}, and F(1) = P(X ≤ 1) = 3/4.", "related": [ "expectation-and-variance", "normal-distribution", "poisson-and-binomial-distributions" ], "source": "Feller / Ross, probability texts" }, { "slug": "real-analysis", "term": "Real Analysis", "aka": [ "analysis of the real line" ], "category": "analysis-calculus", "short": "The rigorous study of real numbers, sequences, limits, continuity, and the foundations of calculus.", "definition": "Real analysis grounds calculus in precise definitions, building from the completeness of the real numbers — every nonempty set bounded above has a least upper bound (supremum). This completeness distinguishes ℝ from ℚ and guarantees that Cauchy sequences converge, that continuous functions attain extrema, and that limits behave as expected. It supplies the proofs behind the rules of differentiation and integration.", "example": "Completeness explains why √2 exists as a real number: the set of rationals whose square is less than 2 is bounded above, so it has a real supremum, namely √2, which is itself irrational.", "related": [ "epsilon-delta-limit", "continuity", "integral", "complex-analysis-cauchy" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "recurrence-relation", "term": "Recurrence Relation", "aka": [ "difference equation" ], "category": "discrete-combinatorics", "short": "An equation that defines each term of a sequence using one or more previous terms, such as aₙ = c₁aₙ₋₁ + c₂aₙ₋₂.", "definition": "A recurrence relation expresses a sequence term aₙ in terms of earlier terms together with initial conditions. A linear homogeneous recurrence with constant coefficients can be solved via its characteristic equation: each distinct root r contributes a term of the form r ⁿ to the general solution. Generating functions provide a complementary route to closed forms.", "example": "For aₙ = 2aₙ₋₁ with a₀ = 1, the solution is aₙ = 2ⁿ. For the Fibonacci recurrence Fₙ = Fₙ₋₁ + Fₙ₋₂, the characteristic equation r² = r + 1 has roots φ and ψ, yielding Binet's closed form.", "related": [ "fibonacci-sequence", "generating-function", "binomial-coefficient" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "regression", "term": "Regression", "aka": [ "least squares regression", "linear regression" ], "category": "probability-statistics", "short": "Regression fits a function to data to model how a response variable depends on one or more predictor variables.", "definition": "Regression estimates the relationship between a dependent variable and one or more independent variables, most simply by fitting a straight line y = a + bx through observed points. The classical method of least squares chooses the coefficients that minimize the sum of squared residuals between observed and predicted values. The fitted model can then be used to summarize trends, test hypotheses about the coefficients, or predict new outcomes.", "example": "Fitting a line to (height, weight) measurements by least squares yields a slope estimating how much weight increases per unit of height, with residuals measuring the scatter around the line.", "related": [ "standard-deviation", "expectation-and-variance", "hypothesis-testing", "linear-algebra" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "riemann-hypothesis", "term": "Riemann Hypothesis", "category": "number-theory", "short": "The conjecture that every nontrivial zero of the Riemann zeta function has real part 1/2.", "definition": "The Riemann zeta function ζ(s) = Σ 1/nˢ, extended to the complex plane, has trivial zeros at the negative even integers; the hypothesis states all other zeros lie on the critical line Re(s) = 1/2. Posed by Bernhard Riemann in 1859, it controls the fine distribution of prime numbers and remains unproved. It is one of the seven Clay Millennium Prize Problems.", "example": "The first nontrivial zero sits at approximately s = 1/2 + 14.1347i, and billions of computed zeros so far all lie exactly on the line Re(s) = 1/2.", "related": [ "prime-number-theorem", "prime-number" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "ring", "term": "Ring", "category": "algebra", "short": "A set with two operations, addition and multiplication, where addition forms an abelian group and multiplication distributes over it.", "definition": "A ring is a set R with two binary operations, + and ·, such that (R, +) is an abelian group, multiplication is associative, and multiplication distributes over addition: a·(b+c) = a·b + a·c and (a+b)·c = a·c + b·c. Many authors also require a multiplicative identity 1. Rings need not be commutative and need not have multiplicative inverses, which distinguishes them from fields.", "example": "The integers ℤ form a commutative ring with identity, but most elements lack multiplicative inverses (e.g. 2 has no integer reciprocal). Polynomials with real coefficients, ℝ[x], also form a ring