Concept

Variable (mathematics)

Overview

In mathematics, a variable (from Latin variabilis 'changeable') is a symbol, typically a letter, that refers to an unspecified mathematical object. One says colloquially that the variable represents or denotes the object, and that any valid candidate for the object is the value of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may form a set, such as the set of real numbers.

The object may not always exist, or it might be uncertain whether any valid candidate exists or not. For example, one could represent two integers by the variables p and q and require that the value of the square of p is twice the square of q, which in algebraic notation can be written p2 = 2 q2. A definitive proof that this relationship is impossible to satisfy when p and q are restricted to nonzero integers isn't obvious, but it has been known since ancient times and has had a big influence on mathematics ever since.

Originally, the term variable was used primarily for the argument of a function, in which case its value could be thought of as varying within the domain of the function. This is the motivation for the choice of the term. Also, variables are used for denoting values of functions, such as the symbol y in the equation y = f(x), where x is the argument and f denotes the function itself.

A variable may represent an unspecified number that remains fixed during the resolution of a problem; in which case, it is often called a parameter. A variable may denote an unknown number that has to be determined; in which case, it is called an unknown; for example, in the quadratic equation ax2 + bx + c = 0, the variables a, b, c are parameters, and x is the unknown.

Sometimes the same symbol can be used to denote both a variable and a constant, that is a well defined mathematical object. For example, the Greek letter π generally represents the number π, but has also been used to denote a projection. Similarly, the letter e often denotes Euler's number, but has been used to denote an unassigned coefficient for quartic function and higher degree polynomials. Even the symbol 1 has been used to denote an identity element of an arbitrary field. These two notions are used almost identically, therefore one usually must be told whether a given symbol denotes a variable or a constant.

Variables are often used for representing matrices, functions, their arguments, sets and their elements, vectors, spaces, etc.

In mathematical logic, a variable is a symbol that either represents an unspecified constant of the theory, or is being quantified over.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

In the original texts

Passages from public-domain books where Variable (mathematics) is discussed.

Multiple parts are variable in number and in structure, perhaps arising from such parts not having been closely specialised for any particular function, so that their modifications have not been closely checked by natural selection. It follows probably from this same cause, that organic beings low in the scale are more variable than those standing higher in the scale, and which have their whole organisation more…
Charles Darwin, The Origin of Species by Means of Natural Selection
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Thus when it is worth saying that something "would be true under all circumstances," the something in question must be a propositional function, i.e. an expression containing a variable, and becoming a proposition when a value is assigned to the variable; the varying "circumstances" alluded to are then the different values of which the variable is capable. Thus if "necessary" means "what is true under all…
Bertrand Russell, Mysticism and Logic and Other Essays
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But if, on the contrary, we suppose the variables to be subjected to any fixed conditions, it will be necessary to take notice of the resulting relation between their variations, so that the number of the equations into which this general equation is then decomposed is always equal to only the number of the variables which remain truly independent. It is thus, for example, that instead of seeking for the shortest…
Auguste Comte, The philosophy of mathematics
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unguiformis_; it is also highly variable in _B. balanoides_, but under a systematic point of view this is unimportant, as the section including this latter species is well defined by the membranous basis. The porosity of the basis is in some degree variable in _B. nubilus_, _improvisus_, and _patellaris_; and in _B. flosculus_ we see signs of a passage from a calcareous to a membranous basis. Characters derived from…
Charles Darwin, A Monograph on the Sub-class Cirripedia
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Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
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Variable (mathematics)

Overview

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