Notation List
For Cambridge International Mathematics
Qualifications
For use from 2020
Notation List for Cambridge International Mathematics Qualifications (For use from 2020)
Mathematical notation
Examinations for CIE syllabuses may use relevant notation from the following list.
1 Set notation
∈
∉
{x1, x2, …}
{x : …}
n(A)
∅
,
A′
ℕ
ℤ
is an element of
is not an element of
the set with elements x1, x2, …
the set of all x such that …
the number of elements in set A
the empty set
the universal set
the universal set (for 0607 IGCSE International Mathematics)
the complement of the set A
the set of natural numbers, {1, 2, 3, …}
the set of integers, {0, ±1, ±2, ±3, …}
ℚ
the set of rational numbers,  : p, q ∈ ℤ, q ≠ 0 
ℝ
ℂ
(x, y)
⊆
⊂
⋃
⋂
[a, b]
[a, b)
(a, b]
(a, b)
(S, ∘)
the set of real numbers
the set of complex numbers
the ordered pair x, y
is a subset of
is a proper subset of
union
intersection
the closed interval {x ∈ ℝ : a ⩽ x ⩽ b}
the interval {x ∈ ℝ : a ⩽ x < b}
the interval {x ∈ ℝ : a < x ⩽ b}
the open interval {x ∈ ℝ : a < x < b}
the group consisting of the set S with binary operation ∘
p
q
2 Miscellaneous symbols
=
≠
≡
≈
~
≅
∝
<
⩽
>
⩾
∞
⇒
⇐
⇔
is equal to
is not equal to
is identical to or is congruent to
is approximately equal to
is distributed as
is isomorphic to
is proportional to
is less than
is less than or equal to
is greater than
is greater than or equal to
infinity
implies
is implied by
implies and is implied by (is equivalent to)
2


Notation List for Cambridge International Mathematics Qualifications (For use from 2020)
3 Operations
a+b
a–b
a × b, ab
a ÷ b,
a
b
n
a plus b
a minus b
a multiplied by b
a divided by b
∑ ai
a1 + a2 + … + an
a
the non-negative square root of a, for a ∈ ℝ, a ⩾ 0
the (real) nth root of a, for a ∈ ℝ, where n a . 0 for a ⩾ 0
the modulus of a
n factorial
i =1
n a
|a|
n!
( nr )
the binomial coefficient
n!
for n, r ∈ ℤ and 0 ⩽ r ⩽ n
r !(n − r )!
4 Functions
f(x)
f:A→B
f:x↦y
f –1
gf
lim f( x)
the value of the function f at x
f is a function under which each element of set A has an image in set B
the function f maps the element x to the element y
the inverse function of the one-one function f
the composite function of f and g which is defined by gf(x) = g(f(x))
the limit of f(x) as x tends to a
∆x, δx
an increment of x
x→a
dy
dx
d ny
dx n
f ′(x), f ″(x), …, f (n)(x)
∫ y dx
b
∫a y dx
x& , &&
x, K
the derivative of y with respect to x
the nth derivative of y with respect to x
the first, second, …, nth derivatives of f(x) with respect to x
the indefinite integral of y with respect to x
the definite integral of y with respect to x between the limits x = a and x = b
the first, second, … derivatives of x with respect to t
5 Exponential and logarithmic functions
e
ex, exp(x)
loga x
ln x
lg x, log10 x
base of natural logarithms
exponential function of x
logarithm to the base a of x
natural logarithm of x
logarithm of x to base 10
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Notation List for Cambridge International Mathematics Qualifications (For use from 2020)
6 Circular and hyperbolic functions
sin, cos, tan
cosec, sec, cot
}
the circular functions
sin −1 , cos −1 , tan −1 

cosec −1 , sec −1 , cot −1 
sinh, cosh, tanh
cosech, sech, coth
}
the inverse circular functions
the hyperbolic functions
sinh −1 , cosh −1 , tanh −1 
 the inverse hyperbolic functions
cosech −1 , sech −1 , coth −1 
7 Complex numbers
i
z
Re z
Im z
|z|
arg z
z*
the imaginary unit, i2 = –1
a complex number, z = x + iy = r(cos θ + i sin θ )
the real part of z, Re z = x
the imaginary part of z, Im z = y
the modulus of z, | z | = x 2 + y 2
the argument of z, arg z = θ where –π < θ ⩽ π
the complex conjugate of z, x – iy
8 Matrices
Μ
Μ−1
det M, | M |
I
a matrix Μ
the inverse of the non-singular square matrix Μ
the determinant of the square matrix Μ
an identity (or unit) matrix
9 Vectors
auuur
AB
â
i, j, k
x
x  
,  y
y  
z
()
the vector a
the vector represented in magnitude and direction by the directed line segment
AB
a unit vector in the direction of a
unit vectors in the directions of the Cartesian coordinate axes
the vectors xi + yj (in 2 dimensions) and xi + yj + zk (in 3 dimensions)
| a |, a
uuur
AB , AB
the magnitude of a
uuur
the magnitude of AB
a.b
a×b
the scalar product of a and b
the vector product of a and b
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Notation List for Cambridge International Mathematics Qualifications (For use from 2020)
10 Probability and statistics
A, B, C, …
A⋃B
A⋂B
P(A)
A′
P(A | B)
events
union of the events A and B
intersection of the events A and B
probability of the event A
complement of the event A
probability of the event A conditional on the event B
n
Cr
the number of combinations of r objects from n, n Cr =
n
Pr
the number of permutations of r objects from n, n Pr =
X, Y, R, …
x, y, r, …
x1, x2, …
f1, f2, …
p(x)
p1, p2, …
f(x)
F(x)
E(X)
E(g(X))
Var(X)
GX(t)
MX(t)
B(n, p)
Geo(p)
Po(λ)
N(µ, σ2)
random variables
values of the random variables X, Y, R, …
observations
frequencies with which the observations x1, x2, … occur
probability function P(X = x) of the discrete random variable X
probabilities of the values x1, x2, … of the discrete random variable X
value of the probability density function of a continuous random variable X
value of the cumulative distribution function of a continuous random variable X
expectation of the random variable X
expectation of g(X)
variance of the random variable X
probability generating function for the discrete random variable X
moment generating function for the random variable X
binomial distribution with parameters n and p
geometric distribution with parameter p
Poisson distribution with parameter λ
normal distribution with mean µ and variance σ2
population mean
population variance
population standard deviation
1 n
sample mean, x =
xi
n i =1
µ
σ2
σ
x
( nr ) = r !(nn−! r )!
n!
( n − r )!
∑
1 n
( xi − x ) 2
n − 1 i =1
∑
s2
unbiased estimate of population variance from a sample, s 2 =
ρ
product moment correlation coefficient for a population
product moment correlation coefficient for a sample
probability density function of the standardised normal variable Z ~ N(0, 1)
cumulative distribution function of the standardised normal variable Z ~ N(0, 1)
null and alternative hypotheses for a hypothesis test
r
φ
Φ
H0, H1
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