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The following table lists many specialized symbols commonly used in mathematics.


Equality, inequality

[edit]
symbol symbol name reads as used in brief description examples
= equality is equal to; equals everywhere x = y means x and y represent the same thing or value. In computing, multiple equals signs are often used for comparison, while single equals signed are used for assignment. 1 + 1 = 2

<>
 !=
inequation is not equal to; does not equal everywhere x ≠ y means that x and y do not represent the same thing or value. The symbols != and <> are primarily from computer science. They are avoided in mathematical texts. 1 ≠ 2
> strict inequality is greater than order theory x > y means x is greater than y. 2 > 1
< strict inequality is less than order theory x < y means x is less than y. 1 < 2
>> strict inequality is much greater than order theory x >> y means x is much greater than y. 1000 >> 1
<< strict inequality is much less than order theory x >> y means x is much less than y. 1 << 1000
inequality is greater than or equal to order theory xy means x is greater than or equal to y. 2 ≥ 1
inequality is less than or equal to order theory xy means x is less than or equal to y. 1 ≤ 1
proportionality is proportional to everywhere yx means that y = kx for some constant k. if y = 2x, then yx
symbol symbol name reads as used in brief description examples


Basic operators

[edit]
symbol symbol name reads as used in brief description examples
+ addition plus arithmetic 4 + 6 means the sum of 4 and 6. 2 + 7 = 9
subtraction minus arithmetic 9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5
negative sign negative ; minus arithmetic − 3 means the negative of the number 3. −(−5) = 5
×
*
·
multiplication times arithmetic 3 × 4 means the multiplication of 3 by 4. 7 × 8 = 56
÷
division divided by arithmetic 6 ÷ 3 or 6 ⁄ 3 means the division of 6 by 3. 2 ÷ 4 = .5

12 ⁄ 4 = 3
symbol symbol name reads as used in brief description examples


Vector algebra

[edit]
symbol symbol name reads as used in brief description examples
×
cross product cross vector algebra u × v means the cross product of vectors u and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
· dot product dot vector algebra u · v means the dot product of vectors u and v (1,2,5) · (3,4,−1) = 6
symbol symbol name reads as used in brief description examples


Set theory

[edit]
symbol symbol name reads as used in brief description examples
+ disjoint union the disjoint union of ... and ... set theory A1 + A2 means the disjoint union of sets A1 and A2. A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒
A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
set-theoretic complement minus; without set theory A − B means the set that contains all the elements of A that are not in B. {1,2,4} − {1,3,4}  =  {2}
× 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)}
symbol symbol name reads as used in brief description examples





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||plus-minus | rowspan=3|6 3 means both 6 + 3 and 6 - 3.
6 3 5 means both 6 + 3 - 5 and 6 - 3 + 5. | rowspan=3|6 3 = 9 or 3
6 3 5 = 4 or 8 |- |align=center|plus-minus; plus-or-minus
minus-plus; minus-or-plus |- |align=right|arithmetic

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| rowspan=6 bgcolor=#d0f0d0 align=center|

||square root | rowspan=3|√x means the positive number whose square is x. | rowspan=3|√4 = 2 |- |align=center|the principal square root of; square root |- |align=right|real numbers

|- ||complex square root | rowspan=3| if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(i φ/2). | rowspan=3|√(-1) = i |- |align=center|the complex square root of …

square root |- |align=right|complex numbers

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| rowspan=6 bgcolor=#d0f0d0 align=center|

|…|

||absolute value | rowspan=3| |x| means the distance along the real line (or across the complex plane) between x and zero. | rowspan=3| |3| = 3

|–5| = |5|

i | = 1

| 3 + 4i | = 5 |- |align=center|absolute value of |- |align=right|numbers

|- ||Euclidean distance | rowspan=3| |x – y| means the Euclidean distance between x and y. | rowspan=3| For x = (1,1), and y = (4,5),
|x – y| = √([1–4]2 + [1–5]2) = 5 |- |align=center|Euclidean distance between; Euclidean norm of |- |align=right|Geometry

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|

||divides | rowspan=3| A single vertical bar is used to denote divisibility.
a|b means a divides b. | rowspan=3| Since 15 = 3×5, it is true that 3|15 and 5|15. |- |align=center|divides |- |align=right|Number Theory

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!

||factorial | rowspan=3|n ! is the product 1 × 2× ... × n. | rowspan=3|4! = 1 × 2 × 3 × 4 = 24 |- |align=center|factorial |- |align=right|combinatorics


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T

||transpose | rowspan=3| Swap rows for columns | rowspan=3| |- |align=center|transpose |- |align=right|matrix operations


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~

||probability distribution | rowspan=3| X ~ D, means the random variable X has the probability distribution D. | rowspan=3|X ~ N(0,1), the standard normal distribution |- |align=center|has distribution |- |align=right|statistics

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||material implication | rowspan=3|AB means if A is true then B is also true; if A is false then nothing is said about B.

