Math Symbols: Meaning, How to Read Them and LaTeX Code

+plus sign

The sign for addition, also placed in front of a positive number. It is thought to come from a quickly written form of the Latin et (and).

3 + 4 = 7read: three plus four equals seven

First useIts first printed use is in Johannes Widmann's commercial arithmetic, Leipzig 1489, where Widmann first uses it to mark a surplus in a warehouse.

LaTeX +U+002B

−minus sign

The sign for subtraction and for negative numbers. The short hyphen on your keyboard (-) and the mathematical minus (−) are not the same character, the real minus is longer and sits at the same height as the plus.

7 − 4 = 3read: seven minus four equals three

First useWidmann, 1489, together with the plus.

LaTeX -U+2212

×multiplication sign, times

The sign for multiplication. In algebra it is usually replaced by a dot or by writing letters side by side, so it isn't confused with the letter x.

6 × 7 = 42read: six times seven equals forty-two

First useWilliam Oughtred, Clavis Mathematicae, 1631. Leibniz dislikes it because it looks too much like the letter x.

LaTeX \timesU+00D7

⋅multiplication dot

A raised dot for multiplication. In countries that use a decimal point, it sits in the middle of the line so it isn't mistaken for the decimal point.

a ⋅ bread: a times b

First useIn a letter Leibniz writes to Johann Bernoulli in 1698.

LaTeX \cdotU+22C5

÷division sign, obelus

The sign for division. It lives on in school books and on calculators, but the international ISO standard recommends a slash or a fraction bar instead.

12 ÷ 4 = 3read: twelve divided by four equals three

First useJohann Rahn, Teutsche Algebra, 1659.

LaTeX \divU+00F7

±plus-minus sign

Writes the added and the subtracted value of an expression at once, or gives the margin of error of a measurement.

x = (−b ± √(b² − 4ac)) / 2aread: x equals minus b plus or minus the square root of b squared minus four a c, all over two a

First useWilliam Oughtred, 1631.

LaTeX \pmU+00B1

√square root, radical sign

The positive number whose square is the given number. The sign is thought to come from a small r, for the Latin radix (root).

√16 = 4read: the square root of sixteen is four

First useChristoff Rudolff, Coss, 1525. Descartes attaches the horizontal bar over the root in 1637.

LaTeX \sqrt{x}U+221A

%percent sign

One percent, that is 1/100. From the 15th century on, Italian merchants' abbreviation of per cento slowly turns into today's shape in handwriting.

25% = 1/4read: twenty-five percent equals one quarter

LaTeX \%U+0025

n!factorial

The product of all whole numbers from 1 to n. Zero factorial is 1 by definition.

5! = 120read: five factorial equals one hundred twenty

First useChristian Kramp, 1808.

LaTeX n!U+0021

|x|absolute value

A number's distance from zero, the size that is left once you drop its sign.

|−3| = 3read: the absolute value of minus three is three

First useKarl Weierstrass, 1841.

LaTeX |x|U+007C

⌊x⌋floor function

The largest whole number not greater than x. Its sibling ⌈x⌉, the ceiling function, is the smallest whole number not less than x.

⌊3.7⌋ = 3, ⌈3.2⌉ = 4read: the floor of three point seven is three, the ceiling of three point two is four

First useKenneth Iverson, 1962. Iverson designs these signs for the APL programming language.

LaTeX \lfloor x \rfloorU+230A U+230B

(ⁿₖ)binomial coefficient

The number of different ways to choose k elements from a set of n. These are the numbers in Pascal's triangle.

(⁵₂) = 10read: five choose two equals ten

First useAndreas von Ettingshausen, 1826.

LaTeX \binom{n}{k}

=equals sign

Says that two expressions have the same value.

2 + 2 = 4read: two plus two equals four

First useRobert Recorde, The Whetstone of Witte, 1557. Recorde says no two things can be more equal than a pair of parallel lines, and he draws his lines much longer than ours.

LaTeX =U+003D

≠not equal to

Says that two expressions have different values.

π ≠ 22/7read: pi is not equal to twenty-two over seven

First useIt appears in Euler's writing, its origin is uncertain.

LaTeX \neqU+2260

<less than

Says the expression on the left is smaller than the one on the right. Turned around, > is greater than. The open end always faces the bigger side.

3 < 5read: three is less than five

First useThomas Harriot, Artis Analyticae Praxis, 1631. The book is printed ten years after Harriot's death.

