Table of mathematical symbols by introduction date
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 | ||
n√ | radical symbol (for nth root) | 1629 | Albert Girard |
< > | strict inequality signs (less-than sign and greater-than sign) | 1631 | Thomas Harriot |
xy | superscript notation (for exponentiation) | 1636 (using Roman numerals as superscripts) | James Hume |
| 1637 (in the modern form) | René Descartes (La Géométrie) | ||
x | 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 | ||
d | 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 |
x′ | prime symbol (for derivative) | 1748 | |
Σ | 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 |
[x] | integral part (a.k.a. floor) | 1808 | |
! | factorial | 1808 | Christian Kramp |
Π | 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 |
∩ ∪ | intersection union | 1888 | Giuseppe Peano |
ℵ | aleph symbol (for transfinite cardinal numbers) | 1893 | Georg Cantor |
∈ | membership sign (is an element of) | 1894 | Giuseppe Peano |
O | 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: X → Y) | 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 |
⌊x⌋ ⌈x⌉ | greatest integer ≤ x (a.k.a. floor) smallest integer ≥ x (a.k.a. ceiling) | 1962[6] | Kenneth E. Iverson |
See also
[edit]- History of mathematical notation
- History of the Hindu–Arabic numeral system
- Glossary of mathematical symbols
- List of mathematical symbols by subject
- Mathematical notation
- Mathematical operators and symbols in Unicode
Sources
[edit]- ^ Cajori, Florian (1993). A History of Mathematical Notations. Mineola, New York: Dover Publications.
- ^ Boyer, Carl B. (1991), A History of Mathematics (Second ed.), John Wiley & Sons, Inc., ISBN 978-0-471-54397-8
- ^ a b c "Earliest Uses of Symbols for Matrices and Vectors". jeff560.tripod.com. Retrieved 18 December 2016.
- ^ Weil, André (1992), The Apprenticeship of a Mathematician, Springer, p. 114, ISBN 9783764326500.
- ^ 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.
- ^ Kenneth E. Iverson (1962), A Programming Language, Wiley, retrieved 20 April 2016