Variable (mathematics)

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.

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.

Sometimes the same symbol can be used to denote both a variable and a constant, that is a well defined mathematical object.

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] 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).

[10] In works of ancient greece such as Euclid's Elements (c. 300 BC), mathematics was described gemoetrically.

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.

[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.

would be written in Diophantus's syncopated notation as: In the 7th century BC, Brahmagupta used different colours to represent the unknowns in algebraic equations in the Brāhmasphuṭasiddhānta.

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.

[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".

[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.

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".

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).

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.

Under the influence of computer science, some variable names in pure mathematics consist of several letters and digits.

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

It is common for variables to play different roles in the same mathematical formula, and names or qualifiers have been introduced to distinguish them.

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.

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.

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.

Considering constants and variables can lead to the concept of moduli spaces.

The set of points (x, y) in the 2D plane satisfying this equation trace out the graph of a parabola.