LaTeX <U+003C

≤less than or equal to

Says the expression on the left is smaller than or equal to the one on the right. Its sibling is ≥, greater than or equal to.

x ≤ 10read: x is less than or equal to ten

First usePierre Bouguer, 1734.

LaTeX \leqU+2264

≈approximately equal to

Says two values are very close to each other but not exactly equal.

π ≈ 3.1416read: pi is approximately three point one four one six

First useAlfred George Greenhill, 1892.

LaTeX \approxU+2248

≡congruent to, identical to

In number theory it says two numbers leave the same remainder, in algebra that two expressions are equal for every value.

17 ≡ 2 (mod 5)read: seventeen is congruent to two mod five

First useGauss, Disquisitiones Arithmeticae, 1801. The mod notation comes from the same book.

LaTeX \equivU+2261

∝proportional to

Says one quantity grows in a fixed ratio with another.

F ∝ 1/r²read: F is proportional to one over r squared

First useWilliam Emerson, 1768.

LaTeX \proptoU+221D

∼similar to, asymptotic to

In geometry it says two shapes are similar, in analysis that the ratio of two functions goes to 1, in probability which distribution a variable follows. You read it by context.

n! ∼ √(2πn) (n/e)ⁿread: n factorial is asymptotic to the square root of two pi n times n over e to the n

First useLeibniz uses it for similarity.

LaTeX \simU+223C

∞infinity

Not a number but a sign for growing without bound. Used in limits, intervals and series.

lim (1/x) = 0, x → ∞read: the limit of one over x as x goes to infinity is zero

First useJohn Wallis, De sectionibus conicis, 1655.

LaTeX \inftyU+221E

limlimit

The value an expression approaches as a variable approaches a given value.

lim (sin x / x) = 1, x → 0read: the limit of sine x over x as x goes to zero is one

First useSimon L'Huilier uses the abbreviation lim. in 1786. The habit of putting an arrow and the approached value underneath is spread by G. H. Hardy's A Course of Pure Mathematics in 1908.

LaTeX \lim_{x \to a}

∑summation sign, sigma

Writes the sum of a run of terms on a single line. Capital Greek sigma, an S, for the Latin summa (sum).

∑ k = n(n+1)/2, k = 1 … nread: the sum of k from one to n equals n times n plus one over two

First useEuler, Institutiones calculi differentialis, 1755.

LaTeX \sum_{k=1}^{n}U+2211

∏product sign

Writes the product of a run of terms on a single line. Capital Greek pi, for the Latin productum.

∏ k = n!, k = 1 … nread: the product of k from one to n is n factorial

First useGauss, 1812.

LaTeX \prod_{k=1}^{n}U+220F

∫integral sign

The area under a curve and the inverse of the derivative. The sign is a stretched letter s, for the Latin summa.

∫ x dx = x²/2 + Cread: the integral of x d x equals x squared over two plus C

First useLeibniz, in a manuscript dated 29 October 1675. First printed in 1686.

LaTeX \int_a^b f(x)\,dxU+222B

dy/dxderivative, Leibniz notation

The rate of change of y with respect to x. The letter d is for the Latin differentia.

d(x²)/dx = 2xread: the derivative of x squared with respect to x is two x

First useLeibniz, 1675.

LaTeX \frac{dy}{dx}U+002F

f′(x)prime, Lagrange notation

The derivative of f. The second derivative is written f″, the third f‴.

f(x) = x³ ⇒ f′(x) = 3x²read: if f of x is x cubed, f prime of x is three x squared

First useJoseph-Louis Lagrange, Théorie des fonctions analytiques, 1797.

LaTeX f'(x)U+2032

ẋdot notation, Newton's fluxion

The derivative with respect to time. Still used everywhere in physics, especially mechanics. Two dots give the second derivative.

ẍ = −ω²xread: x double dot equals minus omega squared x

First useNewton uses these dotted letters in his notebooks in the 1660s, they first appear in print in Wallis's Algebra in 1693.

LaTeX \dot{x}U+1E8B

∂partial derivative, del

The derivative of a function of several variables with respect to just one of them, holding the others fixed.

∂f/∂xread: the partial derivative of f with respect to x

First useAdrien-Marie Legendre uses it in 1786 and then drops it. Carl Jacobi brings it back in 1841, and this time it stays.

LaTeX \partialU+2202

∇nabla, del operator

A vector operator made of partial derivatives. Gradient, divergence and curl are all written with it.

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)read: the gradient of f

First useHamilton uses it as a triangle on its side in 1837. The name nabla is suggested by William Robertson Smith, from Peter Guthrie Tait's circle, because the sign looks like an old kind of harp.