→ may mean the same as ⇒, or it may have the meaning for functions given below.

⊃ may mean the same as ⇒, or it may have the meaning for superset given below. | rowspan=3|x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2). |- |align=center|implies; if … then |- |align=right|propositional logic

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||material equivalence | rowspan=3|A ⇔ B means A is true if B is true and A is false if B is false. | rowspan=3|x + 5 = y +2  ⇔  x + 3 = y |- |align=center|if and only if; iff |- |align=right|propositional logic

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¬

˜

||logical negation | rowspan=3|The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.

(The symbol ~ has many other uses, so ¬ or the slash notation is preferred.) | rowspan=3|¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y) |- |align=center|not |- |align=right|propositional logic

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||logical conjunction or meet in a lattice | rowspan=3|The statement A B is true if A and B are both true; else it is false.

For functions A(x) and B(x), A(x) B(x) is used to mean min(A(x), B(x)). | rowspan=3|n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number. |- |align=center|and; min |- |align=right|propositional logic, lattice theory

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||logical disjunction or join in a lattice | rowspan=3|The statement AB is true if A or B (or both) are true; if both are false, the statement is false.

For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)). | rowspan=3|n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number. |- |align=center|or; max |- |align=right|propositional logic, lattice theory

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||exclusive or

| rowspan=3| The statement AB is true when either A or B, but not both, are true. AB means the same. | rowspan=3| (¬A) ⊕ A is always true, AA is always false. |- |align=center|xor |- |align=right|propositional logic, Boolean algebra |- ||direct sum |rowspan=3|The direct sum is a special way of combining several one modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).

|rowspan=3|Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = VW ⇔ (U = V + W) ∧ (VW = ∅) |- |align=center|direct sum of |- |align=right|Abstract algebra

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||universal quantification | rowspan=3|∀ x: P(x) means P(x) is true for all x. | rowspan=3|∀ n ∈ ℕ: n2 ≥ n. |- |align=center|for all; for any; for each |- |align=right|predicate logic

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||existential quantification | rowspan=3|∃ x: P(x) means there is at least one x such that P(x) is true. | rowspan=3|∃ n ∈ ℕ: n is even. |- |align=center|there exists |- |align=right|predicate logic

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∃!

||uniqueness quantification | rowspan=3|∃! x: P(x) means there is exactly one x such that P(x) is true. | rowspan=3|∃! n ∈ ℕ: n + 5 = 2n. |- |align=center|there exists exactly one |- |align=right|predicate logic

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:=



:⇔

||definition | rowspan=3|x := y or x ≡ y means x is defined to be another name for y

(Some writers useto mean congruence).

P :⇔ Q means P is defined to be logically equivalent to Q. | rowspan=3|cosh x := (1/2)(exp x + exp (−x))

A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B) |- |align=center|is defined as |- |align=right|everywhere

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||congruence | rowspan=3|△ABC ≅ △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. | rowspan=3| |- |align=center|is congruent to |- |align=right|geometry

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{ , }

||set brackets | rowspan=3|{a,b,c} means the set consisting of a, b, and c. | rowspan=3|ℕ = { 1, 2, 3, …} |- |align=center|the set of … |- |align=right|set theory

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{ : }

{ | }

||set builder notation | rowspan=3|{x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. | rowspan=3|{n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4} |- |align=center|the set of … such that |- |align=right|set theory

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{ }

||empty set

| rowspan=3|∅ means the set with no elements. { } means the same. | rowspan=3|{n ∈ ℕ : 1 < n2 < 4} = ∅ |- |align=center| the empty set |- |align=right|set theory

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||set membership | rowspan=3|a ∈ S means a is an element of the set S; a  S means a is not an element of S. | rowspan=3|(1/2)−1 ∈ ℕ

2−1  ℕ |- |align=center|is an element of; is not an element of |- |align=right|everywhere, set theory

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||subset | rowspan=3|(subset) A ⊆ B means every element of A is also element of B.

(proper subset) A ⊂ B means A ⊆ B but A ≠ B.

(Some writers use the symbol ⊂ as if it were the same as ⊆.) | rowspan=3|(A ∩ B) ⊆ A

ℕ ⊂ ℚ

ℚ ⊂ ℝ |- |align=center|is a subset of |- |align=right|set theory

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||superset | rowspan=3|A ⊇ B means every element of B is also element of A.

A ⊃ B means A ⊇ B but A ≠ B.

