Category: Math Dictionary

Zermelo-Fraenkel set theory: the standard form of set theory and the most common foundation of modern mathematics, based on a list of nine axioms (usually modified by a tenth, the axiom of choice) about what kinds of sets exist, commonly abbreviated together as ZFC

Zeta function: A function based on an infinite series of reciprocals of exponents (Riemann’s zeta function is the extension of Euler’s simple zeta function into the domain of complex numbers)

Venn diagram: a diagram where sets are represented as simple geometric figures (often circles), and overlapping and similar sets are represented by intersections and unions of the figures

Vector space: a three-dimensional area where vectors can be plotted, or a mathematical structure formed by a collection of vectors

Vector: a physical quantity having magnitude and direction, represented by a directed arrow indicating its orientation in space

Type theory: an alternative to naive set theory in which all mathematical entities are assigned to a type within a hierarchy of types, so that objects of a given type are built exclusively from objects of preceding types lower in the hierarchy, thus preventing loops and paradoxes

Trinomial: an algebraic equation with 3 terms, e.g. 3x + 5y + 8z; 3x³ + 2x² + x; etc

Trigonometry: the branch of mathematics that studies the relationships between the sides and the angles of right triangles, and deals with and with the trigonometric functions (sine, cosine, tangent and their reciprocals)

Triangular number: a number which can be represented as an equilateral triangle of dots, and is the sum of all the consecutive numbers up to its largest prime factor – it can also be calculated as ^{n(n + 1)}⁄₂, e.g. 15 = 1 + 2 + 3 + 4 + 5 = ^{5(5 + 1)}⁄₂

Transfinite numbers: cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite

Transcendental number: an irrational number that is “not algebraic”, i.e. no finite sequence of algebraic operations on integers (such as powers, roots, sums, etc.) can be equal to its value, examples being π and e. For example, √2 is irrational but not transcendental because it is the solution to the polynomial x² = 2.

Topology: the field of mathematics concerned with spatial properties that are preserved under continuous deformations of objects (such as stretching, bending and morphing, but not tearing or gluing)

Term: in an algebraic expression or equation, either a single number or variable, or the product of several numbers and variables separated from another term by a + or – sign, e.g. in the expression 3 + 4x + 5yzw, the 3, the 4x and the 5yzw are all separate terms

Theorem: a mathematical statement or hypothesis which has been proved on the basis of previously established theorems and previously accepted axioms, effectively the proof of the truth of a statement or expression

Tensor: a collection of numbers at every point in space which describe how much the space is curved, e.g. in four spatial dimensions, a collection of ten numbers is needed at each point to describe the properties of the mathematical space or manifold, no matter how distorted it may be

Symmetry: the correspondence in size, form or arrangement of parts on a plane or line (line symmetry is where each point on one side of a line has a corresponding point on the opposite side, e.g. a picture a butterfly with wings that are identical on either side; plane symmetry refers to similar figures being repeated at different but regular locations on the plane)

Subset: a subsidiary collection of objects that all belong to, or is contained in, an original given set, e.g. subsets of {a, b} could include: {a}, {b}, {a, b} and {}

Surd: the n-th root a number, such as √5, the cube root of 7, etc

Spherical geometry: a type of non-Euclidean (elliptic) geometry using the two-dimensional surface of a sphere, where a curved geodesic (not a straight line) is the shortest paths between points

Spherical trigonometry: a branch of spherical geometry which deals with polygons (especially triangles) on the sphere, and the relationships between their sides and angles

Simultaneous equations: a set or system of equations containing multiple variables which has a solution that simultaneously satisfies all of the equations (e.g. the set of simultaneous linear equations 2x + y = 8 and x + y = 6, has a solution x = 2 and y = 4)

Slope: the steepness or incline of a line, determined by reference to two points on the line, e.g. the slope of the line y = mx + b is m, and represents the rate at which y is changing per unit of change in x

Significant digits: the number of digits to consider when using measuring numbers, those digits that carry meaning contributing to its precision (i.e. ignoring leading and trailing zeros)

Set: a collection of distinct objects or numbers, without regard to their order, considered as an object in its own right

