Mathematical notation

Last updated December 13, 2024

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.

Symbols

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 ⁠ $\aleph$ ⁠, Cyrillic $Ш$ , and Hiragana $よ$ . Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces also provide different symbols. For example, $r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},$ and ${\mathfrak {R}}$ could theoretically appear in the same mathematical text with six different meanings. Normally, roman upright typeface is not used for symbols, except for symbols representing a standard function, such as the symbol " $\sin$ " of the sine function.^[2]

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, ${\hat {f'_{1}}}$ may denote the Fourier transform of the derivative of a function called $f_{1}.$

Other symbols

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

Expressions

An expression is a finite combination of symbols that is well-formed according to rules that depend on the context. In general, an expression denotes or names a mathematical object, and plays therefore in the language of mathematics the role of a noun phrase in the natural language.

An expression contains often some operators, and may therefore be evaluated by the action of the operators in it. For example, $3+2$ is an expression in which the operator $+$ can be evaluated for giving the result $5.$ So, $3+2$ and $5$ are two different expressions that represent the same number. This is the meaning of the equality $3+2=5.$

A more complicated example is given by the expression ${\textstyle \int _{a}^{b}xdx}$ that can be evaluated to ${\textstyle {\frac {b^{2}}{2}}-{\frac {a^{2}}{2}}.}$ Although the resulting expression contains the operators of division, subtraction and exponentiation, it cannot be evaluated further because $a$ and $b$ denote unspecified numbers.

History

Numbers

It is believed that a notation to represent numbers was first developed at least 50,000 years ago.^[3] Early mathematical ideas such as finger counting ^[4] 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 $x,y,z$ for unknown quantities and $a,b,c$ 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 $f(x),$ $e$ for the base of the natural logarithm, ${\textstyle \sum }$ for summation, etc.^[5] He also popularized the use of $π$ for the Archimedes constant (proposed by William Jones, based on an earlier notation of William Oughtred).^[6]

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, 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.

International standard mathematical notation

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., E = mc²) and roman (upright) fonts for mathematical constants (e.g., e or π).

Non-Latin-based mathematical notation

Modern Arabic mathematical notation is based mostly on the Arabic alphabet and is used widely in the Arab world, especially in pre-tertiary education. (Western notation uses Arabic numerals, but the Arabic notation also replaces Latin letters and related symbols with Arabic script.)

In addition to Arabic notation, mathematics also makes use of Greek letters to denote a wide variety of mathematical objects and variables. On some occasions, certain Hebrew letters are also used (such as in the context of infinite cardinals).

Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are Penrose graphical notation and Coxeter–Dynkin diagrams.

Braille-based mathematical notations used by blind people include Nemeth Braille and GS8 Braille.

Related Research Articles

Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.

In mathematics, a product is the result of multiplication, or an expression that identifies objects to be multiplied, called factors. For example, 21 is the product of 3 and 7, and $is the product of and . When one factor is an integer, the product is called a multiple .$

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.

Blackboard bold is a style of writing bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most commonly used to represent the number sets $, (integers),,, and .$

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object. Equality between $A$ and $B$ is written $A = B$ , and pronounced " $A$ equals $B$ ". In this equality, $A$ and $B$ are distinguished by calling them left-hand side (LHS), and right-hand side (RHS). Two objects that are not equal are said to be distinct.

In mathematics, a function from a set $X$ to a set $Y$ assigns to each element of $X$ exactly one element of $Y$ . The set $X$ is called the domain of the function and the set $Y$ is called the codomain of the function.

The tilde is a grapheme ⟨˜⟩ or ⟨~⟩ with a number of uses. The name of the character came into English from Spanish tilde, which in turn came from the Latin titulus, meaning 'title' or 'superscription'. Its primary use is as a diacritic (accent) in combination with a base letter. Its freestanding form is used in modern texts mainly to indicate approximation.

In mathematics, 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.

The prime symbol ′, double prime symbol ″, triple prime symbol ‴, and quadruple prime symbol ⁗ are used to designate units and for other purposes in mathematics, science, linguistics and music.

In mathematics, 0.999... denotes the smallest number greater than every number in the sequence (0.9, 0.99, 0.999, ...). It can be proved that this number is 1; that is,

ISO 31-0 is the introductory part of international standard ISO 31 on quantities and units. It provides guidelines for using physical quantities, quantity and unit symbols, and coherent unit systems, especially the SI. It was intended for use in all fields of science and technology and is augmented by more specialized conventions defined in other parts of the ISO 31 standard. ISO 31-0 was withdrawn on 17 November 2009. It is superseded by ISO 80000-1. Other parts of ISO 31 have also been withdrawn and replaced by parts of ISO 80000.

Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.

In mathematics, a variable is a symbol, typically a letter, that holds a place for constants, often numbers. One say colloquially that the variable represents or denotes the object, and that the object is the value of the variable.

In formal language theory, an alphabet, sometimes called a vocabulary, is a non-empty set of indivisible symbols/characters/glyphs, typically thought of as representing letters, characters, digits, phonemes, or even words. Alphabets in this technical sense of a set are used in a diverse range of fields including logic, mathematics, computer science, and linguistics. An alphabet may have any cardinality ("size") and, depending on its purpose, may be finite, countable, or even uncountable.

An overline, overscore, or overbar, is a typographical feature of a horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a vinculum, a notation for grouping symbols which is expressed in modern notation by parentheses, though it persists for symbols under a radical sign. The original use in Ancient Greek was to indicate compositions of Greek letters as Greek numerals. In Latin, it indicates Roman numerals multiplied by a thousand and it forms medieval abbreviations (sigla). Marking one or more words with a continuous line above the characters is sometimes called overstriking, though overstriking generally refers to printing one character on top of an already-printed character.

In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots. These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields. An algebraic operation may also be defined more generally as a function from a Cartesian power of a given set to the same set.

The history of mathematical notation includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators. The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.

In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets [ ], braces { } and angle brackets ⟨ ⟩, are frequently used in mathematical notation. Generally, such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed sub-expression, the operators in the sub-expression take precedence over those surrounding it. Sometimes, for the clarity of reading, different kinds of brackets are used to express the same meaning of precedence in a single expression with deep nesting of sub-expressions.

In probability and statistics, a factorial moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both. Moment measures generalize the idea of factorial moments, which are useful for studying non-negative integer-valued random variables.

References

↑ Einstein, Albert (1905). "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?". Annalen der Physik (in German). 323 (13): 639–641. doi:10.1002/andp.19053231314. ISSN 0003-3804.
↑ ISO 80000-2:2019
↑ Eves, Howard (1990). An Introduction to the History of Mathematics (6 ed.). p. 9. ISBN 978-0-03-029558-4.
↑ Ifrah, Georges (2000). The Universal History of Numbers: From prehistory to the invention of the computer. Translated by Bellos, David; Harding, E. F.; Wood, Sophie; Monk, Ian. John Wiley and Sons. p. 48. ISBN 0-471-39340-1. (NB. Ifrah supports his thesis by quoting idiomatic phrases from languages across the entire world. He notes that humans learned to count on their hands. He shows, for example, a picture of Boethius (who lived 480–524 or 525) reckoning on his fingers.)
↑ Boyer, Carl Benjamin; Merzbach, Uta C. (1991). A History of Mathematics. John Wiley & Sons. pp. 442–443. ISBN 978-0-471-54397-8.
↑ Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. p. 166. ISBN 978-3-540-66572-4.

External links

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.

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[1] Einstein, Albert (1905). "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?". Annalen der Physik (in German). 323 (13): 639–641. doi:10.1002/andp.19053231314. ISSN 0003-3804.

[2] ISO 80000-2:2019

[Eves_1990-3] Eves, Howard (1990). An Introduction to the History of Mathematics (6 ed.). p. 9. ISBN 978-0-03-029558-4.

[Ifrah_2000-4] Ifrah, Georges (2000). The Universal History of Numbers: From prehistory to the invention of the computer. Translated by Bellos, David; Harding, E. F.; Wood, Sophie; Monk, Ian. John Wiley and Sons. p. 48. ISBN 0-471-39340-1. (NB. Ifrah supports his thesis by quoting idiomatic phrases from languages across the entire world. He notes that humans learned to count on their hands. He shows, for example, a picture of Boethius (who lived 480–524 or 525) reckoning on his fingers.)

[Boyer-Merzbach_1991-5] Boyer, Carl Benjamin; Merzbach, Uta C. (1991). A History of Mathematics. John Wiley & Sons. pp. 442–443. ISBN 978-0-471-54397-8.

[Arndt-Haenel_2006-6] Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. p. 166. ISBN 978-3-540-66572-4.

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