Symbols
| Symbol |
Symbol |
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 |
||
|
≦ ≧ |
congruence relation ...is less than ... is greater than... modular arithmetic | 7k ≡ 28 (mod 2) is only true if k is an even integer. Assume that the problem requires k to be non-negative; the domain is defined as 0 ≦ k ≦ ∞. | 10a ≡ 5 (mod 5) for 1 ≦ a ≦ 10 | |
| vector inequality ... is less than or equal... is greater than or equal... order theory | x ≦ y means that each component of vector x is less than or equal to each corresponding component of vector y. x ≧ y means that each component of vector x is greater than or equal to each corresponding component of vector y. It is important to note that x ≦ y remains true if every element is equal. However, if the operator is changed, x ≤ y is true if and only if x ≠ y is also true. |
|||
| ≺ |
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. | 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
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} | ||
| ± | 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). | |
| × |
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 |
a * b means the product of a and b. (Multiplication can also be denoted with × or ⋅, or even simple juxtaposition. * is generally used where ease of typing and use of ASCII text is preferred, such as programming languages.) |
4 * 3 means the product of 4 and 3, or 12. | |
|
convolution
convolution; convolved with functional analysis |
f * g means the convolution of f and g. | . | ||
| complex conjugate conjugate complex numbers | z* means the complex conjugate of z. ( can also be used for the conjugate of z, as described below.) |
. | ||
| 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 above, or U(R). |
|||
| hyperreal numbers the (set of) hyperreals non-standard analysis | *R means the set of hyperreal numbers. Other sets can be used in place of R. | *N is the hypernatural numbers. | ||
|
Hodge dual
Hodge dual; Hodge star linear algebra |
*v means the Hodge dual of a vector v. If v is a k-vector within an n-dimensional oriented inner product space, then *v is an (n−k)-vector. | If are the standard basis vectors of, | ||
| · |
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 | ||
| placeholder (silent) functional analysis | A · means a placeholder for an argument of a function. Indicates the functional nature of an expression without assigning a specific symbol for an argument. | |||
| ⊗ | tensor product, tensor product of modules tensor product of linear algebra | means the tensor product of V and U. means the tensor product of modules V and U over the ring R. | {1, 2, 3, 4} ⊗ {1, 1, 2} = {{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} |
|
| Kulkarni–Nomizu product Kulkarni–Nomizu product tensor algebra | Derived from the tensor product of two symmetric type (0,2) tensors; it has the algebraic symmetries of the Riemann tensor. has components . | |||
|
÷ ⁄ |
division (Obelus)
divided by; over arithmetic |
6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. | 2 ÷ 4 = 0.5 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) } |
||
| √ | square root the (principal) square root of real numbers | means the nonnegative number whose square is . | ||
| complex square root the (complex) square root of complex numbers | if is represented in polar coordinates with, then . | |||
| x |
mean
overbar; … bar statistics |
(often read as “x bar”) is the mean (average value of ). | . | |
| complex conjugate conjugate complex numbers | means the complex conjugate of z. (z* can also be used for the conjugate of z, as described above.) |
. | ||
| finite sequence, tuple finite sequence, tuple model theory | means the finite sequence/tuple . | . | ||
| algebraic closure algebraic closure of field theory | is the algebraic closure of the field F. | The field of algebraic numbers is sometimes denoted as because it is the algebraic closure of the rational numbers . | ||
| topological closure (topological) closure of topology | is the topological closure of the set S. This may also be denoted as cl(S) or Cl(S). |
In the space of the real numbers, (the rational numbers are dense in the real numbers). | ||
| â | unit vector hat geometry | (pronounced "a hat") is the normalized version of vector, having length 1. | ||
| |…| |
absolute value; 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. | || x + y || ≤ || x || + || y || | |
|
nearest integer function
nearest integer to
List Of Mathematical Symbols
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