Sequence: an ordered set whose elements are usually determined based on some function of the counting numbers, e.g. a geometric sequence is a set where each element is a multiple of the previous element; an arithmetic sequence is a set where each element is the previous element plus or minus a number

Self-similarity: object is exactly or approximately similar to a part of itself (in fractals, the shapes of lines at different iterations look like smaller versions of earlier shapes)

Reciprocal: a number which, when multiplied by x yields the multiplicative identity 1, and can therefore be thought of as the inverse of multiplication, e.g. the reciprocal of x is ¹⁄_x, the reciprocal of ³⁄₅ is ⁵⁄₃

Riemannian geometry: a non-Euclidean geometry that studies curved surfaces and differentiable manifolds in higher dimensional spaces

Real numbers: all numbers (including natural numbers, integers, decimals, rational numbers and irrational numbers) which do not involve imaginary numbers (multiples of the imaginary unit i, or the square root of -1), may be thought of as all points on an infinitely long number line

Rational numbers: numbers that can be expressed as a fraction (or ratio) ^a⁄_b of two integers (the integers are therefore a subset of the rationals), or alternatively a decimal which terminates after a finite number of digits or begins to repeat a sequence

Quintic equation: a polynomial having a degree of 5 (i.e. the highest power is 5), of the form ax⁵ + bx⁴ + cx³ + dx² + ex + f = 0, not solvable by factorization into radicals for all rational numbers

Quaternions: a number system that extends complex numbers to four dimensions (so that an object is described by a real number and three complex numbers, all mutually perpendicular to each other), which can be used to represent a three-dimensional rotation by just an angle and a vector

Quartic equation: a polynomial having a degree of 4 (i.e. the highest power is 4), of the form ax⁴ + bx³ + cx² + dx + e = 0, the highest order polynomial equation that can be solved by factorization into radicals by a general formula

Quadrature: the act of squaring, or finding a square equal in area to a given figure, or finding the area of a geometrical figure or the area under a curve (such as by a process of numerical integration)

Quadratic equation: a polynomial equation with a degree of 2 (i.e. the highest power is 2), of the form ax² + bx + c = 0, which can be solved by various methods including factoring, completing the square, graphing, Newton’s method and the quadratic formula

Pythagorean triples: groups of three positive integers a, b and c such that the a² + b² = c² equation of Pythagoras’ theorem, e.g. ( 3, 4, 5), ( 5, 12, 13), ( 7, 24, 25), ( 8, 15, 17), etc

Pythagoras’ (Pythagorean) theorem: the square of the hypotenuse of a right angled triangle is equal to the sum of the squares of the two sides (a² + b² = c²)

Probability theory: the branch of mathematics concerned with analysis of random variables and events, and with the interpretation of probabilities (the likelihood of an event happening)

Plane: a flat two-dimensional surface (physical or theoretical) with infinite width and length, zero thickness and zero curvature

Prime numbers: integers greater than 1 which are only divisible by themselves and 1

Projective geometry: a kind of non-Euclidean geometry which considers what happens to shapes when they are projected on to a non-parallel plane, e.g. a circle may be projected into an ellipse or a hyperbola

Polar coordinates: a two-dimensional coordinate system in which each point on a plane is determined by its distance r from a fixed point (e.g. the origin) and its angle θ (theta) from a fixed direction (e.g the x axis)

Polynomial: an algebraic expression or equation with more than one term, constructed from variables and constants using only the operations of addition, subtraction, multiplication and non-negative whole-number exponents, e.g. 5x² – 4x + 4y + 7

Platonic solids: the five regular convex polyhedra (symmetrical 3-dimensional shapes): the tetrahedron (made up of 4 regular triangles), the octahedron (made up of 8 triangles), the icosahedron (made up of 20 triangles), the cube (made up of 6 squares) and the dodecahedron (made up of 12 pentagons)

Permutation: a particular ordering of a set of objects, e.g. given the set {1, 2, 3}, there are six permutations: {1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, and {3, 2, 1}

Pi (π): the ratio of a circumference of a circle to its diameter, an irrational (and transcendental) number approximately equal to 3.141593…

