Category: Math Dictionary

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)

Distributive property: property whereby summing two numbers and then multiplying by another number yields the same value as multiplying both values by the other value and then adding them together, e.g. a(b + c) = ab + ac

Element: a member of, or an object in, a set

Diophantine equation: a polynomial equation with integer coefficients that also allows the variables and solutions to be integers only

Differentiation: the operation in calculus (inverse to the operation of integration) of finding the derivative of a function or equation

Differential geometry: a field of mathematics that uses the methods of differential and integral calculus (as well as linear and multilinear algebra) to study the geometry of curves and surfaces

Differential equation: an equation that expresses a relationship between a function and its derivative, the solution of which is not a single value but a function (has many applications in engineering, physics economics, etc)

Deductive reasoning or logic: a type of reasoning where the truth of a conclusion necessarily follows from, or is a logical consequence of, the truth of the premises (as opposed to inductive reasoning)

Derivative: a measure of how a function or curve changes as its input changes, i.e. the best linear approximation of the function at a particular input value, as represented by the slope of the tangent line to the graph of the function at that point, found by the operation of differentiation

Decimal number: a real number which expresses fractions on the base 10 standard numbering system using place value, e.g. ³⁷⁄₁₀₀ = 0.37

Cubic equation: a polynomial having a degree of 3 (i.e. the highest power is 3), of the form ax³ + bx² + cx + d = 0, which can be solved by factorization or formula to find its three roots

Correlation: a measure of relationship between two variables or sets of data, a positive correlation coefficient indicating that one variable tends to increase or decrease as the other does, and a negative correlation coefficient indicating that one variable tends to increase as the other decreases and vice versa

Coordinate plane: a plane with two scaled perpendicular lines that intersect at the origin, usually designated x (horizontal axis) and y (vertical axis)

Coordinate: the ordered pair that gives the location or position of a point on a coordinate plane, determined by the point’s distance from the x and y axes, e.g. (2, 3.7) or (-5, 4)

Continued fraction: a fraction whose denominator contains a fraction, whose denominator in turn contains a fraction, etc, etc

Conic section: the section or curve formed by the intersection of a plane and a cone (or conical surface), depending on the angle of the plane it could be an ellipse, a hyperbola or a parabola

Congruence: two geometrical figures are congruent to one another if they have the same size and shape, and so one can be transformed into the other by a combination of translation, rotation and reflection

Composite number: a number with at least one other factor besides itself and one, i.e. not a prime number

Complex number: a number expressed as an ordered pair comprising a real number and an imaginary number, written in the form a + bi, where a and b are real numbers, and i is the imaginary unit (equal to the square root of -1)

Complex dynamics: the study of mathematical models and dynamical systems defined by iteration of functions on complex number spaces

Combinatorics: the study of different combinations and groupings of numbers, often used in probability and statistics, as well as in scheduling problems and Sudoku puzzles

Coefficients: the factors of the terms (i.e. the numbers in front of the letters) in a mathematical expression or equation, e.g. in the expression 4x + 5y² + 3z, the coefficients for x, y² and z are 4, 5 and 3 respectively

Cartesian coordinates: a pair of numerical coordinates which specify the position of a point on a plane based on its distance from the the two fixed perpendicular axes (which, with their positive and negative values, split the plane up into four quadrants)

Cardinal numbers: numbers used to measure the cardinality or size (but not the order) of sets – the cardinality of a finite set is just a natural number indicating the number of elements in the set; the sizes of infinite sets are described by transfinite cardinal numbers, r₀ (aleph-null), r₁ (aleph-one), etc

Calculus of variations: an extension of calculus used to search for a function which minimizes a certain functional (a functional is a function of a function)

Boolean algebra or logic: a type of algebra which can be applied to the solution of logical problems and mathematical functions, in which the variables are logical rather than numerical, and in which the only operators are AND, OR and NOT

Calculus (infinitesimal calculus): a branch of mathematics involving derivatives and integrals, used to study motion and changing values

Bnomial coefficients: the coefficients of the polynomial expansion of a binomial power of the form (x + y) ⁿ, which can be arranged geometrically according to the binomioal theorem as a symmetrical triangle of numbers known as Pascal’s Triangle, e.g. (x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴ the coefficients are 1, 4, 6, 4, 1

Binomial: a polynomial algebraic expression or equation with just two terms, e.g. 2x³ – 3y = 7; x² + 4x; etc

Bijection: a one-to-one comparison or correspondence of the members of two sets, so that there are no unmapped elements in either set, which are therefore of the same size and cardinality

Bell curve: the shape of the graph that indicates a normal distribution in probability and statistics

Bayesian probability: a popular interpretation of probability which evaluates the probability of a hypothesis by specifying some prior probability, and then updating in the light of new relevant data

Base n: the number of unique digits (including zero) that a positional numeral system uses to represent numbers, e.g. base 10 (decimal) uses 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 in each place value position; base 2 (binary) uses just 0 and 1; base 60 (sexagesimal, as used in ancient Mesopotamia) uses all the numbers from 0 to 59; etc

Axiom: a proposition that is not actually proved or demonstrated, but is considered to be self-evident and universally accepted as a starting point for deducing and inferring other truths and theorems, without any need of proof

Asymptote: a line that the curve of a function tends towards as the independent variable of the curve approaches some limit (usually infinity) i.e. the distance between the curve and the line approaches zero

Arithmetic: the part of mathematics that studies quantity, especially as the result of combining numbers (as opposed to variables) using the traditional operations of addition, subtraction, multiplication and division (the more advanced manipulation of numbers is usually known as number theory)

Associative property: property (which applies both to multiplication and addition) by which numbers can be added or multiplied in any order and still yield the same value, e.g. (a + b) + c = a + (b + c) or (ab)c = a(bc)

Analysis (mathematical analysis): grounded in the rigorous formulation of calculus, analysis is the branch of pure mathematics concerned with the notion of a limit (whether of a sequence or of a function)

Analytic (Cartesian) geometry: the study of geometry using a coordinate system and the principles of algebra and analysis, thus defining geometrical shapes in a numerical way and extracting numerical information from that representation

Algorithm: a step by step procedure by which an operation can be carried out

Amicable numbers: pairs of numbers for which the sum of the divisors of one number equals the other number, e.g. 220 and 284, 1184 and 1210

Algebraic equation: a combination of numbers and letters equivalent to a sentence in language, e.g. y = x² + 3x – 4

Abstract algebra: the area of modern mathematics that considers algebraic structures to be sets with operations defined on them, and extends algebraic concepts usually associated with the real number system to other more general systems, such as groups, rings, fields, modules and vector spaces

Algebra: a branch of mathematics that uses symbols or letters to represent variables, values or numbers, which can then be used to express operations and relationships and to solve equations