Reference Guide Official Cambridge Notation Updated for 2026

Cambridge Mathematics Notation Guide

The complete guide to mathematical symbols used in Cambridge International examinations. This notation is used across IGCSE Mathematics, Additional Mathematics, A-Level Mathematics, and Further Mathematics.

Why Learn Mathematical Notation?

Mathematical notation is the universal language of mathematics. Cambridge International uses a standardized set of symbols across all their mathematics qualifications. Understanding these symbols is essential for:

• Reading exam questions correctly without confusion
• Writing solutions that examiners will accept
• Understanding textbooks and mark schemes
• Preparing for university mathematics

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Set Notation = Miscellaneous + Operations f Functions e Exponential sin Trigonometry i Complex Numbers M Matrices a Vectors P Statistics

Set Notation

Used in IGCSE, A-Level, and Further Mathematics

∈

is an element of

Shows membership in a set

3 ∈ 4

∉

is not an element of

Shows non-membership in a set

5 ∉ 4

{x₁, x₂, ...}

set with elements

Lists the elements of a set

A = 8

{x : ...}

set of all x such that

Defines a set by a condition

{x : x > 0}

n(A)

number of elements

The cardinality of set A

n({a, b, c}) = 3

∅

the empty set

A set with no elements

∅ or ''

ℰ

the universal set

The set containing all elements under consideration

ℰ = {1,2,3,...,100}

A′

complement of A

All elements NOT in set A

If A = {1,2}, A′ = ℰ - A

ℕ

natural numbers

Counting numbers: 1, 2, 3, ...

ℕ = {1, 2, 3, ...}

ℤ

integers

Whole numbers including negatives

ℤ = {..., -2, -1, 0, 1, 2, ...}

ℚ

rational numbers

Numbers that can be written as fractions

½, 0.75, -3 ∈ ℚ

ℝ

real numbers

All numbers on the number line

π, √2, -5 ∈ ℝ

ℂ

complex numbers

Numbers with real and imaginary parts

3 + 2i ∈ ℂ

(x, y)

ordered pair

A pair where order matters

(2, 5) ≠ (5, 2)

⊆

is a subset of

All elements of A are in B

{1, 2} ⊆ {1, 2, 3}

⊂

is a proper subset of

Subset but not equal

{1, 2} ⊂ {1, 2, 3}

∪

union

Elements in A or B (or both)

{1,2} ∪ {2,3} = {1,2,3}

∩

intersection

Elements in both A and B

{1,2,3} ∩ {2,3,4} = {2,3}

[a, b]

closed interval

All x where a ≤ x ≤ b

[0, 5] includes 0 and 5

(a, b)

open interval

All x where a < x < b

(0, 5) excludes 0 and 5

Miscellaneous Symbols

Common mathematical relationships and comparisons

is equal to

Two expressions have the same value

2 + 3 = 5

≠

is not equal to

Two expressions have different values

3 ≠ 5

≡

is identical to / congruent to

True for all values, or congruent shapes

(x+1)² ≡ x² + 2x + 1

≈

is approximately equal to

Close to but not exactly equal

π ≈ 3.14

∝

is proportional to

Direct proportionality relationship

y ∝ x means y = kx

is less than

Strict inequality

3 < 5

⩽

is less than or equal to

Non-strict inequality

x ⩽ 5 means x can be 5

is greater than

Strict inequality

7 > 4

⩾

is greater than or equal to

Non-strict inequality

x ⩾ 0 includes zero

∞

infinity

Unbounded value, larger than any number

x → ∞

⇒

implies

If left is true, right must be true

x = 2 ⇒ x² = 4

⇐

is implied by

Reverse implication

x² = 4 ⇐ x = 2

⇔

if and only if (equivalent)

Both directions of implication

x > 0 ⇔ -x < 0

is distributed as

Random variable follows a distribution

X ~ N(0, 1)

≅

is isomorphic to

Structurally identical (Further Maths)

G ≅ H (groups)