Periodic function: a function that repeats its values in regular intervals or periods, such as the trigonometric functions of sine, cosine, tangent, etc

Perfect number: a number that is the sum of its divisors (excluding the number itself), e.g. 28 = 1 + 2 + 4 + 7 + 14

Pascal’s Triangle: a geometrical arrangement of the coefficients of the polynomial expansion of a binomial power of the form (x + y)ⁿ as a symmetrical triangle of numbers

Partial differential equation: a relation involving an unknown function with several independent variables and its partial derivatives with respect to those variables

Paradox: a statement that appears to contradict itself, suggesting a solution which is actually impossible

Parabola: a type of conic section curve, any point of which is equally distant from a fixed focus point and a fixed straight line

Ordinal numbers: an extension of the natural numbers (different from integers and from cardinal numbers) used to describe the order type of sets i.e. the order of elements within a set or series

Number theory: the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular

Number line: a line on which all points correspond to real numbers (a simple number line may only mark integers, but in theory all real numbers to +/- infinity can be shown on a number line)

Normal (Gaussian) distribution: a continuous probability distribution in probability theory and statistics that describes data which clusters around the mean in a curved “bell curve”, highest in the middle and quickly tapering off to each side

Non-Euclidean geometry: geometry based on a curved plane, whether elliptic (spherical) or hyperbolic (saddle-shaped), in which there are no parallel lines and the angles of a triangle do not sum to 180°

Non-commutative algebra: an algebra in which a x b does not always equal b x a, such as that used by quaternions

Negative numbers: any integer, ration or real number which is less than 0, e.g. -743, -1.4, -√5 (but not √-1, which is an imaginary or complex number)

Natural numbers: the set of positive integers (regular whole counting numbers), sometimes including zero

Monomial: an algebraic expression consisting of a single term (although that term could be an exponent), e.g. y = 7x, y = 2x³

Modulus: a number by which two given numbers can be divided by integer division, and produce the same remainder, e.g. 38 ÷ 12 = 3 remainder 2, and 26 ÷ 12 = 2 remainder 2, therefore 38 and 26 are congruent modulo 12, or (38 ≡ 26) mod 12

Method of exhaustion: a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape (a precursor to the methods of calculus)

Modular arithmetic: a system of arithmetic for integers, where numbers “wrap around” after they reach a certain value (the modulus), e.g. on a 12-hour clock, 15 o’clock is actually 3 o’clock (15 = 3 mod 12)

Mersenne primes: prime numbers that are one less than a power of 2, e.g. 3 (2² – 1); 7 (2³ – 1); 31 (2⁵ – 1); 127 (2⁷ – 1); 8,191 (2¹³ – 1); etc – many, but not all, Mersenne numbers are primes, e.g. 2,047 = 2¹¹ – 1 = 23 x 89, so 2,047 is a Mersenne number but not a Mersenne prime

Mersenne number: numbers that are one less than 2 to the power of a prime number, e.g. 3 (2² – 1); 7 (2³ – 1); 31 (2⁵ – 1); 127 (2⁷ – 1); 8,191 (2¹³ – 1); etc

Matrix: a rectangular array of numbers, which can be added, subtracted and multiplied, and used to represent linear transformations and vectors, solve equations, etc

Manifold: a topological space or surface which, on a small enough scale, resembles the Euclidean space of a specific dimension (called the dimension of the manifold), e.g. a line and a circle are one-dimensional manifolds; a plane and the surface of a sphere are two-dimensional manifolds; etc

Mandelbrot set: a set of points in the complex plane, the boundary of which forms a fractal, based on all the possible c points and Julia sets of a function of the form z² + c (where c is a complex parameter)

Magic square: a square array of numbers where each row, column and diagonal added up to the same total, known as the magic sum or constant (a semi-magic square is a square numbers where just the rows and columns, but not both diagonals, sum to a constant)

Logicism: the theory that mathematics is just an extension of logic, and that therefore some or all mathematics is reducible to logic

Logic: the study of the formal laws of reasoning (mathematical logic the application of the techniques of formal logic to mathematics and mathematical reasoning, and vice versa)

