Identities (Edexcel GCSE Maths): Revision Notes
Identities
What is an identity?
An identity is a mathematical statement that is always true, regardless of what values you substitute for the variables. Unlike equations that are only true for specific values, identities remain equal for all possible values.
The key features of identities are:
- The left-hand side always equals the right-hand side
- This equality holds for any value you substitute
- We use the symbol '≡' to represent an identity (not '=')
- You might need to prove that an identity is true in your exam
Think of an identity as a mathematical fact that never changes, no matter what numbers you put in place of the variables. This is what makes identities so powerful in mathematics - their universal truth.
Golden rule for working with identities
When proving an identity, remember this important rule:
An identity is not like an equation - do not solve it using the balance method.
Instead, you should:
- Manipulate each side separately ✓
- Do not apply the same operation to both sides ✗
This means you work on one side at a time until both sides look identical.
How to prove an identity
When proving an identity in an exam, follow these steps:
- Show every line of your working - you get marks for the process
- Start with the more complex side (usually the left-hand side)
- Use algebraic techniques such as:
- Multiplying out brackets
- Simplifying expressions
- Factorising
- Work towards the expression on the right-hand side
Worked Example: Proving an Identity
Prove that
Starting with the left-hand side:
(n - 1)² + (n + 1)²
= (n - 1)(n - 1) + (n + 1)(n + 1)
= n² - 2n + 1 + n² + 2n + 1
= 2n² + 2
= 2(n² + 1)
This matches the right-hand side, so the identity is proven.
Using algebra to prove number properties
You can use algebraic identities to prove facts about different types of numbers. Algebra allows you to represent any integer and show properties that apply to all numbers of that type.
Representing different types of numbers
| Number type | Algebraic representation |
|---|---|
| Even number | |
| Odd number | or |
| Multiple of 3 | |
| Consecutive numbers | |
| Consecutive even numbers | |
| Consecutive odd numbers | |
| Consecutive square numbers |
Understanding multiples
If is an integer, then we can make important observations about multiples. For example, 3n is always a multiple of 3, while 3n + 1 and 3n + 2 are never multiples of 3.
This is because any integer can be written as , , or , where only the first form is divisible by 3. This principle helps us understand why certain number patterns always work.
Worked Example: Consecutive Integers
Show that the sum of any three consecutive integers is always a multiple of 3.
Let the three consecutive integers be: , , and
Sum = = = =
Since is an integer, is always a multiple of 3.
Exam tips
- Always show your working step by step
- Start with the more complex side when proving identities
- Use brackets carefully when multiplying out expressions
- Check your final answer matches the target expression
- Remember to state your conclusion clearly
Key Points to Remember:
- An identity uses the symbol ≡ and is always true for any value
- Work on each side separately - don't treat it like an equation
- Show every step of your working in exam questions
- Use algebraic representations to prove properties of number types
- Start with the complex side and work towards the simpler expression