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Main Notation

Special symbols
= equal to
≡ identically equal to
≠ not equal to
≈ approximately equal to
∼ of same order as (used in comparisons of infinitesimals or infinites)
< less than; “a less than b” is written as a < b (or, equivalently, b > a)
≤ less than or equal to; a less than or equal to b is written as a ≤ b

much less than; a much less than b is written as a b
> greater than; a greater than b is written as a > b (or, equivalently, b < a)
≥ greater than or equal to; a greater than or equal to b is written as a ≥ b

much greater than; a much greater than b is written as a b
+ plus sign; the sum of numbers a and b is denoted by a + b and has the property
a + b = b + a
– minus sign; the difference of numbers a and b is denoted by a – b
⋅ multiplication sign; the product of numbers a and b is denoted by either ab
or a ⋅ b (sometimes a × b) and has the property ab = ba; the inner product of
vectors a and b is denoted by a ⋅ b
× multiplication sign; the product of numbers a and b is sometimes denoted by
a × b; the cross-product of vectors a and b is denoted by a × b
: division sign; the ratio of numbers a and b is denoted by a:b or a/b
! factorial sign: 0!= 1!= 1, n!= 1 ⋅ 2 ⋅ 3 ... (n – 1)n, n = 2, 3, 4, ...
!! double factorial sign: 0!! = 1!! = 1,(2n)!! = 2 ⋅ 4 ⋅ 6 ... (2n), (2n + 1)!! =
1 ⋅ 3 ⋅ 5 ... (2n + 1), where n = 1, 2, 3, ...
% percent sign; 1% is one hundredth of the entire quantity
∞ infinity
→ tends (infinitely approaches) to; x → a means that x tends to a
=⇒ implies; consequently
⇐⇒ is equivalent to (if and only if ... )
∀ for all, for any
∃ there exists
belongs to; a A means that a is an element of the set A
does not belong to; a A means that a is not an element of the set A
∪ union (Boolean addition); A ∪ B stands for the union of sets A and B
∩ intersection (Boolean multiplication); A∩B stands for the intersection (com-
mon part) of sets A and B
⊂ inclusion; A ⊂ B means that the set A is part of the set B
⊆ nonstrict inclusion; A ⊆ B means that the set A is part of the set B or coincides
with B
∅ empty set
n
sum, a k = a 1 + a 2 + ··· + a n

k=1
n

product, a k = a 1 ⋅ a 2 ⋅ ... ⋅ a n
k=1
∂ symbol used to denote partial derivatives and differential operators; ∂ x is the
operator of differentiation with respect to x
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