Variable (mathematics)

Template:Short description Template:Distinguish Template:Use mdy dates 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 Template:Mvar and Template:Mvar and require that the value of the square of Template:Mvar is twice the square of Template:Mvar, which in algebraic notation can be written Template:Math. A definitive proof that this relationship is impossible to satisfy when Template:Mvar and Template:Mvar are restricted to integer numbers 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 can vary in 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 Template:Math in Template:Math.

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 Template:Math, the variables Template:Math, Template:Math, Template:Math are parameters, and Template:Math 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 Template:Math generally represents the number [[Pi|Template:Math]], but has also been used to denote a projection. Similarly, the letter Template:Math often denotes Euler's number, but has been used to denote an unassigned coefficient for quartic function and higher degree polynomials. Even the symbol Template:Math 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.

History

Early history

Rhind Mathematical Papyrus

The earliest uses of an "unknown quantity" date back to at least the Ancient Egyptians with the Moscow Mathematical Papyrus (c. 1500 BC) which described problems with unknowns rhetorically, called the "Aha problems". The "Aha problems" involve finding unknown quantities (referred to as aha, "stack") if the sum of the quantity and part(s) of it are given (The Rhind Mathematical Papyrus also contains four of these type of problems). For example, problem 19 asks one to calculate a quantity taken Template:Frac times and added to 4 to make 10. In modern mathematical notation: Template:Math. Around the same time in Mesopotamia, mathematics of the Old Babylonian period (c. 2000 BC – 1500 BC) was more advanced, also studying quadratic and cubic equations.

A page from Euclid's Elements

In works of ancient greece such as Euclid's Elements (c. 300 BC), mathematics was described gemoetrically. For example, The Elements, proposition 1 of Book II, Euclid includes the proposition:

"If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments."

This corresponds to the algebraic identity Template:Math (distributivity), but is described entirely geometrically. Euclid, and other greek geometers, also used single letters refer to geometric points and shapes. This kind of algebra is now sometimes called Greek geometric algebra.

Diophantus of Alexandria, pioneered a form of syncopated algebra in his Arithmetica (c. 200 AD), which introduced symbolic manipulation of expressions with unknowns and powers, but without modern symbols for relations (such as equality or inequality) or exponents. An unknown number was called $\zeta$. The square of $\zeta$ was ${\mathrm{\Delta }}^v$; the cube was $K^v$; the fourth power was ${\mathrm{\Delta }}^v\mathrm{\Delta }$; and the fifth power was $\mathrm{\Delta }{K}^v$. So for example, what would be written in modern notation as: $$x^3-2x^2+10x-1,$$ would be written in Diophantus's syncopated notation as:

${\mathrm{K}}^{\upsilon }\bar{\alpha }\zeta \bar{\iota }⋔{\mathrm{\Delta }}^{\upsilon }\bar{\beta }\mathrm{M}\bar{\alpha }$

In the 7th century BC, Brahmagupta used different colours to represent the unknowns in algebraic equations in the Brāhmasphuṭasiddhānta. One section of this book is called "Equations of Several Colours". Greek and other ancient mathematical advances, were often trapped in long periods of stagnation, and so there were few revolutions in notation, but this began to change by the early modern period.

Early modern period

At the end of the 16th century, François Viète introduced the idea of representing known and unknown numbers by letters, nowadays called variables, and the idea 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 Template:Math, Template:Math, and Template:Math, and knowns by Template:Math, Template:Math, and Template:Math". Contrarily to Viète's convention, Descartes' is still commonly in use. The history of the letter x in math was discussed in an 1887 Scientific American article.

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 time-varying quantity, called a Fluent, induces a corresponding variation of another quantity which is a function of the first variable. Almost a century later, Leonhard Euler fixed the terminology of infinitesimal calculus, and introduced the notation Template:Math for a function Template:Math, its variable Template:Math and its value Template:Math. 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 nowhere differentiable continuous function. 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 Template:Math varies and tends toward Template:Math, then Template:Math tends toward Template:Math", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula

$(\mathrm{\forall }ϵ>0)(\mathrm{\exists }\eta >0)(\mathrm{\forall }x)|x-a|<\eta$$\Rightarrow |L-f(x)|<ϵ,$

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 that either is unknown, or may be replaced by any element of a given set (e.g., the set of real numbers).

Notation

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 Template:Math), another variable (Template:Math), a word or abbreviation of a word as a label (Template:Math) or a mathematical expression (Template:Math). 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 Template:Math, Template:Math, Template:Math are commonly used for known values and parameters, and letters at the end of the alphabet such as Template:Math, Template:Math, Template:Math 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 Template:Math, where Template:Math, Template:Math and Template:Math are parameters (also called constants, because they are constant functions), while Template:Math is the variable of the function. A more explicit way to denote this function is Template:Math, which clarifies the function-argument status of Template:Math and the constant status of Template:Math, Template:Math and Template:Math. Since Template:Math occurs in a term that is a constant function of Template:Math, it is called the constant term.

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 Template:Math, Template:Math, and Template:Math. 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 Template:Math, Template:Math, Template:Math for the names of random variables, keeping Template:Math, Template:Math, Template:Math for variables representing corresponding better-defined values.

