thumb|Highlighted [[LaTeX mathematical notation]]

Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations, and any other mathematical objects and assembling them into expressions and formulas. Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way.

For example, the physicist Albert Einstein's formula <math>E=mc^2</math> is the quantitative representation in mathematical notation of mass–energy equivalence.

Mathematical notation was first introduced by François Viète at the end of the 16th century and largely expanded during the 17th and 18th centuries by René Descartes, Isaac Newton, Gottfried Wilhelm Leibniz, and overall Leonhard Euler.

Symbols and typeface

The use of many symbols is the basis of mathematical notation. They play a similar role as words in natural languages. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence.

Letters as symbols

Letters are typically used for naming—in mathematical jargon, one says representing—mathematical objects. The Latin and Greek alphabets are used extensively, but a few letters of other alphabets are also used sporadically, such as the Hebrew , Cyrillic , and Hiragana . Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces also provide different symbols. For example, <math>r, R, \R, \mathcal R, \mathfrak r,</math> and <math>\mathfrak R</math> could theoretically appear in the same mathematical text with six different meanings. Normally, roman upright typeface is not used for symbols, except for symbols representing a standard function, such as the symbol "<math>\sin</math>" 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, <math>\hat {f'_1}</math> may denote the Fourier transform of the derivative of a function called <math>f_1.</math>

Other symbols

Symbols are not only used for naming mathematical objects. They can be used for operations <math>(+, -, /, \oplus, \ldots),</math> for relations <math>(=, <, \le, \sim, \equiv, \ldots),</math> for logical connectives <math>(\implies, \land, \lor, \ldots),</math> for quantifiers <math>(\forall, \exists),</math> 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.

International standard mathematical notation

The International Organization for Standardization (ISO) is an international standard development organization composed of representatives from the national standards organizations of member countries. The international standard ISO 80000-2 (previously, ISO 31-11) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., ) and roman (upright) fonts for mathematical constants (e.g., e or π).

Expressions and formulas

An expression is a written arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers, variables, operations, and functions. Other symbols include punctuation marks and brackets, used for grouping where there is not a well-defined order of operations.

Expressions are commonly distinguished from formulas: expressions are a kind of mathematical object, whereas formulas are statements about mathematical objects. This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, <math>8x-5</math> is an expression, while the inequality <math>8x-5 \geq 3 </math> is a formula.

To evaluate an expression means to find a numerical value equivalent to the expression. Expressions can be evaluated or simplified by replacing operations that appear in them with their result. For example, the expression <math>8\times 2-5</math> simplifies to <math>16-5</math>, and evaluates to <math>11.</math>

History

Numbers

It is believed that a notation to represent numbers was first developed at least 50,000 years ago.

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Further reading

  • Florian Cajori, A History of Mathematical Notations (1929), Vol. 1, Vol. 2. (Dover reprint 2011, )
  • Mazur, Joseph (2014), Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton, New Jersey: Princeton University Press.
  • Earliest Uses of Various Mathematical Symbols
  • Mathematical ASCII Notation how to type math notation in any text editor.
  • Mathematics as a Language at Cut-the-Knot
  • Stephen Wolfram: Mathematical Notation: Past and Future. October 2000. Transcript of a keynote address presented at MathML and Math on the Web: MathML International Conference.