Table of mathematical symbols

From Wikipedia, the free encyclopedia Source : 

http://en.wikipedia.org/wiki/Table_of_mathematical_symbols

Common symbols

This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts to be installed, and in TEX, as an image.

This list is incomplete; you can help by expanding it.

See also

• Greek letters used in mathematics, science, and engineering
• ISO 31-11
• List of mathematical abbreviations
• Mathematical alphanumeric symbols
• Mathematical notation
• Notation in probability and statistics
• Physical constants
• Roman letters used in mathematics
• Table of logic symbols
• Table of mathematical symbols by introduction date
• Unicode mathematical operators

Variations

In mathematics written in Arabic, some symbols may be reversed to make right-to-left reading easier. ^[11]

References

1. ^ Rónyai, Lajos (1998), Algoritmusok(Algorithms), TYPOTEX, ISBN 963-9132-16-0
2. ^ Berman, Kenneth A; Paul, Jerome L. (2005), Algorithms: Sequential, Parallel, and Distributed, Boston: Course Technology, p. 822, ISBN 0-534-42057-5
3. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, p. 66, ISBN 0-521-63503-9, OCLC 43641333
4. ^ Copi, Irving M.; Cohen, Carl (1990) [1953], "Chapter 8.3: Conditional Statements and Material Implication", Introduction to Logic (8th ed.), New York: Macmillan, pp. 268–269,ISBN 0023250356, LCCN 89-37742
5. ^ ^{a b c d e} Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 3, ISBN 0-412-60610-0
6. ^ ^a b c d Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 4, ISBN 0-412-60610-0
7. ^ Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 5, ISBN 0-412-60610-0
8. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, p. 62, ISBN 0-521-63503-9, OCLC 43641333
9. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, pp. 69–70, ISBN 0-521-63503-9, OCLC 43641333
10. ^ Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: Cambridge University Press, pp. 71–72, ISBN 0-521-63503-9, OCLC 43641333
11. ^ M. Benatia, A. Lazrik, and K. Sami, "Arabic mathematical symbols in Unicode", 27th Internationalization and Unicode Conference, 2005.

External links

• The complete set of mathematics Unicode characters
• Jeff Miller: Earliest Uses of Various Mathematical Symbols
• Numericana: Scientific Symbols and Icons
• TCAEP - Institute of Physics
• GIF and PNG Images for Math Symbols
• Mathematical Symbols in Unicode
• Using Greek and special characters from Symbol font in HTML
• Unicode Math Symbols - a quick form for using unicode math symbols.
• DeTeXify handwritten symbol recognition — doodle a symbol in the box, and the program will tell you what its name is

Some Unicode charts of mathematical operators:

• Index of Unicode symbols
• Range 2100 – 214F: Letterlike Symbols
• Range 2190 – 21FF: Arrows
• Range 2200 – 22FF: Unicode Mathematical Operators

Some Unicode cross-references:

• Short list of commonly used LaTeX symbols and Comprehensive LaTeX Symbol List
• MathML Characters - sorts out Unicode, HTML and MathML/TeX names on one page
• Unicode values and MathML names
• Unicode values and Postscript names from the source code for Ghostscript

Contents

 [hide]

