Numbers It is believed that a notation to represent
numbers was first developed at least 50,000 years ago. Early mathematical ideas such as
finger counting have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The
tally stick is a way of counting dating back to the
Upper Paleolithic. Perhaps the oldest known mathematical texts are those of ancient
Sumer. The
Census Quipu of the Andes and the
Ishango Bone from Africa both used the
tally mark method of accounting for numerical concepts. The concept of
zero and the introduction of a notation for it are important developments in early mathematics, which predates for centuries the concept of zero as a number. It was used as a placeholder by the
Babylonians and
Greek Egyptians, and then as an
integer by the
Mayans,
Indians and
Muslims (see
the history of zero).
Modern notation Until the 16th century, mathematics was essentially
rhetorical, in the sense that everything but explicit numbers was expressed in words. However, some authors, such as
Diophantus, used some symbols as abbreviations.
Michael Stifel's (1487–1567) most important work, (1544), contains important innovations in mathematical notation. It has the first use of multiplication by juxtaposition (with no symbol between the terms) in Europe. He is the first to use the term
exponent. The first systematic use of formulas, and, in particular, the use of symbols (
variables) for unspecified numbers, is generally attributed to
François Viète (1540–1603). However, he used different symbols than those that are now standard. Later,
René Descartes (1596–1650) introduced the modern notation for variables and
equations; in particular, the use of x,y,z for
unknown quantities and a,b,c for known ones (
constants). He introduced also the notation and the term
imaginary for the
imaginary unit. The 18th and 19th centuries saw the standardization of mathematical notation as used today.
Leonhard Euler (1707–1783) was responsible for many of the notations currently in use: the
functional notation f(x), for the base of the
natural logarithm, \sum for
summation, etc. He also popularized the use of for the
Archimedes constant (proposed by
William Jones, based on an earlier notation of
William Oughtred). Since then many new notations have been introduced, often specific to a particular area of mathematics. Some notations are named after their inventors, such as
Leibniz's notation,
Legendre symbol, the
Einstein summation convention, etc.
Typesetting General
typesetting systems are generally not well suited for mathematical notation. One of the reasons is that, in mathematical notation, the symbols are often arranged in two-dimensional figures, such as in: : \sum_{n=0}^\infty \frac {\begin{bmatrix}a&b\\c&d\end{bmatrix}^n}{n!}.
TeX () is a mathematically oriented typesetting system that was created in 1978 by
Donald Knuth. It is widely used in mathematics, through its extension called
LaTeX ( or ), and is a
de facto standard. (The above expression is written in LaTeX.) More recently, another approach for mathematical typesetting is provided by
MathML. However, it is not well supported in web browsers, which are its primary target. == Non-Latin-based mathematical notation ==