Operations

Basic arithmetic and advanced operations

a + b

a plus b

Addition of two values

3 + 4 = 7

a − b

a minus b

Subtraction of two values

7 − 3 = 4

a × b, ab

a multiplied by b

Multiplication (× or juxtaposition)

3 × 4 = 12 or 3(4) = 12

a ÷ b, a/b

a divided by b

Division of two values

12 ÷ 3 = 4

summation

Sum of a series of terms

Σᵢ₌₁³ i = 1+2+3 = 6

√a

square root of a

Non-negative square root

√16 = 4 (not −4)

ⁿ√a

nth root of a

The real nth root

³√8 = 2

|a|

modulus of a

Absolute value (distance from zero)

|−5| = 5, |3| = 3

n factorial

Product of 1 to n

5! = 5×4×3×2×1 = 120

(ⁿᵣ) or ⁿCᵣ

binomial coefficient

Number of ways to choose r from n

(⁵₂) = 10

Functions

Function notation and calculus symbols

f(x)

value of function f at x

The output when x is the input

If f(x) = x², then f(3) = 9

f : A → B

function from A to B

Maps elements from set A to set B

f : ℝ → ℝ

f : x ↦ y

f maps x to y

Shows what each element maps to

f : x ↦ x²

f⁻¹

inverse function

Reverses the function

If f(x) = 2x, f⁻¹(x) = x/2

gf or g∘f

composite function

Apply f first, then g

gf(x) = g(f(x))

lim

limit

Value as x approaches a

lim(x→0) sin(x)/x = 1

dy/dx

derivative of y with respect to x

Rate of change / gradient

If y = x², dy/dx = 2x

f′(x), f″(x)

first, second derivatives

Alternative derivative notation

f(x) = x³, f′(x) = 3x²

dⁿy/dxⁿ

nth derivative

Differentiate n times

d²y/dx² is the second derivative

∫y dx

indefinite integral

Antiderivative / reverse of differentiation

∫2x dx = x² + c

∫ₐᵇ y dx

definite integral

Area under the curve from a to b

∫₀² x dx = 2

Δx, δx

increment of x

A small change in x

δx → 0

Exponential and Logarithmic Functions

Used in A-Level and Further Mathematics

base of natural logarithms

Euler's number ≈ 2.71828...

e = lim(n→∞)(1 + 1/n)ⁿ

eˣ, exp(x)

exponential function

e raised to the power x

e² ≈ 7.389

logₐ x

logarithm base a of x

The power to which a must be raised to get x

log₂ 8 = 3 because 2³ = 8

ln x

natural logarithm

Logarithm base e

ln e = 1, ln 1 = 0

lg x, log₁₀ x

logarithm base 10

Common logarithm

lg 100 = 2

sin

Circular and Hyperbolic Functions

Trigonometric functions used across all qualifications

sin, cos, tan

circular functions

Sine, cosine, and tangent

sin 30° = 0.5

cosec, sec, cot

reciprocal functions

Cosecant, secant, cotangent

sec θ = 1/cos θ

sin⁻¹, cos⁻¹, tan⁻¹

inverse circular functions

Also written as arcsin, arccos, arctan

sin⁻¹(0.5) = 30°

sinh, cosh, tanh

hyperbolic functions

Further Mathematics only

sinh x = (eˣ - e⁻ˣ)/2

sinh⁻¹, cosh⁻¹, tanh⁻¹

inverse hyperbolic functions

Further Mathematics only

sinh⁻¹ x = ln(x + √(x²+1))

cosech, sech, coth

reciprocal hyperbolic

Further Mathematics only

sech x = 1/cosh x

Complex Numbers

A-Level and Further Mathematics

imaginary unit

Defined as i² = −1

√(−1) = i

a complex number

z = x + iy = r(cos θ + i sin θ)

z = 3 + 4i

Re z

real part of z

The x component

Re(3 + 4i) = 3

Im z

imaginary part of z

The y component (coefficient of i)

