Number · 59 signs
Mathematical symbols
Most of these signs are younger than printing. Each was proposed by one writer, competed with rivals for decades, and won because it was easy to set in type and hard to misread. The table gives each sign’s meaning, an example, its exact Unicode character, and its first known use where the history is well documented.
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| Sign | Meaning | Example | Unicode | First use |
|---|---|---|---|---|
| Arithmetic and relations | ||||
| + | plus, add | 3 + 4 = 7 | U+002B plus sign | printed by Johann Widman, 1489 |
| − | minus, subtract (not the hyphen) | 7 − 4 = 3 | U+2212 minus sign | printed by Johann Widman, 1489 |
| × | times; also the vector product | 3 × 4 = 12 | U+00D7 multiplication sign | William Oughtred, Clavis mathematicae, 1631 |
| ⋅ | times; the scalar (dot) product | a ⋅ b | U+22C5 dot operator | |
| ÷ | divided by | 12 ÷ 4 = 3 | U+00F7 division sign | Johann Heinrich Rahn, Teutsche Algebra, 1659 |
| / | divided by, ratio (the IAU and ISO prefer it to ÷) | m/s | U+002F solidus | |
| = | is equal to | 2 + 2 = 4 | U+003D equals sign | Robert Recorde, The Whetstone of Witte, 1557 |
| ≠ | is not equal to | 1 ≠ 2 | U+2260 not equal to | |
| ≡ | is identically equal to; congruent to (modulo) | −16 ≡ 9 (mod 5) | U+2261 identical to | congruence: Carl Friedrich Gauss, 1801 |
| ≈ | is approximately equal to | π ≈ 3.14 | U+2248 almost equal to | |
| ≅ | is congruent to (geometry); also asymptotically equal | △ABC ≅ △DEF | U+2245 approximately equal to | |
| ∼ | is of the order of; is similar to | n! ∼ √(2πn)(n/e)ⁿ | U+223C tilde operator | |
| ∝ | is proportional to | F ∝ 1/r² | U+221D proportional to | |
| < | is less than | 2 < 3 | U+003C less-than sign | Thomas Harriot, Artis analyticae praxis, 1631 |
| > | is greater than | 3 > 2 | U+003E greater-than sign | Thomas Harriot, Artis analyticae praxis, 1631 |
| ≤ | is less than or equal to | x ≤ 1 | U+2264 less-than or equal to | |
| ≥ | is greater than or equal to | n ≥ 0 | U+2265 greater-than or equal to | |
| ≪ | is much less than | m ≪ M | U+226A much less-than | |
| ≫ | is much greater than | R ≫ r | U+226B much greater-than | |
| ± | plus or minus | 9.81 ± 0.02 | U+00B1 plus-minus sign | Albert Girard 1626 (with “ou”); William Oughtred 1631 |
| → | tends to, approaches; maps to | x → 0 | U+2192 rightwards arrow | |
| ∞ | infinity | 1/x → ∞ | U+221E infinity | John Wallis, De sectionibus conicis, 1655 |
| √ | square root | √9 = 3 | U+221A square root | Christoff Rudolff, Coss, 1525 |
| ! | factorial: 1 × 2 × … × n | 5! = 120 | U+0021 exclamation mark | Christian Kramp, 1808 |
| % | per hundred | 5 % = 0.05 | U+0025 percent sign | |
| ‰ | per thousand (salinity, gradients) | 35 ‰ | U+2030 per mille sign | |
| Analysis | ||||
| ∑ | sum of terms | Σ k = n(n+1)/2 | U+2211 n-ary summation | Leonhard Euler, 1755 (as the Greek Σ) |
| ∏ | product of terms | n! = Π k | U+220F n-ary product | |
| ∫ | integral (a long S for summa) | ∫ x dx = x²/2 + C | U+222B integral | Gottfried Leibniz, manuscript of 29 October 1675 |
| ∂ | partial derivative | ∂f/∂x | U+2202 partial differential | |
| ∇ | gradient (del) operator | ∇T | U+2207 nabla | |
