Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation. A typical example is the quadratic formula, which allows one to solve every quadratic equation by simply substituting the numeric values of the coefficients of the given equation for the variables that represent them.
In mathematical logic, a variable is either a symbol representing an unspecified term of the theory, or a basic object of the theory, which is manipulated without referring to its possible intuitive interpretation.
Genesis and evolution of the concept
At the end of the 16th century François Viète introduced the idea of representing known and unknown numbers by letters, nowadays called variables, and of computing with them as if they were numbers, in order to obtain the result by a simple replacement. Viète's convention was to use consonants for known values and vowels for unknowns.
In 1637, René Descartes "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c". Contrarily to Viète's convention, Descartes' is still commonly in use.
Starting in the 1660s, Isaac Newton and Gottfried Wilhelm Leibniz independently developed the infinitesimal calculus, which essentially consists of studying how an infinitesimal variation of a variable quantity induces a corresponding variation of another quantity which is a function of the first variable (quantity). Almost a century later Leonhard Euler fixed the terminology of infinitesimal calculus and introduced the notation y = f(x) for a function f, its variable x and its value y. Until the end of the 19th century, the word variable referred almost exclusively to the arguments and the values of functions.
In the second half of the 19th century, it appeared that the foundation of infinitesimal calculus was not formalized enough to deal with apparent paradoxes such as a continuous function which is nowhere differentiable. To solve this problem, Karl Weierstrass introduced a new formalism consisting of replacing the intuitive notion of limit by a formal definition. The older notion of limit was "when the variable x varies and tends toward a, then f(x) tends toward L", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula
in which none of the five variables is considered as varying.
This static formulation led to the modern notion of variable which is simply a symbol representing a mathematical object which either is unknown or may be replaced by any element of a given set; for example, the set of real numbers.
Specific kinds of variables
It is common for variables to play different roles in the same mathematical formula and names or qualifiers have been introduced to distinguish them. For example, the general cubic equation
is interpreted as having five variables: four, a, b, c, d, which are taken to be given numbers and the fifth variable, x, is understood to be an unknown number. To distinguish them, the variable x is called an unknown, and the other variables are called parameters or coefficients, or sometimes constants, although this last terminology is incorrect for an equation and should be reserved for the function defined by the left-hand side of this equation.
In the context of functions, the term variable refers commonly to the arguments of the functions. This is typically the case in sentences like "function of a real variable", "x is the variable of the function f: x ↦ f(x)", "f is a function of the variable x" (meaning that the argument of the function is referred to by the variable x).
In the same context, variables that are independent of x define constant functions and are therefore called constant. For example, a constant of integration is an arbitrary constant function that is added to a particular antiderivative to obtain the other antiderivatives. Because the strong relationship between polynomials and polynomial function, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.
This use of "constant" as an abbreviation of "constant function" must be distinguished from the normal meaning of the word in mathematics. A constant, or mathematical constant is a well and unambiguously defined number or other mathematical object, as, for example, the numbers 0, 1, π and the identity element of a group.
Other specific names for variables are:
- An unknown is a variable in an equation which has to be solved for.
- An indeterminate 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.
- A parameter 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.
- Free variables and bound variables
- A random variable is a kind of variable that is used in probability theory and its applications.
Dependent and independent variables
In calculus and its application to physics and other sciences, it is rather common to consider a variable, say y, whose possible values depend on the value of another variable, say x. In mathematical terms, the dependent variable y represents the value of a function of x. To simplify formulas, it is often useful to use the same symbol for the dependent variable y and the function mapping x onto y. For example, the state of a physical system depends on measurable quantities such as the pressure, the temperature, the spatial position, ..., and all these quantities vary when the system evolves, that is, they are function of the time. In the formulas describing the system, these quantities are represented by variables which are dependent on the time, and thus considered implicitly as functions of the time.
The property of a variable to be dependent or independent depends often of the point of view and is not intrinsic. For example, in the notation f(x, y, z), the three variables may be all independent and the notation represents a function of three variables. On the other hand, if y and z depend on x (are dependent variables) then the notation represents a function of the single independent variable x.
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.
In mathematics, the variables are generally denoted by a single letter. However, this letter is frequently followed by a subscript, as in x2, and this subscript may be a number, another variable (xi), a word or the abbreviation of a word (xin and xout), and even a mathematical expression. Under the influence of computer science, one may encounter in pure mathematics some variable names consisting in several letters and digits.
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 known values and parameters, and letters at the end of the alphabet, e.g. x, y, z, and t are commonly used for unknowns and variables of functions. In printed mathematics, the norm is to set variables and constants in an italic typeface.
For example, a general quadratic function is conventionally written as:
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
which makes the function-argument status of x clear, and thereby implicitly 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.:18
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. 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 actual values.
There are many other notational usages. Usually, variables that play a similar role are represented by consecutive letters or by the same letter with different subscript. Below are some of the most common usages.
- a, b, c, and d (sometimes extended to e and f) often represent parameters or coefficients.
- a0, a1, a2, ... play a similar role, when otherwise too many different letters would be needed.
- ai or ui is often used to denote the i-th term of a sequence or the i-th coefficient of a series.
- 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. They may also be used to denote unit vectors.
- l and w are often used to represent the length and width of a figure.
- l is also used to denote a line. In number theory, l often denotes a prime number not equal to p.
- n usually denotes a fixed integer, such as a count of objects or the degree of an equation.
- When two integers are needed, for example for the dimensions of a matrix, one uses commonly m and n.
- p often denotes a prime numbers or a probability.
- q often denotes a prime power or a quotient
- r often denotes a radius, a remainder or a correlation coefficient.
- t often denotes time.
- 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 variable.
- α, β, γ, θ and φ commonly denote angle measures.
- ε usually represents an arbitrarily small positive number.
- ε and δ commonly denote two small positives.
- λ is used for eigenvalues.
- σ often denotes a sum, or, in statistics, the standard deviation.
- J. Edwards (1892). Differential Calculus. London: MacMillan and Co. pp. 1 ff.
- Karl Menger, "On Variables in Mathematics and in Natural Science", The British Journal for the Philosophy of Science 5:18:134–142 (August 1954) JSTOR 685170
- Jaroslav Peregrin, "Variables in Natural Language: Where do they come from?", in M. Boettner, W. Thümmel, eds., Variable-Free Semantics, 2000, pp. 46–65.
- W.V. Quine, "Variables Explained Away", Proceedings of the American Philosophical Society 104:343–347 (1960).
- ""Variable" Origin". dictionary.com. Retrieved 18 May 2015.
- Tabak, John (2014). Algebra: Sets, Symbols, and the Language of Thought. Infobase Publishing. p. 40. ISBN 978-0-8160-6875-3.
- Fraleigh, John B. (1989). A First Course in Abstract Algebra (4 ed.). United States: Addison-Wesley. p. 276. ISBN 0-201-52821-5.
- Tom Sorell, Descartes: A Very Short Introduction, (2000). New York: Oxford University Press. p. 19.
- Edwards Art. 5
- Edwards Art. 6
- Edwards Art. 4
- William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, The Rosen Publishing Group, 2010, ISBN 1-61530-219-0, 978-1-61530-219-2, p. 71
- Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-165711-9.