Logarithm: the inverse operation to exponentiation, the exponent of a power to which a base (usually 10 or e for natural logarithms) must be raised to produce a given number, e.g. because 1,000 = 10³, the log₁₀ 100 = 3

Linear regression: a technique in statistics and probability theory for modelling scattered data by assuming an approximate linear relationship between the dependent and independent variables

Line: in geometry, a one-dimensional figure following a continuous straight path joining two or more points, whether infinite in both directions or just a line segment bounded by two distinct end points

Linear equation: an algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable, and whose graph is therefore a straight line, e.g. y = 4, y = 5x + 3

Least squares method: a method of regression analysis used in probability theory and statistics to fit a curve-of-best-fit to observed data by minimizing the sum of the squares of the differences between the observed values and the values provided by the model

Limit: the point towards which a series or function converges, e.g. as x becomes closer and closer to zero, ^(sin x)⁄_x becomes closer and closer to the limit of 1

Knot theory: an area of topology that studies mathematical knots (a knot is a closed curve in space formed by interlacing a piece of “string” and joining the ends)

Julia set: the set of points for a function of the form z² + c (where c is a complex parameter), such that a small perturbation can cause drastic changes in the sequence of iterated function values and iterations will either approach zero, approach infinity or get trapped in loop

Integration: the operation in calculus (inverse to the operation of differentiation) of finding the integral of a function or equation

Irrational numbers: numbers that can not be represented as decimals (because they would contain an infinite number of non-repeating digits) or as fractions of one integer over another, e.g. π, √2, e

Integers: whole numbers, both positive (natural numbers) and negative, including zero

Integral: the area bounded by a graph or curve of a function and the x axis, between two given values of x (definite integral), found by the operation of integration

Infinity: a quantity or set of numbers without bound, limit or end, whether countably infinite like the set of integers, or uncountably infinite like the set of real numbers (represented by the symbol ∞)

Infinitesimal: quantities or objects so small that there is no way to see them or to measure them, so that for all practical purposes they approach zero as a limit (an idea used in the developement of infinitesimal calculus)

Infinite series: the sum of an infinite sequence of numbers (which are usually produced according to a certain rule, formula or algorithm)

Imaginary numbers: numbers in the form bi, where b is a real number and i is the “imaginary unit”, equal to √-1 (i.e. i² = -1)

Identity: an equality that remains true regardless of the values of any variables that appear within it, e.g. for multiplication, the identity is one; for addition, the identity is zero

Hyperbolic geometry: a non-Euclidean geometry based on a saddle-shaped plane, in which there are no parallel lines and the angles of a triangle sum to less than 180°

Hyperbola: a smooth symmetrical curve with two branches produced by the section of a conical surface

Hilbert problems: an influential list of 23 open (unsolved) problems in mathematics described by David Hilbert in 1900

Group theory: the mathematical field that studies the algebraic structures and properties of groups and the mappings between them

Group: a mathematical structure consisting of a set together with an operation that combines any two of its elements to form a third element, e.g. the set of integers and the addition operation form a group

Graph theory: a branch of mathematics focusing on the properties of a variety of graphs (meaning visual representations of data and their relationships, as opposed to graphs of functions on a Cartesian plane)

Geometry: the part of mathematics concerned with the size, shape and relative position of figures, or the study of lines, angles, shapes and their properties

Golden ratio (golden mean, divine proportion): the ratio of two quantities (equivalent to approximately 1 : 1.6180339887) where the ratio of the sum of the quantities to the larger quantity equals the ratio of the larger quantity to the smaller one, usually denoted by the Greek letter phi φ (phi)

Gaussian curvature: an intrinsic measure of the curvature of a point on a surface, dependent only on how distances are measured on the surface and not on the way it is embedded in space

Game theory: a branch of mathematics that attempts to mathematically capture behaviour in strategic situations, in which an individual’s success in making choices depends on the choices of others, with applications in the areas of economics, politics, biology, engineering, etc

Function: a relation or correspondence between two sets in which one element of the second (codomain or range) set ƒ(x) is assigned to each element of the first (domain) set x, e.g. ƒ(x) = x² or y = x² assigns a value to ƒ(x) or y based on the square of each value of x