(Some writers use the symbol ⊃ as if it were the same as ⊇.) | rowspan=3|(A ∪ B) ⊇ B

ℝ ⊃ ℚ |- |align=center|is a superset of |- |align=right|set theory

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||set-theoretic union | rowspan=3|(exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both.
"A or B, but not both."

(inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B.
"A or B or both". | rowspan=3|A ⊆ B  ⇔  (A ∪ B) = B (inclusive) |- |align=center|the union of … and

union |- |align=right|set theory

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||set-theoretic intersection | rowspan=3|A ∩ B means the set that contains all those elements that A and B have in common. | rowspan=3|{x ∈ ℝ : x2 = 1} ∩ ℕ = {1} |- |align=center|intersected with; intersect |- |align=right|set theory

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||symmetric difference | rowspan=3| means the set of elements in exactly one of A or B. | rowspan=3|{1,5,6,8} {2,5,8} = {1,2,6} |- |align=center|symmetric difference |- |align=right|set theory

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||set-theoretic complement | rowspan=3|A ∖ B means the set that contains all those elements of A that are not in B. | rowspan=3|{1,2,3,4} ∖ {3,4,5,6} = {1,2} |- |align=center|minus; without |- |align=right|set theory

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( )

||function application | rowspan=3|f(x) means the value of the function f at the element x. | rowspan=3|If f(x) := x2, then f(3) = 32 = 9. |- |align=center|of |- |align=right|set theory

|- |precedence grouping | rowspan=3|Perform the operations inside the parentheses first. | rowspan=3|(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. |- |align=center|parentheses |- |align=right|everywhere

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f:XY

||function arrow | rowspan=3|fX → Y means the function f maps the set X into the set Y. | rowspan=3|Let f: ℤ → ℕ be defined by f(x) := x2. |- |align=center|from … to |- |align=right|set theory

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o

||function composition | rowspan=3|fog is the function, such that (fog)(x) = f(g(x)). | rowspan=3|if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3). |- |align=center|composed with |- |align=right|set theory

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N

||natural numbers | rowspan=3|N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention. | rowspan=3|ℕ = {|a| : a ∈ ℤ, a ≠ 0} |- |align=center|N |- |align=right|numbers

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Z

||integers

| rowspan=3|ℤ means {..., −3, −2, −1, 0, 1, 2, 3, ...} and ℤ+ means {1, 2, 3, ...} = ℕ. | rowspan=3|ℤ = {p, -p : p ∈ ℕ} ∪ {0} |- |align=center|Z |- |align=right|numbers

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Q

||rational numbers

| rowspan=3|ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}. | rowspan=3|3.14000... ∈ ℚ

π ∉ ℚ |- |align=center|Q |- |align=right|numbers

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R

||real numbers

| rowspan=3|ℝ means the set of real numbers. | rowspan=3|π ∈ ℝ

√(−1) ∉ ℝ |- |align=center|R |- |align=right|numbers

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C

||complex numbers

| rowspan=3|ℂ means {a + b i : a,b ∈ ℝ}. | rowspan=3|i = √(−1) ∈ ℂ |- |align=center|C |- |align=right|numbers |- ||arbitrary constant | rowspan=3| C can be any number, most likely unknown; usually occurs when calculating antiderivatives. | rowspan=3|if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C |- |align=center|C |- |align=right|integral calculus

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||infinity | rowspan=3|∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. | rowspan=3|limx→0 1/|x| = ∞ |- |align=center|infinity |- |align=right|numbers

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π

||pi

| rowspan=3|π is the ratio of a circle's circumference to its diameter. Its value is 3.14159265... . | rowspan=3|A = π r² is the area of a circle with radius r

π radians = 180°

π ≈ 22 / 7 |- |align=center|pi |- |align=right|Euclidean geometry

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| rowspan=3 bgcolor=#d0f0d0 align=center|

||…||

||norm | rowspan=3| || x || is the norm of the element x of a normed vector space. | rowspan=3| || x  + y || ≤  || x ||  +  || y || |- |align=center|norm of

length of |- |align=right| linear algebra

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| rowspan=3 bgcolor=#d0f0d0 align=center|

||summation | rowspan=3| means a1 + a2 + … + an. | rowspan=3| = 12 + 22 + 32 + 42 

= 1 + 4 + 9 + 16 = 30

|- |align=center|sum over … from … to … of |- |align=right|arithmetic

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||product | rowspan=3| means a1a2···an. | rowspan=3| = (1+2)(2+2)(3+2)(4+2)

= 3 × 4 × 5 × 6 = 360

|- |align=center|product over … from … to … of |- |align=right|arithmetic

|- ||Cartesian product | rowspan=3| means the set of all (n+1)-tuples

(y0, …, yn).

