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.
algebraic expression: a combination of numbers and letters equivalent to a phrase in language, e.g. x 2 + 3 x – 4.
algebraic equation: a combination of numbers and letters equivalent to a sentence in language, e.g. y = x 2 + 3 x – 4.
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.
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.
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).
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 ).
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.
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.
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.
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.
bell curve: the shape of the graph that indicates a normal distribution in probability and statistics.
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.
binomial: a polynomial algebraic expression or equation with just two terms, e.g. 2 x 3 – 3 y = 7; x 2 + 4 x ; etc.
binomial 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 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 xy 3 + y 4 the coefficients are 1, 4, 6, 4, 1.
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.
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).
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, 0 (aleph-null), 1 (aleph-one), etc.
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).
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 4 x + 5 y 2 + 3 z , the coefficients for x , y 2 and z are 4, 5 and 3 respectively.
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 .
complex dynamics: the study of mathematical models and dynamical systems defined by iteration of functions on complex number spaces.
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).
composite number: a number with at least one other factor besides itself and one, i.e. not a prime number.
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.
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.
continued fraction: a fraction whose denominator contains a fraction, whose denominator in turn contains a fraction, etc, etc.
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).
coordinate plane: a plane with two scaled perpendicular lines that intersect at the origin, usually designated x (horizontal axis) and y (vertical axis).
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.
cubic equation: a polynomial having a degree of 3 (i.e. the highest power is 3), of the form ax 3 + bx 2 + cx + d = 0, which can be solved by factorization or formula to find its three roots.