LaTeX \nablaU+2207

Δdelta, change

The change in a quantity, final value minus initial value. Capital delta is also used for the Laplace operator.

Δx = x₂ − x₁read: delta x equals x two minus x one

LaTeX \DeltaU+0394

f(x)function notation

The value of the function f at x.

f(x) = x² + 1read: f of x equals x squared plus one

First useEuler, 1734. Euler is also the one who puts the function at the centre of analysis, in his Introductio of 1748.

LaTeX f(x)

loglogarithm

The power you have to raise the base to in order to get a number. ln is the natural logarithm, the logarithm to base e.

log₁₀ 1000 = 3read: log base ten of a thousand is three

First useThe word logarithm is John Napier's, 1614. Kepler uses the abbreviation Log. in 1624.

LaTeX \log

∈element of

Says an object is an element of a set. Not an element of is written ∉.

3 ∈ ℕread: three is an element of the natural numbers

First useGiuseppe Peano, 1889. Peano picks epsilon, the first letter of the Greek ἐστί (is).

LaTeX \inU+2208

⊆subset of

Says every element of one set is also in another. ⊂ means subset in some books and proper subset in others, so check the author's definition.

ℕ ⊆ ℤread: the natural numbers are a subset of the integers

LaTeX \subseteqU+2286

∪union

All the elements that are in at least one of two sets.

{1, 2} ∪ {2, 3} = {1, 2, 3}read: the set one two union the set two three is the set one two three

First useGiuseppe Peano, 1888.

LaTeX \cupU+222A

∩intersection

The elements that are in both of two sets.

{1, 2} ∩ {2, 3} = {2}read: the set one two intersect the set two three is the set two

First useGiuseppe Peano, 1888.

LaTeX \capU+2229

∖set difference

The elements in the first set that are not in the second. A backslash is used so it isn't confused with a minus sign.

{1, 2, 3} ∖ {2} = {1, 3}read: the set one two three minus the set two is the set one three

LaTeX \setminusU+2216

∅empty set

The set with no elements. Not a zero with a stroke through it but the letter Ø of the Norwegian alphabet.

A ∩ ∅ = ∅read: A intersect the empty set is the empty set

First useAndré Weil, 1939, for the Bourbaki group's books. Weil says in his memoirs that the choice was his.

LaTeX \varnothingU+2205

ℵ₀aleph null

The size of the infinity of the natural numbers, countable infinity. The infinity of the real numbers is bigger.

|ℕ| = ℵ₀read: the cardinality of the natural numbers is aleph null

First useGeorg Cantor, 1895. Aleph is the first letter of the Hebrew alphabet.

LaTeX \aleph_0U+2135 U+2080

∀for all, universal quantifier

Says a statement holds for every element of a set. An upside-down A, for the German Alle and the English All.

∀x ∈ ℝ, x² ≥ 0read: for every real x, x squared is greater than or equal to zero

First useGerhard Gentzen, 1935, modelled on Peano's ∃.

LaTeX \forallU+2200

∃there exists, existential quantifier

Says at least one element of a set makes the statement true. To say there is exactly one, write ∃!.

∃x ∈ ℤ, x + 5 = 2read: there exists an integer x with x plus five equal to two

First useGiuseppe Peano, 1897. A turned E, for the Latin existit.

LaTeX \existsU+2203

¬negation, not

The negation of a statement.

¬(p ∧ q)read: not p and q

First useArend Heyting, 1930.

LaTeX \negU+00AC

∧logical and, conjunction

True when both statements are true. Its sibling ∨, or, is true when at least one of them is.

p ∧ q, p ∨ qread: p and q, p or q

First use∨ comes from the Latin vel (or) and is used in Principia Mathematica, 1910. Heyting uses ∧ in 1930.

LaTeX \wedgeU+2227

⇒implies

Says that if the statement on the left is true, the one on the right is true too. If the left is false, the whole statement counts as true automatically.

x = 2 ⇒ x² = 4read: x equals two implies x squared equals four

LaTeX \RightarrowU+21D2

⇔if and only if

Says two statements are either both true or both false. In writing it is often shortened to iff.

x² = 0 ⇔ x = 0read: x squared is zero if and only if x is zero

LaTeX \LeftrightarrowU+21D4

⊢turnstile, proves

Says the statement on the right can be proved from the assumptions on the left.

A, A ⇒ B ⊢ Bread: from A and A implies B, B can be proved

First useGottlob Frege, Begriffsschrift, 1879. It grows out of Frege's judgement stroke.

