Mathematical variables

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and its applications. A "constant", in this context, should not be confused with a mathematical constant, which is a specific number independent of the scope of the given problem.

Dependent and independent variables
Variables are further distinguished as being either a dependent variable or an independent variable. Independent variables are regarded as inputs to a system and may take on different values freely. Dependent variables are those values that change as a consequence of changes in other values in the system.

When one value is completely determined by another or several others, then it is called a function of the other value or values. In this case the value of the function is a dependent variable and the other values are independent variables. The notation f(x) is used for the value of the function f with x representing the independent variable. Similarly, notation such as f(x, y, z) may be used when there are several independent variables that are not the same.

What it means for a variable to vary
Varying, in the context of mathematical variables, does not mean change in the course of time, but rather dependence on the context in which the variable is used. This can be the immediate context of the expression in which the variable occurs, as in the case of summation variables or variables that designate the argument of a function being defined. The context can also be larger, for instance when a variable is used to designate a value occurring in a hypothesis of the discussion at hand. In some cases nothing varies at all, and alternative names can be used instead of "variable": a parameter is a value that is fixed in the statement of the problem being studied (although its value may not be explicitly known), an unknown is a variable that is introduced to stand for a constant value that is not initially known, but which may become known by solving some equation(s) for it, and an indeterminate is a symbol that need not stand for anything else but is an abstract value in itself. In all these cases the term "variable" is often still used because the rules for the manipulation of these symbols are the same.

Examples
If one defines a function f from the real numbers to the real numbers by


 * $$f(x) = x^2+\sin(x+4)\ $$

then x is a variable standing for the argument of the function being defined, which can be any real number. In the identity


 * $$\sum_{i=1}^n i = \frac{n^2+n}2\ $$

the variable i is a summation variable which designates in turn each of the integers 1, 2, ..., n (it is also called index because its variation is over a discrete set of values) while n is a parameter (it does not vary within the formula).

In the theory of polynomials, a polynomial of degree 2 is generally denoted as ax2 + bx + c, where a, b and c are called coefficients (they are assumed to be fixed, i.e., parameters of the problem considered) while x is called a variable. When studying this polynomial for its polynomial function this x stands for the function argument. When studying the polynomial as an object in itself, x is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status.

Formulas from physics such as E = mc2 or PV = nRT (the ideal gas law) do not involve the mathematical notion of a variable, because the quantities E, m, P, V, n, and T are instead used to designate certain properties (energy, mass, pressure, volume, quantity, temperature) of the physical system.

Notation
In mathematics, single-symbol names for variables are the norm. Following the 17th century French philosopher and mathematician, René Descartes, letters at the beginning of the alphabet, e.g. a, b, c are commonly used for constants and letters at the end of the alphabet, e.g. x, y, z, and t are commonly used for variables. Earlier, François Viète 's convention was to use consonants as known constants and vowels for unknown quantities. In printed mathematics, variables and constants are usually set in an italic typeface.

Specific branches and applications of mathematics usually have specific naming conventions for variables. Variables with similar roles or meanings are often assigned consecutive letters. For example, the three axes in 3D coordinate space are conventionally called x, y, and z, while random variables in statistics are usually named X, Y, 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, with these being replaced by x, y, z for observations or sample outcomes of those random variables. These latter (lower case) symbols are ordinary mathematical variables. The former (upper case) symbols actually stand for functions from the sample space (set of atomic outcomes) of the experiment to (typically) the real numbers. Another convention sometimes used in statistics is to denote population values of particular statistics by lower (or upper) case Greek letters, with sample-based estimates of those quantities being denoted by the corresponding lower (or upper) case letters from the ordinary alphabet.

General introduction
Variables are used in open sentences. For instance, in the formula x + 1 = 5, x is a variable which represents an "unknown" number. Variables are often represented by Greek or Roman letters and may be used with other special symbols.

In mathematics, variables are essential because they allow quantitative relationships to be stated in a general way. If we were forced to use actual values, then the relationships would only apply in a more narrow set of situations. For example:


 * State a mathematical definition for finding the number twice that of ANY other finite number:


 * 2(x) = x + x or x * 2


 * Now, all we need to do to find the double of a number is replace x with any number we want.


 * 2(1) = 1 + 1 = 2 or 1 * 2
 * 2(3) = 3 + 3 = 6 or 3 * 2
 * 2(55) = 55 + 55 = 110 or 55 * 2
 * etc.

So in this example, the variable x is a "placeholder" for any number—that is to say, a variable. One important thing we assume is that the value of x does not change, even though we do not know what x is. But in some algorithms, obviously, will change x, and there are various ways to then denote if we mean its old or new value—again, generally not knowing either, but perhaps (for example) that one is less than the other.

Naming conventions
Mathematics has many conventions. Below are some of the more common ones. Many of the symbols have other conventional uses, but they may actually represent a constant or a specific function rather than a variable.


 * a, b, c, and d (sometimes extended to e and f) usually play similar roles or are made to represent parallel notions in a mathematical context. They often represent constants or coefficients, for example in a polynomial or an equation, which are not completely specified.
 * a0, a1, a2, ... play a similar role, when otherwise too many different letters would be needed.
 * f and g (sometimes h) commonly denote functions.
 * i, j, and k (sometimes l or h) are often used to denote varying integers or indices in an indexed family.
 * ai is often used to denote the i-th term of a sequence.
 * l and w are often used to represent the length and width of a figure.
 * m and n usually denote integers and usually play similar roles or are made to represent parallel notions in a mathematical context, such a pair of dimensions.
 * n commonly denotes a fixed integer like a count of objects or the degree of an equation.
 * p, q, and r usually play similar roles or are made to represent parallel notions in a mathematical context.
 * p and q often denote prime numbers or relatively prime numbers, or, in statistics, probabilities.
 * r often denotes a remainder or a modulus.
 * r, s, and t usually play similar roles or are made to represent parallel notions in a mathematical context.
 * u and v usually play similar roles or are made to represent parallel notions in a mathematical context, such as denoting a vertex (graph theory).
 * w, x, y, and z usually play similar roles or are made to represent parallel notions in a mathematical context, such as representing unknowns in an equation.
 * x, y and z usually denote the three Cartesian coordinates of a point in Euclidean geometry. By extension, they are used to name the corresponding axes.
 * z typically denotes a complex number, or, in statistics, a normal random variate.
 * $$\alpha$$, $$\beta$$, $$\gamma$$, $$\theta$$ and $$\phi$$ commonly denote angle measures.
 * $$\epsilon$$ usually represents an arbitrarily small positive number.
 * $$\epsilon$$ and $$\delta$$ commonly denote two small positives.
 * $$\lambda$$ is used for eigenvalues.
 * $$\sigma$$ often denotes a sum, or, in statistics, the standard deviation.