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Mathematical notation consists of usingsymbols for representingoperations, unspecifiednumbers,relations, and any othermathematical objects and assembling them intoexpressions andformulas. Mathematical notation is widely used inmathematics,science, andengineering for representing complexconcepts andproperties in a concise, unambiguous, and accurate way.
For example, the physicistAlbert Einstein's formula is the quantitative representation in mathematical notation ofmass–energy equivalence.[1]
Mathematical notation was first introduced byFrançois Viète at the end of the 16th century and largely expanded during the 17th and 18th centuries byRené Descartes,Isaac Newton,Gottfried Wilhelm Leibniz, and overallLeonhard Euler.
The use of many symbols is the basis of mathematical notation. They play a similar role as words innatural languages. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence.
Letters are typically used for naming—inmathematical jargon, one saysrepresenting—mathematical objects. TheLatin andGreek alphabets are used extensively, but a few letters of other alphabets are also used sporadically, such as theHebrew,CyrillicШ, andHiraganaよ. Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces also provide different symbols. For example, and 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 "" of thesine function.[2]
In order to have more symbols, and for allowing related mathematical objects to be represented by related symbols,diacritics,subscripts andsuperscripts are often used. For example, may denote theFourier transform of thederivative of afunction called
Symbols are not only used for naming mathematical objects. They can be used foroperations forrelations forlogical connectives forquantifiers and for other purposes.
Some symbols are similar to Latin or Greek letters, some are obtained by deforming letters, some are traditionaltypographic symbols, but many have been specially designed for mathematics.
TheInternational Organization for Standardization (ISO) is aninternational standard development organization composed of representatives from the nationalstandards organizations of member countries. The international standardISO 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).
Anexpression is a written arrangement ofsymbols following the context-dependent,syntactic conventions of mathematical notation. Symbols can denotenumbers,variables,operations, andfunctions.[3] Other symbols includepunctuation marks andbrackets, used forgrouping where there is not a well-definedorder of operations.
Expressions are commonly distinguished fromformulas: expressions are a kind ofmathematical object, whereas formulas are statementsabout mathematical objects.[4] This is analogous tonatural language, where anoun phrase refers to an object, and a wholesentence refers to afact. For example, is an expression, while theinequality is a formula.
Toevaluate an expression means to find a numericalvalue equivalent to the expression.[5][6] Expressions can beevaluated orsimplified by replacingoperations that appear in them with their result. For example, the expression simplifies to, and evaluates to
It is believed that a notation to representnumbers was first developed at least 50,000 years ago.[7] Early mathematical ideas such asfinger counting[8] have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. Thetally stick is a way of counting dating back to theUpper Paleolithic. Perhaps the oldest known mathematical texts are those of ancientSumer. TheCensus Quipu of the Andes and theIshango Bone from Africa both used thetally mark method of accounting for numerical concepts.
The concept ofzero 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 theBabylonians andGreek Egyptians, and then as aninteger by theMayans,Indians andArabs (seethe history of zero).
Until the 16th century, mathematics was essentiallyrhetorical, in the sense that everything but explicit numbers was expressed in words. However, some authors such asDiophantus 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 toFrançois 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 andequations; in particular, the use of forunknown quantities and for known ones (constants). He introduced also the notationi and the term "imaginary" for theimaginary 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: thefunctional notatione for the base of thenatural logarithm, forsummation, etc.[9] He also popularized the use ofπ for theArchimedes constant (proposed byWilliam Jones, based on an earlier notation ofWilliam Oughtred).[10]
Since then many new notations have been introduced, often specific to a particular area of mathematics. Some notations are named after their inventors, such asLeibniz's notation,Legendre symbol, theEinstein summation convention, etc.
Generaltypesetting 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:
TeX is a mathematically oriented typesetting system that was created in 1978 byDonald Knuth. It is widely used in mathematics, through its extension calledLaTeX, and is ade facto standard. (The above expression is written in LaTeX.)
More recently, another approach for mathematical typesetting is provided byMathML. However, it is not well supported in web browsers, which is its primary target.
Modern Arabic mathematical notation is based mostly on theArabic alphabet and is used widely in theArab world, especially in pre-tertiary education. (Western notation usesArabic numerals, but the Arabic notation also replaces Latin letters and related symbols with Arabic script.)
In addition to Arabic notation, mathematics also makes use ofGreek letters to denote a wide variety of mathematical objects and variables. On some occasions, certainHebrew letters are also used (such as in the context ofinfinite cardinals).
Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples arePenrose graphical notation andCoxeter–Dynkin diagrams.
Braille-based mathematical notations used by blind people includeNemeth Braille andGS8 Braille.
Thesyntax of notation defines how symbols can be combined to makewell-formed expressions, without any given meaning or interpretation. Thesemantics of notation interprets what the symbols represent and assigns a meaning to the expressions and formulas. The reverse process of taking a statement and writing it in logical or mathematical notation is calledtranslation.
Given aformal language, aninterpretation assigns adomain of discourse to the language. Specifically, it assigns each of the constant symbols to objects of the domain, function letters to functions within the domain, predicate letters to statements, and vairiables are assumed to range over the domain.
Themap–territory relation describes the relationship between an object and the representation of that object, such as theEarth and amap of it. In mathematics, this is how the number 4 relates to its representation "4". The quotation marks are the formally correct usage, distinguishing the number from its name. However, it is fairly common practice in math to commit this fallacy saying "Let x denote...", rather than "Let "x" denote..." which is generally harmless.
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