Variable (mathematics)

In mathematics, a variable (from Latin variabilis, "changeable") is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.^[1]

For variables in computer science, see Variable (computer science). For other uses, see Variable (disambiguation).

Algebraic computations with variables as if they were explicit numbers solve a range of problems in a single computation. For example, the quadratic formula solves any quadratic equation by substituting the numeric values of the coefficients of that equation for the variables that represent them in the quadratic formula. In mathematical logic, a variable is either a symbol representing an unspecified term of the theory (a meta-variable), or a basic object of the theory that is manipulated without referring to its possible intuitive interpretation.

Notation[edit]

Variables are generally denoted by a single letter, most often from the Latin alphabet and less often from the Greek, which may be lowercase or capitalized. The letter may be followed by a subscript: a number (as in $x 2$ ), another variable ( $x i$ ), a word or abbreviation of a word ( $x total$ ) or a mathematical expression ( $x 2 i + 1$ ). Under the influence of computer science, some variable names in pure mathematics consist of several letters and digits. Following René Descartes (1596–1650), letters at the beginning of the alphabet such as a, b, c are commonly used for known values and parameters, and letters at the end of the alphabet such as (x, y, z) are commonly used for unknowns and variables of functions.^[6] In printed mathematics, the norm is to set variables and constants in an italic typeface.^[7]

For example, a general quadratic function is conventionally written as ${\textstyle ax^{2}+bx+c\,}$ , where a, b and c are parameters (also called constants, because they are constant functions), while x is the variable of the function. A more explicit way to denote this function is ${\textstyle x\mapsto ax^{2}+bx+c\,}$ , which clarifies the function-argument status of x and the constant status of a, b and c. Since c occurs in a term that is a constant function of x, it is called the constant term.^[8]

Specific branches and applications of mathematics have specific naming conventions for variables. Variables with similar roles or meanings are often assigned consecutive letters or the same letter with different subscripts. For example, the three axes in 3D coordinate space are conventionally called x, y, and z. In physics, the names of variables are largely determined by the physical quantity they describe, but various naming conventions exist. A convention often followed in probability and statistics is to use X, Y, Z for the names of random variables, keeping x, y, z for variables representing corresponding better-defined values.

An unknown is a variable in an which has to be solved for.

equation

An is a symbol, commonly called variable, that appears in a polynomial or a formal power series. Formally speaking, an indeterminate is not a variable, but a constant in the polynomial ring or the ring of formal power series. However, because of the strong relationship between polynomials or power series and the functions that they define, many authors consider indeterminates as a special kind of variables.

indeterminate

A is a quantity (usually a number) which is a part of the input of a problem, and remains constant during the whole solution of this problem. For example, in mechanics the mass and the size of a solid body are parameters for the study of its movement. In computer science, parameter has a different meaning and denotes an argument of a function.

parameter

Free variables and bound variables

A is a kind of variable that is used in probability theory and its applications.

random variable

a, b, c, d (sometimes extended to e, f) for parameters or

coefficients

a₀, a₁, a₂, ... for situations where distinct letters are inconvenient

a_i or u_i for the i-th term of a or the i-th coefficient of a series

sequence

e for

Euler's number

f, g, h for (as in $f(x)$ )

functions

i for the

imaginary unit

i, j, k (sometimes l or h) for varying or indices in an indexed family, or unit vectors

integers

l and w for the length and width of a figure

l also for a line, or in number theory for a prime number not equal to p

n (with m as a second choice) for a fixed integer, such as a count of objects or the of an equation

degree

p for a or a probability

prime number

q for a or a quotient

prime power

r for a , a remainder or a correlation coefficient

radius

t for

time

x, y, z for the three of a point in Euclidean geometry or the corresponding axes

Cartesian coordinates

z for a , or in statistics a normal random variable

complex number

α, β, γ, θ, φ for

angle measures

ε (with δ as a second choice) for an arbitrarily small positive number

λ for an

eigenvalue

Σ (capital sigma) for a sum, or σ (lowercase sigma) in statistics for the ^[11]