Algebraic Notation & Vocabulary | CIE IGCSE Maths: Core Revision Notes 2025

Algebraic Notation

What is algebra?

Algebra is a topic in mathematics that uses letters to represent general (or unknown) numbers
- x and y are two unknown numbers
  - More information is needed to find their values
Letters are also called variables

How do I write calculations in algebra?

Writing mathematical ideas in letters is called using algebraic notation
You can use + and – to mean add and subtract
- a + b
  c + d – e
However no symbol is used for multiplication
- ab (means a × b)
  3ab (means 3 × a × b)
Fractions are used for division
- $\frac{a}{b}$ (means a ÷ b)
You can combine the ideas above
- $a b + \frac{c}{3}$ (means a×b + c÷3)
- The order of operations for numbers still works
  - work out a×b and c÷d before adding them together
Powers (indices) and roots are the same as with numbers
- a² means a × a
  4a² means 4 × a²
  - With the order of operations, a² happens before multiplying by 4
- $\sqrt{a}$ means the square root of $a$
Brackets work in the same way as they do with numbers
- $3 (a + b)$ means $3 \times (a + b)$
  - Add the a and b first, then multiply the result by 3

Worked example

Raheem is playing a game and has q points.
He then earns a further 6 points.

(a)

Use algebra to write down the number of points that Raheem has.

Raheem has q + 6 points

At this point, the game doubles his number of points.

(b)

Use algebra to write down the number of points that he now has.

Raheem has 2(q + 6) points

Algebraic Vocabulary

What is a term?

A term is either:
- a letter (variable) on its own, or a variable raised to a power
  - For example, x or x²
- a number on its own
  - For example, 20
  - These are also called constants as they can't change value
- or a number multiplied by a letter
  - For example, 5x
The number in front of a letter is called a coefficient
- The coefficient of x in 6x is 6
- The coefficient of y in -5y is -5
Terms can include powers and more than one letter,
- 6xy
  4x²ab³c

What is a factor?

A factor is any number or letter that divides a term exactly
- There is no remainder
  - The factors of 3x are 1, 3, x and 3x
  - The factors of 4xy are 1, 2, 4, x, 2x, 4x, y, 2y, 4y, xy, 2xy and 4xy
A term can be separated into factors that multiply together to give that term
- Two factors of 5x are 5 and x
  - 5 × x = 5x
To factorise means to write something as a multiplication of factors
When comparing two terms, a common factor is one that divides both
- Compare 6xy with 4x
  - Common factors are 1, 2, x and 2x
  - The highest (or greatest) common factor is 2x

What is an expression?

An expression is an algebraic statement that does not have an equals sign

There is nothing to solve

An expression is made by adding, subtracting, multiplying or dividing terms

2x + 5y
b² – 2cd
$\frac{6 y}{5 t}$
A single term is still an expression

Expressions can be simplified (made easier)

x + x + x simplifies to 3x

What is an equation?

An equation is an algebraic statement with an equals sign between a left-hand side and a right-hand side
- Both sides are equal in value
- For example, if 2x has the same value as 10, then 2x = 10
An equation can be solved by finding the values of the letters that make both sides equal
- The equation 2x = 10 is solved when x has the value of 5
  - x = 5 is called the solution

What is a formula?

A formula is a rule, definition or relationship between different quantities, written in shorthand using letters
- For example, weight, w, is mass, m, multiplied by gravitational acceleration, g
  - The formula is w = mg
It is common to substitute numbers into a formula
- But a formula on its own cannot be solved
To turn a formula into an equation, more information is needed
- For example, if w = 50 and m = 5, the formula w = mg becomes the equation 50 = 5g

Examiner Tip

Knowing the differences between an expression, an equation and a formula will help you to understand the wording of exam questions.

Worked example

(a)

From the list below, write down

(i)

an expression,

(ii)

an equation.

2x + 5 = 4 7x – 9 x = vt – w

(i)

An expression does not have an equals sign

7x – 9 is the expression

(ii)

An equation has an equals sign and can be solved

2x + 5 = 4 is the equation

(b)

If x = 10, v = 2 and w = 3, use the formula shown to write an equation in t.

x = vt – w is the formula shown (a group of different quantities forming a relationship)

Substitute x = 10, v = 2 and w = 3 into the formula

10 = 2t – 3

Algebraic Notation & Vocabulary (CIE IGCSE Maths: Core)

Revision Note