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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.
If one defines a function f from the real numbers to the real numbers by
then x is a variable standing for the argument of the function being defined, which can be any real number. In the identity
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.
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, Francois Viete'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
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.
- 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.
- , , , and commonly denote angle measures.
- usually represents an arbitrarily small positive number.
- and commonly denote two small positives.
- is used for eigenvalues.
- often denotes a sum, or, in statistics, the standard deviation.
Applied statistics 
In statistics, variables refer to measurable attributes, as these typically vary over time or between individuals. Variables can be discrete (taking values from a finite or countable set), continuous (having a continuous distribution function), or neither. Temperature is a continuous variable, while the number of legs of an animal is a discrete variable. This concept of a variable is widely used in the natural, medical, and social sciences.
In causal models, a distinction is made between "independent variables" and "dependent variables", the latter being expected to vary in value in response to changes in the former. In other words, an independent variable is presumed to potentially affect a dependent one. In experiments, independent variables include factors that can be altered or chosen by the researcher independent of other factors.
So, in an experiment to test if the boiling point of water changes with altitude, the altitude is under direct control and is the independent variable, and the boiling point is presumed to depend upon it, so being the dependent variable. The results of an experiment, or information to be used to draw conclusions, are known as data. It is often important to consider which variables to allow, or directly control or eliminate, in the design of experiments.
There are also quasi-independent variables, which are used by researchers to group things without affecting the variable itself. For example, to separate people into groups by their sex does not change whether they are male or female. Or a researcher may separate people, arbitrarily, on the amount of coffee they had drunk before beginning an experiment. The researcher cannot change the past, but can use it to split people into groups.
While independent variables can refer to quantities and qualities that are under experimental control, they can also include extraneous factors that influence results in a confusing or undesired manner. In statistics the technique to work this out is called correlation.
If strongly confounding variables exist that can substantially change the result, it makes it harder to interpret. For example, a study on cancer against age will also have to take into account variables such as income, location, stress, and lifestyle. Without considering these, the results could be grossly inaccurate deductions. Because of this, controlling unwanted variables is important in research.
See also 
- Free variables and bound variables (Bound variables are also known as dummy variables)
- Variable (programming)
- Mathematical expression
- Physical constant
- Constant of integration
- Constant term of a polynomial
- Indeterminate (variable)
- Syracuse University. "Appendix One Review of Constants and Variables". cstl.syr.edu.
- Edwards Art. 4
- Edwards Art. 5
- Edwards Art. 6
- Fraleigh, John B. (1989). A First Course in Abstract Algebra (4 ed.). United States: Addison-Wesley. p. 276. ISBN 0-201-52821-5.
- William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, The Rosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, page 71
- J. Edwards (1892). Differential Calculus. London: MacMillan and Co. pp. 1 ff.