Variable (mathematics)

Last updated January 18, 2025

In mathematics, a variable (from Latin variabilis , "changeable") is a symbol, typically a letter, that refers to an unspecified mathematical object.^[1]^[2]^[3] 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 $p 2 = 2 q 2$ . A definitive proof that this relationship is impossible to satisfy when $p$ and $q$ 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 $y$ in $y = f (x)$ .

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 $ax 2 + bx + c = 0$ , 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.^[4]

Variables are often used for representing matrices, functions, their arguments, sets and their elements, vectors, spaces, etc.^[5]

In mathematical logic, a variable is a symbol that either represents an unspecified constant of the theory, or is being quantified over.^[6]^[7]^[8]

History

Early history

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 1+1⁄2 times and added to 4 to make 10.^[9] In modern mathematical notation: $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠3/2⁠x + 4 = 10$ . 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.^[10]

A page from Euclid's Elements Euclid-proof-white-background.svg — 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 $a (b + c) = ab + ac$ (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.^[10]

Diophantus of Alexandria,^[11] 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.^[12] An unknown number was called $\zeta$ .^[13] The square of $\zeta$ was $\Delta ^{v}$ ; the cube was $K^{v}$ ; the fourth power was $\Delta ^{v}\Delta$ ; and the fifth power was $\Delta K^{v}$ .^[14] 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 }{\overline {\alpha }}\;\zeta {\overline {\iota }}\;\,\pitchfork \;\,\Delta ^{\upsilon }{\overline {\beta }}\;\mathrm {M} {\overline {\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".^[15] 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.^[16]

In 1637, René Descartes "invented the convention of representing unknowns in equations by $x$ , $y$ , and $z$ , and knowns by $a$ , $b$ , and $c$ ".^[17] 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.^[18]

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 $y = f (x)$ 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 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 $x$ varies and tends toward $a$ , then $f (x)$ tends toward $L$ ", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula

(\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).

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 $x 2$ ), another variable ( $x i$ ), a word or abbreviation of a word as a label ( $x total$ ) or a mathematical expression ( $x 2 i +1$ ). 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 $a$ , $b$ , $c$ are commonly used for known values and parameters, and letters at the end of the alphabet such as $x$ , $y$ , $z$ are commonly used for unknowns and variables of functions.^[19] In printed mathematics, the norm is to set variables and constants in an italic typeface.^[20]

For example, a general quadratic function is conventionally written as $ax 2 + bx + c$ , 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 $x \mapsto ax 2 + bx + c$ , which clarifies the function-argument status of $x$ and 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.^[21]

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 $x$ , $y$ , 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 better-defined values.

Conventional variable names

$a$ , $b$ , $c$ , $d$ (sometimes extended to $e$ , $f$ ) for parameters or coefficients
$a 0$ , $a 1$ , $a 2$ , ... for situations where distinct letters are inconvenient
$a i$ or $u i$ for the $i$ th term of a sequence or the $i$ th coefficient of a series
$f$ , $g$ , $h$ for functions (as in $f (x)$ )
$i$ , $j$ , $k$ (sometimes $l$ or $h$ ) for varying integers or indices in an indexed family, or unit vectors
$l$ and $w$ for the length and width of a figure
$l$ also for a line, or in number theory for a prime number not equal to $p$
$n$ (with $m$ as a second choice) for a fixed integer, such as a count of objects or the degree of a polynomial
$p$ for a prime number or a probability
$q$ for a prime power or a quotient
$r$ for a radius, a remainder or a correlation coefficient
$t$ for time
$x$ , $y$ , $z$ for the three Cartesian coordinates of a point in Euclidean geometry or the corresponding axes
$z$ for a complex number, or in statistics a normal random variable
$α$ , $β$ , $γ$ , $θ$ , $φ$ for angle measures
$ε$ (with $δ$ as a second choice) for an arbitrarily small positive number
$λ$ for an eigenvalue
$Σ$ (capital sigma) for a sum, or $σ$ (lowercase sigma) in statistics for the standard deviation ^[22]
$μ$ 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

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 $f : x \mapsto f (x)$ ", " $f$ 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 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

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.^[23]

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 $x$ .^[24]

Examples

If one defines a function $f$ 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 $i$ 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 $ax 2 + bx + c$ , 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.