under ordinary addition and multiplication.", "related": [ "group", "field", "polynomial" ], "source": "Dummit & Foote, Abstract Algebra" }, { "slug": "rsa-cryptosystem", "term": "RSA Cryptosystem", "aka": [ "RSA encryption" ], "category": "number-theory", "short": "A public-key cryptosystem whose security rests on the difficulty of factoring large integers.", "definition": "In RSA one picks two large primes p and q, sets the modulus n = p·q, and chooses a public exponent e together with a private exponent d satisfying e·d ≡ 1 (mod λ), where λ = lcm(p−1, q−1). Encryption sends a message m to mᵉ mod n, and decryption recovers it via (mᵉ)ᵈ ≡ m (mod n), a fact guaranteed by Euler's and Fermat's theorems. Knowing n alone does not reveal d unless one can factor n, which is believed hard for large n.", "example": "A toy case uses p = 61, q = 53, so n = 3233; with e = 17 and d = 413, encrypting m = 65 gives 65¹⁷ mod 3233 = 2790, and 2790⁴¹³ mod 3233 returns 65.", "related": [ "modular-arithmetic", "fermats-little-theorem", "prime-number" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "russells-paradox", "term": "Russell's Paradox", "aka": [ "Russell paradox", "barber paradox (informal)" ], "category": "foundations-logic", "short": "Russell's Paradox shows that the naive set of all sets that do not contain themselves cannot consistently exist.", "definition": "Discovered by Bertrand Russell in 1901, the paradox considers the set R = {x : x ∉ x}, the collection of all sets that are not members of themselves. Asking whether R ∈ R yields a contradiction: R ∈ R holds if and only if R ∉ R. This demolished Frege's unrestricted comprehension principle and motivated the careful, restricted set-formation axioms of ZFC.", "example": "By analogy: a village barber who shaves exactly those who do not shave themselves cannot consistently exist — does he shave himself? Either answer contradicts the rule.", "related": [ "zfc-axioms", "cantor-diagonal-argument", "godels-incompleteness-theorems" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "schrodinger-equation", "term": "Schrödinger Equation", "aka": [ "wave equation of quantum mechanics" ], "category": "mathematical-physics", "short": "The fundamental equation governing how a quantum state evolves in time.", "definition": "The time-dependent Schrödinger equation is iℏ ∂Ψ/∂t = ĤΨ, where Ψ is the wavefunction, ℏ is the reduced Planck constant, and Ĥ is the Hamiltonian operator. For a single nonrelativistic particle, Ĥ = −(ℏ²/2m)∇² + V, combining kinetic and potential energy. Solutions are complex-valued, and |Ψ|² is interpreted as a probability density.", "example": "For a stationary state of energy E, Ψ separates as Ψ = ψ e^(−iEt/ℏ), reducing the equation to the time-independent form Ĥψ = Eψ, an eigenvalue problem.", "related": [ "wave-equation", "dirac-equation", "hamiltonian-lagrangian" ], "source": "Griffiths, Introduction to Quantum Mechanics" }, { "slug": "set", "term": "Set", "aka": [ "collection" ], "category": "foundations-logic", "short": "A set is a well-defined collection of distinct objects, considered as a single whole and characterized entirely by which objects it contains.", "definition": "A set is determined solely by its members (its extension), so {1, 2, 3} and {3, 2, 1} are the same set and repetition is ignored. Membership is written x ∈ A, and sets combine through union ∪, intersection ∩, complement, and the formation of subsets. Sets are the basic objects of modern mathematics: in the ZFC axioms essentially every mathematical object is built from sets.", "example": "The empty set ∅ has no elements, yet it is a subset of every set; from it, ZFC constructs the natural numbers as 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, and so on.", "related": [ "zfc-axioms", "cardinality", "russells-paradox", "boolean-algebra", "infinity" ], "source": "Rudin, Principles of Mathematical Analysis" }, { "slug": "set-theory", "term": "Set Theory", "category": "disciplines-fields", "short": "The study of collections of objects, which serves as the common foundation on which the rest of mathematics is built.", "definition": "Set theory studies sets — well-defined collections — and the operations, relations, and sizes (cardinalities) among them. Georg Cantor founded it in the 1870s and proved that infinities come in different sizes, while the ZFC axioms later gave it a paradox-free footing. Nearly every mathematical object can be defined in terms of sets, making the field a foundation for mathematics itself.", "example": "Cantor's diagonal argument proves the real numbers are uncountable — a strictly larger infinity than the counting numbers ℵ₀, showing not all infinities are equal.", "related": [ "cardinality", "cantor-diagonal-argument", "aleph-null", "zfc-axioms", "continuum-hypothesis" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "sqrt2", "term": "Square