| rowspan=3| |- |align=center|the Cartesian product of; the direct product of |- |align=right|set theory

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||coproduct | rowspan=3| | rowspan=3| |- |align=center|coproduct over … from … to … of |- |align=right|category theory

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||derivative | rowspan=3|f ′(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. | rowspan=3|If f(x) := x2, then f ′(x) = 2x |- |align=center|… prime

derivative of |- |align=right|calculus

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| rowspan=6 bgcolor=#d0f0d0 align=center|

||indefinite integral or antiderivative | rowspan=3|∫ f(x) dx means a function whose derivative is f. | rowspan=3| ∫x2 dx = x3/3 + C |- |align=center|indefinite integral of

the antiderivative of |- |align=right|calculus

|- ||definite integral | rowspan=3|∫ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. | rowspan=3|∫0b x2  dx = b3/3; |- |align=center|integral from … to … of … with respect to |- |align=right|calculus

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||gradient | rowspan=3|∇f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn). | rowspan=3|If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z) |- |align=center|del, nabla, gradient of |- |align=right|calculus

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||partial derivative | rowspan=3| With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant. | rowspan=3| If f(x,y) := x2y, then ∂f/∂x = 2xy |- |align=center|partial derivative of |- |align=right|calculus

|- |boundary | rowspan=3| ∂M means the boundary of M | rowspan=3| ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2} |- |align=center|boundary of |- |align=right|topology

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||perpendicular | rowspan=3|xy means x is perpendicular to y; or more generally x is orthogonal to y. | rowspan=3|If lm and mn then l || n. |- |align=center|is perpendicular to |- |align=right|geometry

|- ||bottom element | rowspan=3|x = ⊥ means x is the smallest element. | rowspan=3|∀x : x ∧ ⊥ = ⊥ |- |align=center|the bottom element |- |align=right|lattice theory

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| rowspan=3 bgcolor=#d0f0d0 align=center|

||

||parallel | rowspan=3|x || y means x is parallel to y. | rowspan=3|If l || m and mn then ln. |- |align=center|is parallel to |- |align=right|geometry

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||entailment | rowspan=3| AB means the sentence A entails the sentence B, that is every model in which A is true, B is also true. | rowspan=3| AA ∨ ¬A |- |align=center|entails |- |align=right| model theory

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| rowspan=3 bgcolor=#d0f0d0 align=center|

||inference | rowspan=3|xy means y is derived from x. | rowspan=3| AB ⊢ ¬B → ¬A |- |align=center|infers or is derived from |- |align=right|propositional logic, predicate logic

|-

| rowspan=3 bgcolor=#d0f0d0 align=center|

||normal subgroup | rowspan=3| NG means that N is a normal subgroup of group G. | rowspan=3| Z(G) ◅ G |- |align=center|is a normal subgroup of |- |align=right|group theory

|-

| rowspan=6 bgcolor=#d0f0d0 align=center|

/

||quotient group | rowspan=3| G/H means the quotient of group G modulo its subgroup H. | rowspan=3| {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}} |- |align=center| mod |- |align=right| group theory

|- |quotient set | rowspan=3| A/~ means the set of all ~ equivalence classes in A. |- |align=center| |- |align=right| set theory

|-

| rowspan=6 bgcolor=#d0f0d0 align=center|

||isomorphism | rowspan=3| GH means that group G is isomorphic to group H | rowspan=3| Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group. |- |align=center | is isomorphic to |- |align=right| group theory |- |approximately equal | rowspan=3|xy means x is approximately equal to y | rowspan=3|π ≈ 3.14159 |- |align=center|is approximately equal to |- |align=right|everywhere |-

| rowspan=3 bgcolor=#d0f0d0 align=center|

~

||same order of magnitude | rowspan=3| m ~ n, means the quantities m and n have the general size.

(Note that ~ is used for an approximation that is poor, otherwise use ≈ .) | rowspan=3|2 ~ 5

8 × 9 ~ 100

but π2 ≈ 10 |- |align=right|roughly similar

poorly approximates |- |align=right|Approximation theory


|-

| rowspan=3 bgcolor=#d0f0d0 align=center|

<,>

||inner product | rowspan=3|<x,y> means the inner product between x and y, as defined in an inner product space. | rowspan=3|The standard inner product between two vectors x = (2, 3) and y = (-1, 5) is:
<x, y> = 2×-1 + 3×5 = 13 |- |align=center|inner product of |- |align=right|vector algebra

|-

| rowspan=3 bgcolor=#d0f0d0 align=center|

||tensor product | rowspan=3| VU means the tensor product of V and U. | rowspan=3| {1, 2, 3, 4} ⊗ {1,1,2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}} |- |align=center| tensor product of |- |align=right| linear algebra |}

See also

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