Example: the ideal gas law

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

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

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 $T$ . This gives a function $P(T)={\frac {Nk_{\text{B}}T}{V}},$ where now $N$ and $V$ are also regarded as constants. Mathematically, this constitutes a partial application of the earlier function $P$ .

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

Moduli spaces

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

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

Related Research Articles

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with 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 well-formed formula consisting of two expressions related with an equals sign is an equation.

In mathematics, a polynomial is a mathematical expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate $x$ is $x 2 - 4 x + 7$ . An example with three indeterminates is $x 3 + 2 xyz 2 - yz + 1$ .

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, a quadratic equation is an equation that can be rearranged in standard form as $where the variable x represents an unknown number, and a, b, and c represent known numbers, where a \neq 0 . The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term .$

In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any expression. When the combination of variables and constants is not necessarily involved in a product, it may be called a parameter. For example, the polynomial $has coefficients 2, -1, and 3, and the powers of the variable in the polynomial have coefficient parameters,, and .$

In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions.

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

In mathematics, the term linear is used in two distinct senses for two different properties:

In mathematics, a function from a set $X$ to a set $Y$ assigns to each element of $X$ exactly one element of $Y$ . The set $X$ is called the domain of the function and the set $Y$ is called the codomain of the function.

<span class="mw-page-title-main">Quadratic function</span> Polynomial function of degree two

In mathematics, a quadratic function of a single variable is a function of the form

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form ⁠ $⁠$ to the form ⁠ $⁠$ for some values of ⁠ $⁠$ and ⁠ $⁠$ . In terms of a new quantity ⁠ $⁠$ , this expression is a quadratic polynomial with no linear term. By subsequently isolating ⁠ $⁠$ and taking the square root, a quadratic problem can be reduced to a linear problem.

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 $where a 0 (x), ..., a n (x) and b (x) are arbitrary differentiable functions that do not need to be linear, and y', ..., y (n) are the successive derivatives of an unknown function y of the variable x .$

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 expression is a written arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers, variables, operations, and functions. Other symbols include punctuation marks and brackets, used for grouping where there is not a well-defined order of operations.

In mathematics, an indeterminate or formal variable is a variable that is used purely formally in a mathematical expression, but does not stand for any value.

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.

Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.

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

In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact.

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

↑ Sobolev, S.K. (originator). "Individual variable". Encyclopedia of Mathematics. Springer. ISBN 1402006098 . Retrieved September 5, 2024. A symbol of a formal language used to denote an arbitrary element (individual) in the structure described by this language.
↑ Beckenbach, Edwin F (1982). College algebra (5th ed.). Wadsworth. ISBN 0-534-01007-5. A variable is a symbol representing an unspecified element of a given set.
↑ Landin, Joseph (1989). An Introduction to Algebraic Structures. New York: Dover Publications. p. 204. ISBN 0-486-65940-2. A variable is a symbol that holds a place for constants.
↑ "ISO 80000-2:2019" (PDF). Quantities and units, Part 2: Mathematics. International Organization for Standardization. Archived from the original on September 15, 2019. Retrieved September 15, 2019.
↑ Stover & Weisstein.
↑ van Dalen, Dirk (2008). "Logic and Structure" (PDF). Springer-Verlag (4th ed.): 57. doi:10.1007/978-3-540-85108-0. ISBN 978-3-540-20879-2.
↑ Feys, Robert; Fitch, Frederic Brenton (1969). Dictionary of symbols of mathematical logic. Amsterdam: North-Holland Pub. Co. LCCN 67030883.
↑ Shapiro, Stewart; Kouri Kissel, Teresa (2024), "Classical Logic", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved September 1, 2024
↑ Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5
1 2 Boyer, Carl B. (Carl Benjamin) (1991). A History of Mathematics. New York: Wiley. ISBN 978-0-471-54397-8.
↑ Diophantine Equations. Submitted by: Aaron Zerhusen, Chris Rakes, & Shasta Meece. MA 330-002. Dr. Carl Eberhart. 16 February 1999.
↑ Boyer (1991). "Revival and Decline of Greek Mathematics". p. 178. "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."
↑ A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg 456
↑ A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg 458
↑ Tabak 2014 , p. 40.
↑ Fraleigh 1989 , p. 276.
↑ Sorell 2000 , p. 19.
↑ Scientific American. Munn & Company. September 3, 1887. p. 148.
↑ Edwards Art. 4
↑ Hosch 2010 , p. 71 .
↑ Foerster 2006 , p. 18.
↑ Weisstein, Eric W. "Sum". mathworld.wolfram.com. Retrieved February 14, 2022.
↑ Edwards Art. 5
↑ Edwards Art. 6

