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Mathematical notation

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See also: List of mathematical symbols

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2; variables such as x, y and z; delimiters such as "(" and "|"; function symbols such as sin; operator symbols such as "+"; relational symbols such as "<"; conceptual symbols such as lim and dy/dx; equations and complex diagrammatic notations such as Penrose graphical notation and Coxeter–Dynkin diagrams.[1]Cite error: A <ref> tag is missing the closing </ref> (see the help page).—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 development of zero as a number is one of the most important developments in early mathematics. 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 for more information).

Geometry becomes analytic

The earliest mathematical viewpoints in geometry did not lend themselves well to counting. The natural numbers, their relationship to fractions, and the identification of continuous quantities actually took millennia to take form, and even longer to allow for the development of notation.

In fact, it was not until the invention of analytic geometry by René Descartes that geometry became more subject to a numerical notation.[3] Some symbolic shortcuts for mathematical concepts came to be used in the publication of geometric proofs. Moreover, the power and authority of geometry's theorem and proof structure greatly influenced non-geometric treatises, such as Principia Mathematica by Isaac Newton for instance.

Modern notation

The 18th and 19th centuries saw the creation and standardization of mathematical notation as used today. Leonhard Euler was responsible for many of the notations currently in use: the use of a, b, c for constants and x, y, z for unknowns, e for the base of the natural logarithm, sigma (Σ) for summation, i for the imaginary unit, and the functional notation f(x). He also popularized the use of π for Archimedes constant (due to William Jones' proposal for the use of π in this way based on the earlier notation of William Oughtred).

In addition, many fields of mathematics bear the imprint of their creators for notation: the differential operator of Leibniz,[4] the cardinal infinities of Georg Cantor (in addition to the lemniscate (∞) of John Wallis), the congruence symbol (≡) of Gauss, and so forth.

Computerized notation

Mathematically oriented markup languages such as TeX, LaTeX and, more recently, MathML, are powerful enough to express a wide variety of mathematical notations.

Theorem-proving software naturally comes with its own notations for mathematics; the OMDoc project seeks to provide an open commons for such notations; and the MMT language provides a basis for interoperability between other notations.

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 alphabets to denote a wide variety of mathematical objects and variables. In some occasions, certain Hebrew alphabets are also used (such as in the context of infinite cardinals).[5]

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.

See also

Notes

  1. ^ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-08.
  2. ^ Georges Ifrah notes that humans learned to count on their hands. Ifrah shows, for example, a picture of Boethius (who lived 480–524 or 525) reckoning on his fingers in Ifrah 2000, p. 48.
  3. ^ Boyer, C. B. (1959), "Descartes and the geometrization of algebra", The American Mathematical Monthly, 66 (5): 390–393, doi:10.2307/2308751, JSTOR 2308751, MR 0105335, The great accomplishment of Descartes in mathematics invariably is described as the arithmetization of geometry.
  4. ^ "Gottfried Wilhelm Leibnitz". Retrieved 5 October 2014.
  5. ^ "Greek/Hebrew/Latin-based Symbols in Mathematics". Math Vault. 2020-03-20. Retrieved 2020-08-08.

References

Wikimedia Commons has media related to Mathematical notation.
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