Root of Two (√2)", "aka": [ "root two", "Pythagoras' constant" ], "category": "constants-special-numbers", "short": "√2 ≈ 1.41421 is the length of a diagonal of a unit square and the first number proved irrational.", "definition": "The square root of two, √2 ≈ 1.41421356, is the positive number whose square is 2. It is irrational: no fraction of integers equals it, a fact provable by a short argument showing any such fraction could be reduced forever. Unlike π and e, √2 is algebraic, since it is a root of the polynomial x² − 2. It is the diagonal of a unit square, by the Pythagorean theorem.", "example": "A square with sides of length 1 has a diagonal of length √2; the classic proof that √2 is irrational assumes √2 = p/q in lowest terms and derives that p and q must both be even, a contradiction.", "related": [ "golden-ratio-phi", "pi", "transcendental-irrational-algebraic" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "standard-deviation", "term": "Standard Deviation", "aka": [ "σ", "SD" ], "category": "probability-statistics", "short": "The standard deviation measures how spread out values are around their mean, equal to the square root of the variance.", "definition": "For a random variable X, the standard deviation σ is √(Var(X)), the positive square root of the variance E[(X − μ)²], where μ is the mean. It is expressed in the same units as the data itself, unlike the variance, which makes it a natural measure of typical deviation from the average. A small σ means values cluster tightly around the mean; a large σ means they are widely dispersed.", "example": "For a normal distribution, about 68% of values fall within one standard deviation of the mean and about 95% within two — the familiar 68–95–99.7 rule.", "related": [ "expectation-and-variance", "normal-distribution", "random-variable", "central-limit-theorem" ], "source": "Ross, A First Course in Probability" }, { "slug": "statistics", "term": "Statistics", "category": "disciplines-fields", "short": "The science of collecting, analyzing, and drawing conclusions from data in the presence of variability and uncertainty.", "definition": "Statistics turns data into inference: it designs experiments and surveys, summarizes observations, and quantifies how much we can trust conclusions drawn from samples. Built on probability theory, it splits into descriptive statistics (summarizing what is seen) and inferential statistics (generalizing from a sample to a population). It is indispensable across medicine, economics, machine learning, and the empirical sciences.", "example": "Bayes' theorem, P(A|B) = P(B|A)·P(A) / P(B), lets a diagnostic test's result be combined with a disease's base rate to update the probability that a patient actually has the disease.", "related": [ "probability-theory", "bayes-theorem", "normal-distribution", "expectation-and-variance", "random-variable" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "tau", "term": "Tau (τ)", "aka": [ "the turn constant", "2π" ], "category": "constants-special-numbers", "short": "τ = 2π ≈ 6.28318 is the ratio of a circle's circumference to its radius, equal to one full turn in radians.", "definition": "Tau (τ) is defined as 2π ≈ 6.28318530718, the ratio of a circle's circumference to its radius rather than its diameter. Because angles are naturally measured against the radius, one full revolution is exactly τ radians, a quarter turn is τ/4, and so on. Tau shares all of π's properties of irrationality and transcendence, since it differs only by the factor 2.", "example": "A full circle is τ radians and a half circle is τ/2; the area of a circle of radius r can be written A = ½ τ r², mirroring the form of ½ k x² and ½ m v² seen elsewhere in mathematics and physics.", "related": [ "pi", "e", "imaginary-unit-i" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "taylor-series", "term": "Taylor Series", "aka": [ "Taylor expansion", "Maclaurin series" ], "category": "analysis-calculus", "short": "A Taylor series represents a smooth function as an infinite power series built from its derivatives at a single point.", "definition": "The Taylor series of f about a is Σ f⁽ⁿ⁾(a)/n! · (x − a)ⁿ, summing over n ≥ 0. When centered at a = 0 it is called a Maclaurin series. The partial sums are the best polynomial approximations of a given degree, with the truncation error controlled by the remainder term involving the next derivative.", "example": "e^x = Σ xⁿ/n! = 1 + x + x²/2 + x³/6 + …, converging for all real x; setting x = 1 gives e = Σ 1/n!.", "related": [ "power-series", "derivative", "fundamental-theorem-of-calculus", "the-e-limit" ], "source": "Spivak, Calculus" }, { "slug": "continuum-hypothesis", "term": "The Continuum Hypothesis", "aka": [ "CH" ], "category": "foundations-logic", "short": "The Continuum Hypothesis conjectures that