Im(3 + 4i) = 4

|z|

modulus of z

Distance from origin: √(x² + y²)

|3 + 4i| = 5

arg z

argument of z

Angle with positive real axis (−π < θ ≤ π)

arg(1 + i) = π/4

complex conjugate

Reflects across real axis: x − iy

(3 + 4i)* = 3 − 4i

Matrices

A-Level and Further Mathematics

a matrix

Rectangular array of numbers

M = [1 2; 3 4]

M⁻¹

inverse matrix

MM⁻¹ = I (identity)

M × M⁻¹ = I

det M, |M|

determinant

Scalar value from square matrix

det[a b; c d] = ad − bc

identity matrix

1s on diagonal, 0s elsewhere

MI = IM = M

Vectors

Used across IGCSE, A-Level, and Further Mathematics

vector a

Bold lowercase letter represents a vector

a = (3, 4)

AB→

vector from A to B

Directed line segment

AB→ = B − A

unit vector

Vector with magnitude 1

â = a/|a|

i, j, k

unit vectors along axes

Unit vectors in x, y, z directions

i = (1,0,0)

(x, y) or [x; y]

column vector

Vector written as column

a = [3; 4] = 3i + 4j

|a|

magnitude of a

Length of the vector

|(3,4)| = 5

a · b

scalar (dot) product

Result is a scalar

a · b = |a||b|cos θ

a × b

vector (cross) product

Result is a vector perpendicular to both

|a × b| = |a||b|sin θ

Probability and Statistics

Extensively used in A-Level Statistics

A, B, C

events

Capital letters represent events

A = "roll a 6"

A ∪ B

A or B

At least one event occurs

P(A ∪ B)

A ∩ B

A and B

Both events occur

P(A ∩ B)

P(A)

probability of A

Likelihood of event A occurring

0 ≤ P(A) ≤ 1

A′

complement of A

Event A does not occur

P(A′) = 1 − P(A)

P(A | B)

conditional probability

Probability of A given B occurred

P(A|B) = P(A∩B)/P(B)

ⁿCᵣ

combinations

Ways to choose r from n (order doesn't matter)

⁵C₂ = 10

ⁿPᵣ

permutations

Ways to arrange r from n (order matters)

⁵P₂ = 20

X, Y

random variables

Capital letters for random variables

X = number of heads

E(X)

expected value

Mean of the random variable

E(X) = μ

Var(X)

variance

Measure of spread

Var(X) = E(X²) − [E(X)]²

standard deviation

Square root of variance

σ = √Var(X)

x̄

sample mean

Average of sample data

x̄ = Σxᵢ/n

B(n, p)

binomial distribution

n trials, probability p of success

X ~ B(10, 0.5)

N(μ, σ²)

normal distribution

Bell curve with mean μ and variance σ²

X ~ N(100, 225)

Po(λ)

Poisson distribution

Events per interval with rate λ

X ~ Po(3.5)

r, ρ

correlation coefficient

r for sample, ρ for population

−1 ≤ r ≤ 1

H₀, H₁

hypotheses

Null and alternative hypotheses

H₀: μ = 50, H₁: μ ≠ 50

Which Qualifications Use Each Section?

Section	IGCSE Maths	Add Maths	A-Level	Further Maths
Set Notation	✓	✓	✓	✓
Miscellaneous	✓	✓	✓	✓
Operations	✓	✓	✓	✓
Functions (basic)	−	✓	✓	✓
Functions (calculus)	−	✓	✓	✓
Exponential & Logarithmic	−	✓	✓	✓
Trigonometry (circular)	✓	✓	✓	✓
Trigonometry (hyperbolic)	−	−	−	✓
Complex Numbers	−	−	✓	✓
Matrices	−	−	−	✓
Vectors (2D)	✓	✓	✓	✓
Vectors (3D, cross product)	−	−	✓	✓
Probability & Statistics	Basic	−	✓	✓

Official Source

This notation list is based on the official Cambridge Assessment International Education document: Mathematics Notation List (for use from 2020) . Always refer to the official syllabus for your specific qualification for the most current information.