Mathematics

Gödel number (Gödel numbering) Gödel number Gödel numbers are numeric encodings of symbols and formulas used to represent syntax of formal systems as natural numbers, central to Gödel's incompleteness theorem and to computab…

Gödel's incompleteness theorems Two fundamental theorems by Kurt Gödel (1931) showing that any sufficiently expressive, consistent formal system cannot be both complete and able to prove its own consistency.

Goldbach's conjecture A major unsolved problem in additive number theory: the strong form asserts every even integer >2 is the sum of two primes; a related weak form concerns sums of three primes.

Golden ratio (phi) Golden ratio The golden ratio is an irrational mathematical constant, approximately 1.618, arising from a unique proportional equation and appearing in geometry, design, nature, and number sequ…

Googol: the number 10^100 and its history, meaning, and significance Googol Googol is the large number 10^100 (a 1 followed by 100 zeros). Coined in the 1930s to illustrate enormous scale, it appears in mathematics, popular culture, and inspired the name o…

Googolplex: definition, history, scale, and cultural significance Googolplex Googolplex is 10 to the power of a googol (10^(10^100)), an unimaginably large number. This article explains its definition, origins, size comparisons, uses, and informal variants.

goto (programming statement) goto A programming statement that transfers control unconditionally to a labeled location; historically common but controversial due to maintainability and structured-program alternativ…

Gradient (multivariable calculus) Gradient The gradient is a vector of partial derivatives that indicates direction and rate of fastest increase of a multivariable function. It links geometry, optimization, and physical fie…

Gradient (disambiguation) Overview of the various meanings of “gradient”: mathematical operator and slope, color and visual transitions, image processing, physical gradients, optics, and machine-learning gr…

Graham's number An extremely large finite natural number arising as an upper bound in a Ramsey theory problem; defined by iterated Knuth up‑arrow operations and notable for its size beyond physica…

Gram: unit of mass in the metric system Gram Overview, definition, history, and common uses of the gram (g), a metric unit of mass equal to one thousandth of a kilogram and widely used in science and everyday life.

Grammatical number: forms, functions, and cross-linguistic variation Grammatical number Grammatical number is the way languages mark quantity (one, two, many) on nouns, pronouns, verbs and modifiers. This article surveys common categories, expression types, exceptions…

Graph (visual representation of data) Graph A graph is a visual method for showing relationships among quantities or categories. This article explains types, components, history, uses, and how graphs differ from other diagra…

Graph (mathematics): structures, types, and applications Graph (mathematics) An accessible overview of mathematical graphs: definitions, main types (directed, undirected, simple, multigraph), basic properties, historical context, and common applications acr…

Graph coloring: concepts, types, history and applications Graph coloring Graph coloring assigns labels (colors) to elements of a graph under constraints. It includes vertex, edge and face coloring; relates to chromatic number, algorithms, complexity and…

Graph theory: concepts, history, problems, and applications Graph theory Graph theory studies vertices and edges as abstract models of relationships. This article explains basic concepts, types of graphs, historical origins, central problems and algorit…

Graphical projection Methods for representing three-dimensional objects on a two-dimensional plane using systematic projection rules; covers types, parts, history, conventions and common applications.

Graphing calculator: functions, features, history and classroom use Graphing calculator A graphing calculator is a handheld computing device designed to plot graphs and perform advanced mathematical operations, often programmable and widely used in secondary and posts…

Grave (historic unit of mass) Grave (mass) Grave was an early name for the kilogram used during the French Revolutionary metric reforms. It denotes a unit of mass equivalent to the modern kilogram and has been discussed in…

Great circle: definition, properties, history and applications Great circle A great circle is the largest circle on a sphere; it divides the sphere into equal hemispheres, is a geodesic, and gives the shortest surface path between points—used in navigation…

Greatest common divisor (gcd) Greatest common divisor The greatest common divisor (gcd) of integers is the largest integer dividing them both. This article explains its definition, properties, computation (Euclidean algorithm) and com…

Greek alphabet: origin, letters, pronunciation, and modern uses Greek alphabet An overview of the 24-letter Greek alphabet: its letters and forms, development from Phoenician script, phonetic changes, diacritics and reforms, and its ongoing role in science, m…

Greek numerals: the Milesian (alphabetic) numeral system Greek numerals Alphabetic numerals used in ancient and modern Greek contexts. An additive system that assigns numeric values to letters, uses obsolete letters for some values, and remains in limi…

Gross (unit) A gross is a traditional counting unit equal to 144 items (12 dozen). Used historically in trade and manufacturing, it has related terms such as great gross (1728) and small gross…