Conventional variable names

• Template:Math, Template:Math, Template:Math, Template:Math (sometimes extended to Template:Math, Template:Math) for parameters or coefficients
• Template:Math, Template:Math, Template:Math, ... for situations where distinct letters are inconvenient
• Template:Math or Template:Math for the Template:Mathth term of a sequence or the Template:Mathth coefficient of a series
• Template:Math, Template:Math, Template:Math for functions (as in Template:Math)
• Template:Math, Template:Math, Template:Math (sometimes Template:Math or Template:Math) for varying integers or indices in an indexed family, or unit vectors
• Template:Math and Template:Math for the length and width of a figure
• Template:Math also for a line, or in number theory for a prime number not equal to Template:Math
• Template:Math (with Template:Math as a second choice) for a fixed integer, such as a count of objects or the degree of a polynomial
• Template:Math for a prime number or a probability
• Template:Math for a prime power or a quotient
• Template:Math for a radius, a remainder or a correlation coefficient
• Template:Math for time
• Template:Math, Template:Math, Template:Math for the three Cartesian coordinates of a point in Euclidean geometry or the corresponding axes
• Template:Math for a complex number, or in statistics a normal random variable
• Template:Math, Template:Math, Template:Math, Template:Math, Template:Math for angle measures
• Template:Math (with Template:Math as a second choice) for an arbitrarily small positive number
• Template:Math for an eigenvalue
• Template:Math (capital sigma) for a sum, or Template:Math (lowercase sigma) in statistics for the standard deviation
• Template:Math for a mean

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

$a{x}^3+b{x}^2+cx+d=0,$

is interpreted as having five variables: four, Template:Math, which are taken to be given numbers and the fifth variable, Template:Math is understood to be an unknown number. To distinguish them, the variable Template:Math 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", "Template:Math is the variable of the function Template:Math", "Template:Math is a function of the variable Template:Math" (meaning that the argument of the function is referred to by the variable Template:Math).

In the same context, variables that are independent of Template:Math 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 of the strong relationship between polynomials and polynomial functions, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.

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.

All these denominations of variables are of semantic nature, and the way of computing with them (syntax) is the same for all.

Dependent and independent variables

Template:Main In calculus and its application to physics and other sciences, it is rather common to consider a variable, say Template:Math, whose possible values depend on the value of another variable, say Template:Math. In mathematical terms, the dependent variable Template:Math represents the value of a function of Template:Math. To simplify formulas, it is often useful to use the same symbol for the dependent variable Template:Math and the function mapping Template:Math onto Template:Math. 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.

Therefore, in a formula, a dependent variable is a variable that is implicitly a function of another (or several other) variables. An independent variable is a variable that is not dependent.

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 Template:Math, the three variables may be all independent and the notation represents a function of three variables. On the other hand, if Template:Math and Template:Math depend on Template:Math (are dependent variables) then the notation represents a function of the single independent variable Template:Math.

Examples

If one defines a function Template:Math 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 Template:Math is a summation variable which designates in turn each of the integers Template:Math (it is also called index because its variation is over a discrete set of values) while Template:Math is a parameter (it does not vary within the formula).

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

Example: the ideal gas law

Consider the equation describing the ideal gas law, $$PV=N{k}_{\text{B}}T.$$ This equation would generally be interpreted to have four variables, and one constant. The constant is Template:Math}, the Boltzmann constant. One of the variables, Template:Math, the number of particles, is a positive integer (and therefore a discrete variable), while the other three, Template:Math, Template:Math and Template:Math, for pressure, volume and temperature, are continuous variables.

One could rearrange this equation to obtain Template:Math as a function of the other variables, $$P(V,N,T)=\frac{N{k}_{\text{B}}T}{V}.$$ Then Template:Math, as a function of the other variables, is the dependent variable, while its arguments, Template:Math, Template:Math and Template:Math, are independent variables. One could approach this function more formally and think about its domain and range: in function notation, here Template:Math is a function $P:{ℝ}_{>0}\times \mathbb{N}\times {ℝ}_{>0}\to ℝ$.

However, in an experiment, in order to determine the dependence of pressure on a single one of the independent variables, it is necessary to fix all but one of the variables, say Template:Math. This gives a function $$P(T)=\frac{N{k}_{\text{B}}T}{V},$$ where now Template:Math and Template:Math are also regarded as constants. Mathematically, this constitutes a partial application of the earlier function Template:Math.

This illustrates how independent variables and constants are largely dependent on the point of view taken. One could even regard Template:Math as a variable to obtain a function $$P(V,N,T,k_{\text{B}})=\frac{N{k}_{\text{B}}T}{V}.$$

Moduli spaces

Considering constants and variables can lead to the concept of moduli spaces. For illustration, consider the equation for a parabola, $$y=a{x}^2+bx+c,$$ where Template:Math, Template:Math, Template:Math, Template:Math and Template:Math are all considered to be real. The set of points Template:Math in the 2D plane satisfying this equation trace out the graph of a parabola. Here, Template:Math, Template:Math and Template:Math are regarded as constants, which specify the parabola, while Template:Math and Template:Math are variables.

Then instead regarding Template:Math, Template:Math and Template:Math as variables, we observe that each set of 3-tuples Template:Math corresponds to a different parabola. That is, they specify coordinates on the 'space of parabolas': this is known as a moduli space of parabolas.

See also

• Lambda calculus
• Observable variable
• Physical constant
• Propositional variable

Bibliography

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