Bibliography

Edwards, Joseph (1892). An Elementary Treatise on the Differential Calculus (2nd ed.). London: MacMillan and Co.
Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications (classics ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-13-165711-3.
Fraleigh, John B. (1989). A First Course in Abstract Algebra (4th ed.). United States: Addison-Wesley. ISBN 978-0-201-52821-3.
Hosch, William L., ed. (2010). The Britannica Guide to Algebra and Trigonometry. Britannica Educational Publishing. ISBN 978-1-61530-219-2.
Menger, Karl (1954). "On Variables in Mathematics and in Natural Science". The British Journal for the Philosophy of Science. 5 (18). University of Chicago Press: 134–142. doi:10.1093/bjps/V.18.134. JSTOR 685170.
Peregrin, Jaroslav (2000). "Variables in Natural Language: Where do they come from?" (PDF). In Böttner, Michael; Thümmel, Wolf (eds.). Variable-Free Semantics. Osnabrück Secolo. pp. 46–65. ISBN 978-3-929979-53-4.
Quine, Willard V. (1960). "Variables Explained Away" (PDF). Proceedings of the American Philosophical Society. 104 (3). American Philosophical Society: 343–347. JSTOR 985250.
Sorell, Tom (2000). Descartes: A Very Short Introduction. New York: Oxford University Press. ISBN 978-0-19-285409-4.
Stover, Christopher; Weisstein, Eric W. "Variable". In Weisstein, Eric W. (ed.). Wolfram MathWorld. Wolfram Research. Retrieved November 22, 2021.
Tabak, John (2014). Algebra: Sets, Symbols, and the Language of Thought. Infobase Publishing. ISBN 978-0-8160-6875-3.

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[1] Sobolev, S.K. (originator). "Individual variable". Encyclopedia of Mathematics. Springer. ISBN 1402006098 . Retrieved September 5, 2024. A symbol of a formal language used to denote an arbitrary element (individual) in the structure described by this language.

[2] Beckenbach, Edwin F (1982). College algebra (5th ed.). Wadsworth. ISBN 0-534-01007-5. A variable is a symbol representing an unspecified element of a given set.

[3] Landin, Joseph (1989). An Introduction to Algebraic Structures. New York: Dover Publications. p. 204. ISBN 0-486-65940-2. A variable is a symbol that holds a place for constants.

[80000-2:2019-4] "ISO 80000-2:2019" (PDF). Quantities and units, Part 2: Mathematics. International Organization for Standardization. Archived from the original on September 15, 2019. Retrieved September 15, 2019.

[5] Stover & Weisstein.

[6] van Dalen, Dirk (2008). "Logic and Structure" (PDF). Springer-Verlag (4th ed.): 57. doi:10.1007/978-3-540-85108-0. ISBN 978-3-540-20879-2.

[7] Feys, Robert; Fitch, Frederic Brenton (1969). Dictionary of symbols of mathematical logic. Amsterdam: North-Holland Pub. Co. LCCN 67030883.

[8] Shapiro, Stewart; Kouri Kissel, Teresa (2024), "Classical Logic", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved September 1, 2024

[Clagett-9] Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5

[:0-10] 1 2 Boyer, Carl B. (Carl Benjamin) (1991). A History of Mathematics. New York: Wiley. ISBN 978-0-471-54397-8.

[11] Diophantine Equations. Submitted by: Aaron Zerhusen, Chris Rakes, & Shasta Meece. MA 330-002. Dr. Carl Eberhart. 16 February 1999.

[Boyer2-12] Boyer (1991). "Revival and Decline of Greek Mathematics". p. 178. "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."

[13] A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg 456

[14] A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg 458

[15] Tabak 2014 , p. 40.

[16] Fraleigh 1989 , p. 276.

[17] Sorell 2000 , p. 19.

[18] Scientific American. Munn & Company. September 3, 1887. p. 148.

[E004-19] Edwards Art. 4

[20] Hosch 2010 , p. 71 .

[21] Foerster 2006 , p. 18.

[22] Weisstein, Eric W. "Sum". mathworld.wolfram.com. Retrieved February 14, 2022.

[23] Edwards Art. 5

[24] Edwards Art. 6

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