there is no set whose size lies strictly between that of the integers and that of the reals.", "definition": "Proposed by Cantor, CH states that the cardinality of the continuum equals the first uncountable cardinal: 2^ℵ₀ = ℵ₁. It was the first of Hilbert's 1900 problems. Goedel (1940) proved CH cannot be disproved from ZFC, and Cohen (1963) proved it cannot be proved from ZFC, so CH is independent of the standard axioms.", "example": "Cohen's proof of CH's independence introduced the method of forcing, for which he received the Fields Medal in 1966 — the only Fields Medal awarded for work in mathematical logic to that date.", "related": [ "cardinality", "cantor-diagonal-argument", "zfc-axioms", "axiom-of-choice", "godels-incompleteness-theorems" ], "source": "MacTutor History of Mathematics (St Andrews)" }, { "slug": "the-e-limit", "term": "The Limit Defining e", "aka": [ "Euler's number", "compound interest limit" ], "category": "analysis-calculus", "short": "The number e is the limit of (1 + 1/n)ⁿ as n grows without bound, approximately 2.71828.", "definition": "Euler's number e is defined by the limit lim(n→∞) (1 + 1/n)ⁿ ≈ 2.718281828, the value approached by continuously compounded growth. Equivalently e = Σ 1/n! over n ≥ 0. The function e^x is the unique function (up to scaling) equal to its own derivative, which makes e the natural base for exponential growth and the logarithm.", "example": "Compounding $1 at 100% annual interest n times per year yields (1 + 1/n)ⁿ; as compounding becomes continuous (n → ∞) the balance approaches e ≈ 2.71828 dollars.", "related": [ "epsilon-delta-limit", "taylor-series", "derivative", "differential-equations" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "topology", "term": "Topology", "aka": [ "rubber-sheet geometry" ], "category": "geometry-topology", "short": "The study of properties of shapes that survive continuous stretching and bending but not tearing or gluing.", "definition": "Topology studies the features of a space that are preserved under homeomorphisms — continuous deformations with continuous inverses — such as stretching and bending without cutting. Properties like the number of holes, connectedness, and the Euler characteristic are topological invariants, while lengths and angles are not. In topology a coffee mug and a doughnut are regarded as the same object because each has exactly one hole and one can be deformed into the other.", "example": "A sphere and a cube are topologically identical, since each deforms into the other with no holes, but neither is the same as a torus, whose single hole no stretching can remove.", "related": [ "euler-characteristic", "moebius-strip", "manifold", "poincare-conjecture" ], "source": "Munkres, Topology" }, { "slug": "transcendental-irrational-algebraic", "term": "Transcendental, Irrational, and Algebraic Numbers", "aka": [ "number classification", "kinds of real numbers" ], "category": "constants-special-numbers", "short": "Algebraic numbers are roots of integer polynomials, transcendental numbers are not, and irrational numbers are simply those that are not fractions.", "definition": "A number is algebraic if it is a root of some nonzero polynomial with integer coefficients; otherwise it is transcendental. Separately, a number is rational if it equals a ratio of integers and irrational if it does not. These classifications overlap: every rational number is algebraic, √2 is irrational yet algebraic, and π and e are both irrational and transcendental. Although almost all real numbers are transcendental, proving any specific number transcendental is typically very hard.", "example": "√2 is irrational but algebraic since it solves x² − 2 = 0, whereas π is transcendental, proved by Lindemann in 1882 to satisfy no polynomial equation with integer coefficients.", "related": [ "pi", "e", "sqrt2", "imaginary-unit-i", "aleph-null" ], "source": "Hardy & Wright, An Introduction to the Theory of Numbers" }, { "slug": "trigonometric-identities", "term": "Trigonometric Identities", "aka": [ "trig identities" ], "category": "geometry-topology", "short": "Trigonometric identities are equations relating the trigonometric functions that hold for all values of the angle.", "definition": "The foundational identity is the Pythagorean relation sin²θ + cos²θ = 1, which follows directly from the unit circle. From it and the angle-addition formulas sin(α + β) = sin α cos β + cos α sin β and cos(α + β) = cos α cos β − sin α sin β flow the double-angle, half-angle, and product-to-sum formulas. These identities let one simplify expressions, solve trigonometric equations, and evaluate integrals.", "example": "From the addition formula, cos(2θ) = cos²θ − sin²θ = 1 − 2 sin²θ, a double-angle identity used constantly to integrate sin²θ and cos²θ.", "related": [ "pythagorean-theorem", "euclidean-geometry", "fourier-series", "pi" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "trigonometry", "term": "Trigonometry", "category": "disciplines-fields", "short": "The study of triangles and the periodic functions — sine, cosine, tangent — that relate their angles to their side lengths.", "definition": "Trigonometry began as the practical study of the ratios between a triangle's sides and angles, then matured into the theory of the trigonometric functions that describe rotation and any repeating wave. These functions link geometry to the unit circle and, through Fourier analysis, to sound, light, and signals. Trigonometry is essential to navigation, surveying, astronomy, and physics.", "example": "On a right triangle the sine of an angle is opposite over hypotenuse, and the identity sin²θ + cos²θ = 1 holds for every angle θ.", "related": [ "geometry", "pythagorean-theorem", "pi", "fourier-series", "euclidean-geometry" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "vector", "term": "Vector", "aka": [ "vectors" ], "category": "algebra", "short": "An object with both magnitude and direction, equivalently a list of numbers that can be added and scaled.", "definition": "A vector is an element of a vector space: an object that can be added to other vectors and multiplied by scalars while obeying rules like associativity and distributivity. Geometrically a vector in the plane or in space is an arrow with a length and a direction; algebraically it is an ordered tuple of components such as (x, y, z). The dot product u · v measures how much two vectors point the same way and yields lengths and angles.", "example": "The vector (3, 4) has length √(3² + 4²) = 5; adding (3, 4) + (1, 2) = (4, 6) places the arrows tip to tail.", "related": [ "vector-space", "matrix", "linear-algebra", "eigenvalues-and-eigenvectors" ], "source": "Strang, Introduction to Linear Algebra" }, { "slug": "vector-space", "term": "Vector space", "aka": [ "linear space" ], "category": "algebra", "short": "A set of vectors that can be added together and scaled by elements of a field, obeying the usual rules of linear arithmetic.", "definition": "A vector space over a field F is a set V with vector addition and scalar multiplication satisfying axioms such as associativity and commutativity of addition, existence of a zero vector and additive inverses, and distributive laws like a(u + v) = au + av. A basis is a linearly independent set that spans V, and the number of basis vectors is the dimension, which is the same for every basis. Vector spaces are the central objects of linear algebra.", "example": "Ordinary 3-dimensional space ℝ³ is a vector space over ℝ with basis {(1,0,0), (0,1,0), (0,0,1)}. The set of all polynomials of degree at most 2 is also a vector space, with basis {1, x, x²} and dimension 3.", "related": [ "field", "linear-algebra", "polynomial" ], "source": "Strang, Introduction to Linear Algebra" }, { "slug": "wave-equation", "term": "Wave Equation", "aka": [ "d'Alembert equation" ], "category": "mathematical-physics", "short": "The second-order linear PDE describing the propagation of waves at a fixed speed c.", "definition": "The classical wave equation is ∂²u/∂t² = c²∇²u, where u is the disturbance and c is the wave speed. In one spatial dimension, d'Alembert's solution is u(x,t) = f(x − ct) + g(x + ct), a superposition of a right-moving and a left-moving wave of arbitrary shape. The equation is hyperbolic and conserves energy in the absence of damping.", "example": "A plucked string fixed at both ends supports standing waves u = sin(nπx/L) cos(nπct/L), whose frequencies are integer multiples of the fundamental — the mathematical basis of musical harmonics.", "related": [ "navier-stokes-equations", "maxwell-equations", "schrodinger-equation" ], "source": "CRC Standard Mathematical Tables and Formulae" }, { "slug": "zfc-axioms", "term": "Zermelo–Fraenkel Set Theory with Choice (ZFC)", "aka": [ "ZFC", "Zermelo–Fraenkel axioms" ], "category": "foundations-logic", "short": "ZFC is the standard axiomatic foundation of modern mathematics, built from a short list of axioms about sets plus the Axiom of Choice.", "definition": "ZFC is a first-order axiom system describing sets and the single relation ∈ (membership). Its axioms — including Extensionality, Pairing, Union, Power Set, Infinity, Replacement, Foundation (Regularity), and Choice — were assembled to avoid the paradoxes of naive set theory. Nearly all of contemporary mathematics can be formalized as theorems derivable from ZFC.", "example": "The Axiom of Infinity asserts the existence of an infinite set; without it, ZFC could not even guarantee that the set of all natural numbers ℕ exists as a completed whole.", "related": [ "russells-paradox", "axiom-of-choice", "cardinality", "continuum-hypothesis" ], "source": "MacTutor History of Mathematics (St Andrews)" } ] }