Letters as symbols

Letters are typically used for naming—in mathematical jargon, one says representing—mathematical objects. This is typically the Latin and Greek alphabets that are used, but some letters of Hebrew alphabet ${\displaystyle (\aleph ,\beth )}$ are sometimes used. Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces provide also different symbols. For example, ${\displaystyle r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}$ and ${\displaystyle {\mathfrak {R}}}$ could theoretically appear in the same mathematical text with six different meanings. Normally, roman upright typeface is not used for symbols, except for symbols that are formed of several letters, such as the symbol "${\displaystyle \sin }$" of the sine function.

In order to have more symbols, and for allowing related mathematical objects to be represented by related symbols, diacritics, subscripts and superscripts are often used. For example, ${\displaystyle {\hat {f'_{1}}}}$ may denote the Fourier transform of the derivative of a function called ${\displaystyle f_{1}.}$

Other symbols

Symbols are not only used for naming mathematical objects. They can be used for operations ${\displaystyle (+,-,/,\oplus ,\ldots ),}$ for relations ${\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}$ for logical connectives ${\displaystyle (\implies ,\land ,\lor ,\ldots ),}$ for quantifiers ${\displaystyle (\forall ,\exists ),}$ and for other purposes.

Some symbols are similar to Latin or Greek letters, some are obtained by deforming letters, some are traditional typographic symbols, but many have been specially designed for mathematics.

Numbers

It is believed that a notation to represent numbers was first developed at least 50,000 years ago^[1]—early mathematical ideas such as finger counting^[2] 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 Arabs (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.

The first systematic use of formulas, and, in particular the use of symbols (variables) for unspecified numbers is generally attributed to François Viète (16th century). However, he used different symbols than those that are now standard.

Later, René Descartes (17th century) introduced the modern notation for variables and equations; in particular, the use of ${\displaystyle x,y,z}$ for unknown quantities and ${\displaystyle a,b,c}$ for known ones (constants). He introduced also the notation i and the term "imaginary" for the imaginary unit.

The 18th and 19th centuries saw the standardization of mathematical notation as used today. Leonhard Euler was responsible for many of the notations currently in use: the functional notation ${\displaystyle f(x),}$ e for the base of the natural logarithm, ${\textstyle \sum }$ for summation, etc.^[3] He also popularized the use of π for the Archimedes constant (proposed by William Jones, based on an earlier notation of William Oughtred).^[4]

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, Einstein's 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:

${\displaystyle \sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.}$

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, 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 is its primary target.