LaTeX \vdashU+22A2

∴therefore

Says a conclusion follows from what came before. Turned upside down, ∵ means because.

a = b, b = c ∴ a = cread: a equals b, b equals c, therefore a equals c

First useJohann Rahn, Teutsche Algebra, 1659, in the same book as the division sign.

LaTeX \thereforeU+2234

∎end of proof, tombstone, halmos

Marks the end of a proof. It replaces the Q.E.D. of older books, the Latin quod erat demonstrandum (which was to be shown).

First usePaul Halmos borrows it in the 1950s from the end-of-article mark in magazines, which is why it is also called a halmos.

LaTeX \blacksquareU+220E

ℕ ℤ ℚ ℝ ℂnumber sets

The natural numbers, integers, rationals, reals and complex numbers. Z comes from the German Zahlen (numbers), Q from quotient.

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂread: the naturals sit inside the integers, which sit inside the rationals, the reals and the complex numbers

First useBooks first print these set names in plain bold. The double-struck forms are born as a practical way to write bold on a blackboard, hence the name blackboard bold.

LaTeX \mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}U+2115 U+2124 U+211A U+211D U+2102

πpi

The ratio of a circle's circumference to its diameter, 3.14159…, a number that goes on forever without repeating.

C = 2πrread: circumference equals two pi r

First useWilliam Jones, 1706. The one who really spreads it, from 1737 on, is Euler.

LaTeX \piU+03C0

eEuler's number

The base of the natural logarithm, 2.71828… It sits inside everything that grows continuously, compound interest, radioactive decay.

e = lim (1 + 1/n)ⁿ, n → ∞read: e is the limit of one plus one over n to the n as n goes to infinity

First useEuler, in a letter of 1731, in print in his Mechanica in 1736.

LaTeX e

iimaginary unit

The number whose square is −1. Electrical engineers write j so it isn't confused with the i of current.

e^(iπ) + 1 = 0read: e to the i pi plus one equals zero

First useEuler, 1777. The paper is printed after Euler's death, in 1794.

LaTeX i

φphi, golden ratio

(1 + √5)/2, that is 1.61803… The ratio you get by cutting a line so that the larger piece is to the smaller as the whole is to the larger.

φ² = φ + 1read: phi squared equals phi plus one

First useThe American mathematician Mark Barr picks it in the early 1900s, from the initial of the sculptor Phidias.

LaTeX \varphiU+03C6

ℏh-bar, reduced Planck constant

Planck's constant divided by 2π. It turns up so often in quantum mechanics that it has a sign of its own.

E = ℏωread: energy equals h-bar omega

First useIt spreads with Paul Dirac, which is why it is also called Dirac's constant.

LaTeX \hbarU+210F

⟨ψ|φ⟩bra-ket notation

The inner product of two states in quantum mechanics. ⟨ψ| is the bra, |φ⟩ the ket, and together they make a bracket.

⟨ψ|ψ⟩ = 1read: bra psi ket psi equals one

First usePaul Dirac, 1939.

LaTeX \langle \psi | \varphi \rangleU+27E8 U+03C8 U+007C U+03C6 U+27E9

∠angle

The opening between two rays from a shared starting point. ∠ABC is the angle with its vertex at B.

∠ABC = 90°read: angle A B C is ninety degrees

First usePierre Hérigone, Cursus mathematicus, 1634.

LaTeX \angleU+2220

⊥perpendicular

Says two lines or vectors meet at 90 degrees.

AB ⊥ CDread: A B is perpendicular to C D

First usePierre Hérigone, 1634.

LaTeX \perpU+22A5

∥parallel

Says two lines never meet. The same mark is used for the norm of a vector, ‖v‖.

AB ∥ CDread: A B is parallel to C D

LaTeX \parallelU+2225

≅congruent

In geometry it says two shapes match in both form and size, in algebra that two structures are isomorphic.

△ABC ≅ △DEFread: triangle A B C is congruent to triangle D E F

LaTeX \congU+2245

°degree

One 360th of a full turn. The number 360 is left over from the Babylonian base-60 number system.

π rad = 180°read: pi radians is one hundred eighty degrees

LaTeX ^\circU+00B0

a × bcross product

In three dimensions, the vector perpendicular to both vectors, whose length equals the area of the parallelogram they span. Its sibling a ⋅ b is the dot product.

i × j = kread: i cross j equals k

First useJosiah Willard Gibbs, 1881, in the vector analysis notes he has printed for his students.

LaTeX \mathbf{a} \times \mathbf{b}U+00D7