• 1 Common symbols
• 2 See also
• 3 Variations
• 4 References
• 5 External links

Symbol in HTML	Symbol in TEX	Name	Explanation	Examples
		Read as
		Category
=		equality is equal to; equals everywhere	x = y means x and y represent the same thing or value.	2 = 2 1 + 1 = 2
≠		inequality is not equal to; does not equal everywhere	x ≠ y means that x and y do not represent the same thing or value. (The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.)	2 + 2 ≠ 5
< >		strict inequality is less than, is greater than order theory	x < y means x is less than y. x > y means x is greater than y.	3 < 4 5 > 4
		proper subgroup is a proper subgroup of group theory	H < G means H is a proper subgroup of G.	5Z < Z A3  < S3
≪ ≫		(very) strict inequality is much less than, is much greater than order theory	x ≪ y means x is much less than y. x ≫ y means x is much greater than y.	0.003 ≪ 1000000
		asymptotic comparison is of smaller order than, is of greater order than analytic number theory	f ≪ g means the growth of f is asymptotically bounded by g. (This is I. M. Vinogradov's notation. Another notation is the Big O notation, which looks like f = O(g).)	x ≪ ex
≤ ≥		inequality is less than or equal to, is greater than or equal to order theory	x ≤ y means x is less than or equal to y. x ≥ y means x is greater than or equal to y. (The forms <= and >= are generally used in programming languages where ease of typing and use of ASCII text is preferred.)	3 ≤ 4 and 5 ≤ 5 5 ≥ 4 and 5 ≥ 5
		subgroup is a subgroup of group theory	H ≤ G means H is a subgroup of G.	Z ≤ Z A3  ≤ S3
		reduction is reducible to computational complexity theory	A ≤ B means the problem A can be reduced to the problem B. Subscripts can be added to the ≤ to indicate what kind of reduction.	If then
≺		Karp reduction is Karp reducible to; is polynomial-time many-one reducible to computational complexity theory	L1 ≺ L2 means that the problem L1 is Karp reducible to L2.[1]	If L1 ≺ L2 and L2 ∈ P, then L1 ∈ P.
∝		proportionality is proportional to; varies as everywhere	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x.
		Karp reduction[2] is Karp reducible to; is polynomial-time many-one reducible to computational complexity theory	A ∝ B means the problem A can be polynomially reduced to the problem B.	If L1 ∝ L2 and L2 ∈ P, then L1 ∈ P.
+		addition plus; add arithmetic	4 + 6 means the sum of 4 and 6.	2 + 7 = 9
		disjoint union the disjoint union of ... and ... set theory	A1 + A2 means the disjoint union of sets A1 and A2.	A1 = {3, 4, 5, 6} ∧ A2 = {7, 8, 9, 10} ⇒ A1 + A2 = {(3,1), (4,1), (5,1), (6,1), (7,2), (8,2), (9,2), (10,2)}
−		subtraction minus; take; subtract arithmetic	9 − 4 means the subtraction of 4 from 9.	8 − 3 = 5
		negative sign negative; minus; the opposite of arithmetic	−3 means the negative of the number 3.	−(−5) = 5
		set-theoretic complement minus; without set theory	A − B means the set that contains all the elements of A that are not in B. (∖ can also be used for set-theoretic complement as described below.)	{1,2,4} − {1,3,4}  =  {2}
×		multiplication times; multiplied by arithmetic	3 × 4 means the multiplication of 3 by 4. (The symbol * is generally used in programming languages, where ease of typing and use of ASCII text is preferred.)	7 × 8 = 56
		Cartesian product the Cartesian product of ... and ...; the direct product of ... and ... set theory	X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y.	{1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
		cross product cross linear algebra	u × v means the cross product of vectors u and v	(1,2,5) × (3,4,−1) = (−22, 16, − 2)
		group of units the group of units of ring theory	R× consists of the set of units of the ring R, along with the operation of multiplication. This may also be written R* as described below, or U(R).
·		multiplication times; multiplied by arithmetic	3 · 4 means the multiplication of 3 by 4.	7 · 8 = 56
		dot product dot linear algebra	u · v means the dot product of vectors u and v	(1,2,5) · (3,4,−1) = 6
÷ ⁄		division (Obelus) divided by; over arithmetic	6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3.	2 ÷ 4 = .5 12 ⁄ 4 = 3
