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Table of mathematical symbols by introduction date

From Wikipedia, the free encyclopedia

The following table lists many specialized symbols commonly used in modern mathematics, ordered by their introduction date. The table can also be ordered alphabetically by clicking on the relevant header title.

Symbol Name Date of earliest use First author to use
horizontal bar for division 14th century (approx.) Nicole Oresme[1]
plus sign 1360 (approx.), abbreviation for Latin et resembling the plus sign Nicole Oresme
minus sign 1489 (first appearance of minus sign, and also first appearance of plus sign in print) Johannes Widmann
radical symbol (for square root) 1525 (without the vinculum above the radicand) Christoff Rudolff
parentheses (for precedence grouping) 1544 (in handwritten notes) Michael Stifel
1556 Niccolò Tartaglia
equals sign 1557 Robert Recorde
decimal separator 1593 Christopher Clavius
multiplication sign 1618 William Oughtred
plus–minus sign 1628
proportion sign
radical symbol (for nth root) 1629 Albert Girard
strict inequality signs (less-than sign and greater-than sign) 1631 Thomas Harriot
superscript notation (for exponentiation) 1636 (using Roman numerals as superscripts) James Hume
1637 (in the modern form) René Descartes (La Géométrie)
Use of the letter x for an independent variable or unknown value. See History of algebra: The symbol x. 1637[2] René Descartes (La Géométrie)
√ ̅
radical symbol (for square root) 1637 (with the vinculum above the radicand) René Descartes (La Géométrie)
percent sign 1650 (approx.) unknown
infinity sign 1655 John Wallis
division sign (a repurposed obelus variant) 1659 Johann Rahn

unstrict inequality signs (less-than or equals to sign and greater-than or equals to sign) 1670 (with the horizontal bar over the inequality sign, rather than below it) John Wallis
1734 (with double horizontal bar below the inequality sign) Pierre Bouguer
differential sign 1675 Gottfried Leibniz
integral sign
colon (for division) 1684 (deriving from use of colon to denote fractions, dating back to 1633)
middle dot (for multiplication) 1698 (perhaps deriving from a much earlier use of middle dot to separate juxtaposed numbers)
division slash (a.k.a. solidus) 1718 (deriving from horizontal fraction bar, invented by Abu Bakr al-Hassar in the 12th century) Thomas Twining
inequality sign (not equal to) unknown Leonhard Euler
prime symbol (for derivative) 1748
summation symbol 1755
proportionality sign 1768 William Emerson
partial differential sign (a.k.a. curly d or Jacobi's delta) 1770 Marquis de Condorcet
identity sign (for congruence relation) 1801 (first appearance in print; used previously in personal writings of Gauss) Carl Friedrich Gauss
integral part (a.k.a. floor) 1808
factorial 1808 Christian Kramp
product symbol 1812 Carl Friedrich Gauss

set inclusion signs (subset of, superset of) 1817 Joseph Gergonne
1890 Ernst Schröder
absolute value notation 1841 Karl Weierstrass
determinant of a matrix 1841 Arthur Cayley
matrix notation 1843[3]
nabla symbol (for vector differential) 1846 (previously used by Hamilton as a general-purpose operator sign) William Rowan Hamilton


1888 Giuseppe Peano
aleph symbol (for transfinite cardinal numbers) 1893 Georg Cantor
membership sign (is an element of) 1894 Giuseppe Peano
Big O Notation 1894 Paul Bachmann
braces, a.k.a. curly brackets (for set notation) 1895 Georg Cantor
Blackboard bold capital N (for natural numbers set) 1895 Giuseppe Peano
Blackboard bold capital Q (for rational numbers set)
existential quantifier (there exists) 1897
middle dot (for dot product) 1902 J. Willard Gibbs
multiplication sign (for cross product)
logical disjunction (a.k.a. OR) 1906 Bertrand Russell
matrix notation 1909[3] Maxime Bôcher
1909[3] Gerhard Kowalewski
contour integral sign 1917 Arnold Sommerfeld
Blackboard bold capital Z (for integer numbers set) 1930 Edmund Landau
universal quantifier (for all) 1935 Gerhard Gentzen
arrow (for function notation) 1936 (to denote images of specific elements) Øystein Ore
1940 (in the present form of f: XY) Witold Hurewicz
empty set sign 1939 André Weil / Nicolas Bourbaki[4]
Blackboard bold capital C (for complex numbers set) 1939 Nathan Jacobson
end of proof sign (a.k.a. tombstone) 1950[5] Paul Halmos
greatest integer ≤x (a.k.a. floor)

smallest integer ≥x (a.k.a. ceiling)
1962[6] Kenneth E. Iverson

See also



  1. ^ Cajori, Florian (1993). A History of Mathematical Notations. Mineola, New York: Dover Publications.
  2. ^ Boyer, Carl B. (1991), A History of Mathematics (Second ed.), John Wiley & Sons, Inc., ISBN 978-0-471-54397-8
  3. ^ a b c "Earliest Uses of Symbols for Matrices and Vectors". jeff560.tripod.com. Retrieved 18 December 2016.
  4. ^ Weil, André (1992), The Apprenticeship of a Mathematician, Springer, p. 114, ISBN 9783764326500.
  5. ^ Halmos, Paul (1950). Measure Theory. New York: Van Nostrand. pp. vi. The symbol ∎ is used throughout the entire book in place of such phrases as "Q.E.D." or "This completes the proof of the theorem" to signal the end of a proof.
  6. ^ Kenneth E. Iverson (1962), A Programming Language, Wiley, retrieved 20 April 2016