variable (mathematics)

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variable (mathematics)
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{{short description|Symbol that represent an indeterminate value}}{{about||variables in computer science|Variable (computer science)|other uses|Variable (disambiguation)}}In elementary mathematics, a variable is a symbol, commonly a single letter, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown. 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.The concept of a variable is also fundamental in calculus.Typically, a function {{math|1=y = f(x)}} involves two variables, {{math|y}} and {{math|x}}, representing respectively the value and the argument of the function. The term "variable" comes from the fact that, when the argument (also called the "variable of the function") varies, then the value varies accordingly.WEB,weblink Appendix One Review of Constants and Variables, Syracuse University,, dead,weblink" title="">weblink 2014-01-16, In more advanced mathematics, a variable is a symbol that denotes a mathematical object, which could be a number, a vector, a matrix, or even a function. In this case, the original property of "variability" of a variable is not kept (except, sometimes, for informal explanations).Similarly, in computer science, a variable is a name (commonly an alphabetic character or a word) representing some value stored in computer memory. 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.


"Variable" comes from a Latin word, variābilis, with "vari(us)"' meaning "various" and "-ābilis"' meaning "-able", meaning "capable of changing".WEB,weblink "Variable" Origin,, 18 May 2015,

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".BOOK, Tabak, John, Algebra: Sets, Symbols, and the Language of Thought, 2014, Infobase Publishing, 978-0-8160-6875-3, 40,weblink en, 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.BOOK
, Fraleigh
, John B.
, A First Course in Abstract Algebra
, Addison-Wesley
, 4
, 1989
, United States
, 276
, 0-201-52821-5
In 1637, René Descartes "invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c".Tom Sorell, Descartes: A Very Short Introduction, (2000). New York: Oxford University Press. p. 19. 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 {{math|1=y = f(x)}} for a function {{math|f}}, its variable {{math|x}} and its value {{math|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 {{math|x}} varies and tends toward {{math|a}}, then {{math|f(x)}} tends toward {{math|L}}", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula
(forall epsilon >0) (exists eta >0) (forall x) ;|x-a|

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