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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

Fractal: a self-similar geometric shape (one that appears similar at all levels of magnification) produced by an equation that undergoes repeated iterative steps or recursion

Fraction: a way of writing rational numbers (numbers that are not whole numbers), also used to represent ratios or division, in the form of a numerator over a denominator, e.g. ³⁄₅ (a unit fraction is a fraction whose numerator is 1)

Fourier series: an approximation of more complex periodic functions (such as square or saw-tooth functions) by adding together various simple trigonometric functions (e.g. sine, cosine, tangent, etc)

Formula: a rule or equation describing the relationship of two or more variables or quantities, e.g. A = πr²

Finite differences: a method of approximating the derivative or slope of a function using approximately equivalent difference quotients (the function difference divided by the point difference) for small differences

Fibonacci numbers (series): a set of numbers formed by adding the last two numbers to get the next in the series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Fermat primes: prime numbers that are one more than a power of 2 (and where the exponent is itself a power of 2), e.g. 3 (2¹ + 1), 5 (2² + 1), 17 (2⁴ + 1), 257 (2⁸ + 1), 65,537 (2¹⁶ + 1), etc

Factor: a number that will divide into another number exactly, e.g. the factors of 10 are 1, 2 and 5

Factorial: the product of all the consecutive integers up to a given number (used to give the number of permutations of a set of objects), denoted by n!, e.g. 5! = 1 x 2 x 3 x 4 x 5 = 120

Exponentiation: the mathematical operation where a number (the base) is multiplied by itself a specified number of times (the exponent), usually written as a superscript aⁿ, where a is the base and n is the exponent, e.g. 4³ = 4 x 4 x 4

Expected value: the amount predicted to be gained, using the calculation for average expected payoff, which can be calculated as the integral of a random variable with respect to its probability measure (the expected value may not actually be the most probable value and may not even exist, e.g. 2.5 children)

Empty (null) set: a set that has no members, and therefore has zero size, usually represented by {} or ø

Euclidean geometry: “normal” geometry based on a flat plane, in which there are parallel lines and the angles of a triangle sum to 180°

Elliptic geometry: a non-Euclidean geometry based (at its simplest) on a spherical plane, in which there are no parallel lines and the angles of a triangle sum to more than 180°

Ellipse: a plane curve resulting from the intersection of a cone by a plane, that looks like a slightly flattened circle (a circle is a special case of an ellipse)