Group (mathematics) A group is an algebraic structure consisting of a set with a single binary operation that is associative, has an identity element, and where every element has an inverse. Key in sy…

Group theory Group theory studies groups—sets with a single associative operation and identity/inverses. It formalizes symmetry and underlies algebra, geometry, physics, chemistry, and many app…

Growth (increase in quantity or complexity over time) Growth Growth is an increase in an amount, size, complexity or capability over time. The concept appears in biology, economics, technology and social systems and is modeled and measured i…

Gumbel distribution (Type I extreme value distribution) Gumbel distribution A continuous probability distribution for modeling the distribution of block maxima (or minima). Often used in hydrology, meteorology, engineering and extreme-value theory.

Halting problem: undecidability of program termination Halting problem The halting problem asks whether a general algorithm can determine if an arbitrary program halts on a given input. It is undecidable: no single program can correctly decide termina…

Hamiltonian path A Hamiltonian path in a graph visits every vertex exactly once; a Hamiltonian cycle returns to its start. Determining existence is NP-complete and underlies problems like the trave…

Hamming code — single-error-correcting block codes Hamming code Hamming codes are binary block codes that detect and correct single-bit errors using multiple parity bits. They were introduced by Richard Hamming in the 1950s and are widely used…

Hand (unit of length) Hand (unit) A traditional unit of length equal to 4 inches (10.16 cm), historically based on the breadth of a human hand and still used to state the height of horses; abbreviated h or hh.

Hardy–Weinberg law (Hardy–Weinberg equilibrium) Hardy–Weinberg law A foundational principle of population genetics describing stable allele and genotype frequencies under idealized conditions and the forces that cause evolutionary change.

Harmonic analysis Harmonic analysis studies how functions and signals decompose into basic oscillatory components, linking Fourier methods, group representations, and tools used across mathematics,…

Harmonic series (mathematics) Infinite series of reciprocal integers 1 + 1/2 + 1/3 + …, notable for diverging slowly; central in analysis, number theory, probability and algorithm analysis.

Hash function A hash function maps input data of arbitrary size to fixed-size values. Used in data structures, integrity checks, and cryptography, with distinct properties and trade-offs for dif…

Haskell: a concise guide to the purely functional language Haskell Haskell is a standardized, purely functional programming language known for lazy evaluation, a strong static type system with type inference, and a focus on abstractions such as mo…

Hasty generalization A logical error that draws broad conclusions from insufficient or unrepresentative evidence; common in everyday reasoning and misapplied statistics.

Heaviside function (unit step function) Heaviside Function A discontinuous function that is zero for negative arguments and one for positive, used in signal processing, control theory and distribution theory; its derivative is the Dirac de…

Hebrew numerals: alphabetic numbering, values, and uses Hebrew numerals Alphabetic numeral system that assigns numeric values to letters of the Hebrew alphabet; used historically and today for dates, chapters, gematria and liturgical numbering.

Hectogon (100-sided polygon) Hectogon A hectogon, also called a hectagon, is a polygon with 100 sides and vertices. This article explains its properties, regular form, formulas, symmetry, history, and common uses.

Hectosecond (100 seconds) Hectosecond A hectosecond is 100 seconds (1 minute 40 seconds). It applies the metric prefix 'hecto-' to the SI second and is used occasionally to describe human-scale and scientific time inte…

Henagon (Monogon) Henagon A henagon, or monogon, is a one-sided, one-vertex figure discussed in geometry. It is degenerate in the Euclidean plane but appears in spherical, topological, and combinatorial con…

Hendecagon (11-sided polygon) Hendecagon A hendecagon, or 11-gon, is a polygon with eleven sides and vertices. This article explains its geometry, regular and star forms, constructibility, algebraic aspects, and typical u…

Heptadecagon (17-gon): definition, properties and constructibility Heptadecagon A heptadecagon is a 17-sided polygon. This article explains its geometry, the regular heptadecagon, Gauss's constructibility result, symmetry, star forms and notable facts.

Heptagon A heptagon (or septagon) is a seven-sided polygon. This article explains its geometry, regular form, constructibility, algebraic background, related star polygons, and common uses.

Hermann Minkowski: Mathematician and Architect of Spacetime Geometry Hermann Minkowski Hermann Minkowski (1864–1909) was a German mathematician who introduced the geometric view of special relativity and made foundational contributions to the geometry of numbers, con…

Hexacontagon (60-sided polygon) Hexacontagon A hexacontagon, or 60-gon, is a polygon with sixty sides. This article summarizes its geometry, formulas, symmetry, constructibility, star forms, history, and common uses.