Variables began as unknown heaps in Egyptian papyri
A mathematical variable is a symbol—usually a letter—standing for an unspecified object that may not even exist. Latin variabilis means changeable, echoing early use as a function's moving argument. Parameters stay fixed through a problem; unknowns wait to be solved.
The Egyptian aha problems, whose name means stack, asked for a quantity when it plus some fraction of itself was given; problem 19 of the Moscow Papyrus amounts to three halves of x plus 4 equals 10, and the Rhind Papyrus holds four more. Old Babylonian scribes of the same era were already handling quadratic and cubic equations. Euclid, by contrast, stated the distributive law entirely as geometry, about rectangles built on cut lines, though Greek geometers did label points and shapes with single letters.
Diophantus, around 200 AD, abbreviated unknowns and powers in his Arithmetica but lacked symbols for equality or exponents. Brahmagupta marked different unknowns with different colours in a chapter called Equations of Several Colours. In the late 1500s François Viète began writing known and unknown numbers as letters and calculating with them directly, using consonants for knowns and vowels for unknowns. In 1637 Descartes switched to x, y and z for unknowns and a, b and c for knowns, the habit that stuck.
Newton and Leibniz, from the 1660s, built calculus on how a small change in one quantity drives change in another, and Euler later introduced the notation y equals f of x. Until the late 19th century, variable meant almost only a function's input or output. Then paradoxes such as continuous functions differentiable nowhere led Karl Weierstrass to replace the loose idea of x tending toward a with a formal definition of a limit in which none of the five symbols actually varies, giving the modern static sense of the word.
Conventions persist: i, j and k for indices, n for a fixed count, ε for a tiny positive number, and capital X, Y and Z for random variables in statistics. Printed variables are normally set in italics.
Source: Variable (mathematics)
