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{{Short description|Symbol representing a mathematical object}}{{for-multi|variables in computer science|Variable (computer science)|other uses|Variable (disambiguation)}}{{Use mdy dates|date=November 2021}}In mathematics, a variable (from Latin (wikt:variabilis|variabilis), “changeable“) is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.Stover & Weisstein.Algebraic computations with variables as if they were explicit numbers solve a range of problems in a single computation. For example, the quadratic formula solves any quadratic equation by substituting the numeric values of the coefficients of that equation for the variables that represent them in the quadratic formula. In mathematical logic, a variable is either a symbol representing an unspecified term of the theory (a meta-variable), or a basic object of the theory that is manipulated without referring to its possible intuitive interpretation.

History

In ancient works such as Euclid’s Elements, single letters refer to geometric points and shapes. In the 7th century, Brahmagupta used different colours to represent the unknowns in algebraic equations in the Brāhmasphuá¹­asiddhānta. One section of this book is called “Equations of Several Colours”.{{Harvnb|Tabak|2014|page=40}}.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.{{Harvnb|Fraleigh|1989|page=276}}.In 1637, René Descartes “invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c”.{{Harvnb|Sorell|2000|page=19}}. 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.BOOK,books.google.com/books?id=moM9AQAAIAAJ, Scientific American, 1887-09-03, Munn & Company, 148, en, 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 {{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 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 {{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