# Variable (mathematics)

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In mathematics, a variable is a symbol which works as a placeholder for expression or quantities that may vary or change; is often used to represent the argument of a function or an arbitrary element of a set. In addition to numbers, variables are commonly used to represent vectors, matrices and functions. [1] [2]

## Contents

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 (i.e., meta-variable), or a basic object of the theory—which is manipulated without referring to its possible intuitive interpretation.

## Etymology

"Variable" comes from a Latin word, variābilis, with "vari(us)"' meaning "various" and "-ābilis"' meaning "-able", meaning "capable of changing". [3]

## Genesis and evolution of the concept

In the 7th century, 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". [4]

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. [5]

In 1637, René Descartes "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c". [6] 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. Almost a century later, Leonhard Euler fixed the terminology of infinitesimal calculus, and introduced the notation y = f(x) for a function f, its variablex 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 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 variablex varies and tends toward a, then f(x) tends toward L", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula

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

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).

## 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

${\displaystyle ax^{3}+bx^{2}+cx+d=0,}$

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: xf(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, and the identity element of a group.

Other specific names for variables are:

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

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.

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. [7]

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 variablex. [8]

### Examples

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

${\displaystyle f(x)=x^{2}+\sin(x+4)}$

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

In the identity

${\displaystyle \sum _{i=1}^{n}i={\frac {n^{2}+n}{2}}}$

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.

## Notation

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. [9] In printed mathematics, the norm is to set variables and constants in an italic typeface. [10]

For example, a general quadratic function is conventionally written as:

${\displaystyle ax^{2}+bx+c\,,}$

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

${\displaystyle x\mapsto ax^{2}+bx+c\,,}$

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. [11] :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.

## Bibliography

• 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 Science5:18:134–142 (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 Society104:343–347 (1960).

## Related Research Articles

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the “imaginary unit”, satisfying the equation i2 = -1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any equality is an equation.

In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2yz + 1.

A parameter, generally, is any characteristic that can help in defining or classifying a particular system. That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc.

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and determines various properties of the roots. It is generally defined as a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. It is often denoted by the symbol .

In mathematics, the term linear function refers to two distinct but related notions:

In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

In mathematics, an implicit equation is a relation of the form R(x1,…, xn) = 0, where R is a function of several variables. For example, the implicit equation of the unit circle is x2 + y2 − 1 = 0.

In mathematics, a rational function is any function which can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L.

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form

In mathematics, to solve an equation is to find its solutions, which are the values that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set.

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

In mathematics, and/or particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, does not stand for anything else except itself, and is often used as a placeholder in objects such as polynomials and formal power series. In particular:

Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities. Those Greek letters which have the same form as Latin letters are rarely used: capital A, B, E, Z, H, I, K, M, N, O, P, T, Y, X. Small ι, ο and υ are also rarely used, since they closely resemble the Latin letters i, o and u. Sometimes font variants of Greek letters are used as distinct symbols in mathematics, in particular for ε/ϵ and π/ϖ. The archaic letter digamma (Ϝ/ϝ/ϛ) is sometimes used.

Many letters of the Latin alphabet, both capital and small, are used in mathematics, science, and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, or physical entities. Certain letters, when combined with special formatting, take on special meaning.

In mathematics, a surface is a generalization of a plane, which is not necessarily flat – that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line. There are many more precise definitions, depending on the context and the mathematical tools that are used to analyze the surface.

In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance ; as a noun, it has two different meanings:

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

## References

1. "Compendium of Mathematical Symbols: Variables". Math Vault. 2020-03-01. Retrieved 2020-08-09.
2. Weisstein, Eric W. "Variable". mathworld.wolfram.com. Retrieved 2020-08-09.
3. ""Variable" Origin". dictionary.com. Archived from the original on 20 May 2015. Retrieved 18 May 2015.
4. Tabak, John (2014). Algebra: Sets, Symbols, and the Language of Thought. Infobase Publishing. p. 40. ISBN   978-0-8160-6875-3.
5. Fraleigh, John B. (1989). A First Course in Abstract Algebra (4 ed.). United States: Addison-Wesley. p. 276. ISBN   0-201-52821-5.
6. Tom Sorell, Descartes: A Very Short Introduction, (2000). New York: Oxford University Press. p. 19.
7. Edwards Art. 5
8. Edwards Art. 6
9. Edwards Art. 4
10. 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
11. Foerster, Paul A. (2006). (Classics ed.). Upper Saddle River, NJ: Prentice Hall. ISBN   0-13-165711-9.