		quotient group mod group theory	G / H means the quotient of group G modulo its subgroup H.	{0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
		quotient set mod set theory	A/~ means the set of all ~ equivalence classes in A.	If we define ~ by x ~ y ⇔ x − y ∈ ℤ, then ℝ/~ = {x + n : n ∈ ℤ : x ∈ (0,1]}
±		plus-minus plus or minus arithmetic	6 ± 3 means both 6 + 3 and 6 − 3.	The equation x = 5 ± √4, has two solutions, x = 7 and x = 3.
		plus-minus plus or minus measurement	10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2.	If a = 100 ± 1 mm, then a ≥ 99 mm and a ≤ 101 mm.
∓		minus-plus minus or plus arithmetic	6 ± (3 ∓ 5) means both 6 + (3 − 5) and 6 − (3 + 5).	cos(x ± y) = cos(x) cos(y) ∓ sin(x) sin(y).
√		square root the (principal) square root of real numbers	means the nonnegative number whose square is x.
		complex square root the (complex) square root of complex numbers	if  is represented in polar coordinates with , then .
|…|		absolute value or modulus absolute value of; modulus of numbers	|x| means the distance along the real line (or across the complex plane) between x and zero.	|3| = 3 |–5| = |5| = 5 | i | = 1 | 3 + 4i | = 5
		Euclidean norm or Euclidean length or magnitude Euclidean norm of geometry	|x| means the (Euclidean) length of vector x.	For x = (3,-4)
		determinant determinant of matrix theory	|A| means the determinant of the matrix A
		cardinality cardinality of; size of; order of set theory	|X| means the cardinality of the set X. (# may be used instead as described below.)	|{3, 5, 7, 9}| = 4.
||…||		norm norm of; length of linear algebra	|| x || means the norm of the element x of a normed vector space.[3]	|| x  + y || ≤  || x ||  +  || y ||
		nearest integer function nearest integer to numbers	||x|| means the nearest integer to x. (This may also be written [x], ⌊x⌉, nint(x) or Round(x).)	||1|| = 1, ||1.6|| = 2, ||−2.4|| = −2, ||3.49|| = 3
∣ ∤		divisor, divides divides number theory	a|b means a divides b. a∤b means a does not divide b. (This symbol can be difficult to type, and its negation is rare, so a regular but slightly shorter vertical bar | character can be used.)	Since 15 = 3×5, it is true that 3|15 and 5|15.
		conditional probability given probability	P(A|B) means the probability of the event a occurring given that boccurs.	if X is a uniformly random day of the year P(X is May 25 | X is in May) = 1/31
		restriction restriction of … to …; restricted to set theory	f|A means the function f restricted to the set A, that is, it is the function with domain A ∩ dom(f) that agrees with f.	The function f : R → R defined by f(x) = x2 is not injective, but f|R+is injective.
		such that such that; so that everywhere	| means “such that”, see ":" (described below).	S = {(x,y) | 0 < y < f(x)} The set of (x,y) such that y is greater than 0 and less than f(x).
||		parallel is parallel to geometry	x || y means x is parallel to y.	If l || m and m ⊥ n then l ⊥ n.
		incomparability is incomparable to order theory	x || y means x is incomparable to y.	{1,2} || {2,3} under set containment.
		exact divisibility exactly divides number theory	pa || n means pa exactly divides n (i.e. pa divides n but pa+1 does not).	23 || 360.
#		cardinality cardinality of; size of; order of set theory	#X means the cardinality of the set X. (|…| may be used instead as described above.)	#{4, 6, 8} = 3
		connected sum connected sum of; knot sum of; knot composition of topology, knot theory	A#B is the connected sum of the manifolds A and B. If A and Bare knots, then this denotes the knot sum, which has a slightly stronger condition.	A#Sm is homeomorphic to A, for any manifold A, and the sphereSm.
ℵ		aleph number aleph set theory	ℵα represents an infinite cardinality (specifically, the α-th one, where α is an ordinal).	|ℕ| = ℵ0, which is called aleph-null.
ℶ		beth number beth set theory	ℶα represents an infinite cardinality (similar to ℵ, but ℶ does not necessarily index all of the numbers indexed by ℵ. ).
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