| ∆ | finite change, difference (written as capital delta) | Δt = t₂ − t₁ | U+2206 increment | |
| Geometry | ||||
| ∠ | angle | ∠ABC = 90° | U+2220 angle | Pierre Hérigone used < for angle, 1634; the open form ∠ came later |
| ⊥ | is perpendicular to | AB ⊥ CD | U+22A5 up tack | Pierre Hérigone, Cursus mathematicus, 1634 |
| ∥ | is parallel to | AB ∥ CD | U+2225 parallel to | vertical bars for parallel from the 1670s (Kersey and others) |
| △ | triangle | △ABC | U+25B3 white up-pointing triangle | |
| ° | degree of arc (1/360 of a turn) | 45° | U+00B0 degree sign | |
| ′ | minute of arc (1/60 degree) | 12′ | U+2032 prime | |
| ″ | second of arc (1/60 minute) | 30″ | U+2033 double prime | |
| ∴ | therefore | U+2234 therefore | ||
| ∵ | because | U+2235 because | ||
| Sets and logic | ||||
| ∈ | is an element of | 3 ∈ ℕ | U+2208 element of | |
| ∉ | is not an element of | ½ ∉ ℤ | U+2209 not an element of | |
| ⊂ | is a subset of | ℕ ⊂ ℤ | U+2282 subset of | |
| ∪ | union | A ∪ B | U+222A union | |
| ∩ | intersection | A ∩ B | U+2229 intersection | |
| ∅ | the empty set | A ∩ ∅ = ∅ | U+2205 empty set | |
| ∀ | for all | ∀x: x² ≥ 0 | U+2200 for all | |
| ∃ | there exists | ∃x: x² = 2 | U+2203 there exists | |
| ¬ | not | ¬p | U+00AC not sign | |
| ∧ | and | p ∧ q | U+2227 logical and | |
| ∨ | or | p ∨ q | U+2228 logical or | |
| ⇒ | implies | p ⇒ q | U+21D2 rightwards double arrow | |
| ⇔ | if and only if | p ⇔ q | U+21D4 left right double arrow | |
| ℕ | natural numbers | U+2115 double-struck capital n | ||
| ℤ | integers (German Zahlen) | U+2124 double-struck capital z | ||
| ℚ | rational numbers (quotients) | U+211A double-struck capital q | ||
| ℝ | real numbers | U+211D double-struck capital r | ||
| ℂ | complex numbers | U+2102 double-struck capital c | ||
Unicode names are checked against the Unicode Character Database when the site is built. First uses: A History of Mathematical Notations, 2 vols (1928–1929); a blank means the origin is uncertain or was not checked. Sign list for scientific writing: The IAU Style Manual (1989), sections 5.15, 5.223 and 5.25, 5.223; current standard: ISO 80000-2:2019, Quantities and units, Part 2: Mathematics.
Constants written as letters
| Letter | Value | Meaning | First use |
|---|---|---|---|
| π | 3.141 592 653 589 793… | ratio of a circle’s circumference to its diameter | William Jones, Synopsis palmariorum matheseos, 1706; made standard by Euler |
| e | 2.718 281 828 459 045… | base of natural logarithms | Leonhard Euler: manuscript of 1727 or 1728, in print 1736 |
| i | √−1 | imaginary unit | Leonhard Euler, memoir of 1777 (published 1794) |
Variables are set in italic type; these fixed constants are printed upright in the ISO convention (π, e, i), as are units and functions such as sin and log.
Until the 1500s most algebra was written out in words: “the thing” for the unknown, “is equal to” spelled in full. The equals sign appeared in 1557 because, as Recorde wrote, no two things can be more equal than a pair of parallel lines. Some signs lost their contest: Harriot’s 1631 “<” for “less than” beat Hérigone’s use of the same mark for an angle three years later, and the angle sign was reshaped. The IAU style manual of 1989 still allows × only between numbers, not in symbolic formulae, where a dot is preferred.