Algebraic ProofEdexcel IGCSE Maths: Revision notes

Section 1

What is the difference between an identity and an equation?

An equation is only true for particular values of the variable, e.g. $2x + 1 = 7$ is true only when $x = 3$ .

An identity is true for every value of the variable. We write this with the symbol $\equiv$ instead of $=$ .

For example: $3(x + 2) \equiv 3x + 6$

This holds for any value you substitute for $x$ — try $x = 0$ , $x = 1$ , $x = -5$ , it always balances. That is what makes it an identity rather than an equation.

Exam questions often say "show that ... is an identity" or ask you to find missing coefficients so that both sides match for all $x$ — this is done by comparing coefficients (matching the numbers in front of each power of $x$ , and the constant terms, on both sides).

Key termsidentityequationcoefficient

Common mistake

Do not write $\equiv$ for a normal equation you are solving — save it strictly for statements true for all values of the variable.

Exam tip

To check if something is an identity, expand both sides fully and see if every term matches. If they do, for all $x$ , write $\equiv$ .

Section 2

How do we represent numbers algebraically for proof?

Algebraic proof uses letters to stand for any whole number, so that a conclusion holds generally rather than just for one example.

$n$ represents any integer
$2n$ represents any even number (since any integer multiplied by 2 is even)
$2n + 1$ or $2n - 1$ represents any odd number
$n$ and $n + 1$ represent two consecutive integers
$n, n+1, n+2$ represent three consecutive integers
$2n, 2n+2, 2n+4$ represent three consecutive even numbers
$n^2$ represents any square number

Using a specific number like $4$ or $7$ never proves a general statement — it only checks one case. Using $n$ shows the result is true for every possible integer.

Key termsintegerconsecutive integers

Example

Prove that the sum of two consecutive integers is always odd. Let the integers be $n$ and $n+1$ . Sum $= n + (n+1) = 2n + 1$ , which is $2 \times \text{(integer)} + 1$ , so it is always odd.

Common mistake

Writing 'let the number be 3' and checking only that case is not a proof — it is a single example, and the exam gives no marks for it.

Section 3

How do we structure a full algebraic proof?

A good proof follows a clear sequence:

Define the unknown(s) using algebra (e.g. "let the integer be $n$ ").
Set up an expression matching the statement in the question.
Expand and simplify the expression using algebraic manipulation.
Interpret the simplified result, linking it back to what had to be shown (e.g. explain why the result must be even, a multiple of a number, etc.).
Conclude with a statement such as "therefore ... is always true" — do not just stop at the algebra.

Example: Prove that the sum of any three consecutive integers is a multiple of 3.

Let the integers be $n$ , $n+1$ , $n+2$ .

Sum $= n + (n+1) + (n+2) = 3n + 3 = 3(n+1)$ .

Since $3(n+1)$ is 3 multiplied by an integer, the sum is always a multiple of 3. $\blacksquare$

Marks are typically awarded for: setting up correct expressions, correct expansion/simplification, and a valid concluding statement — not just a correct final line of algebra with no explanation.

Key termsmultipleexpandsimplify

Exam tip

Always finish with a sentence explaining WHY the algebra proves the statement — the final line of working is not enough on its own for full marks.

Section 4

How do counter-examples disprove a statement?

Not every statement given in an exam is true. If a statement is false, you do not need general algebra — you only need to find one single example where it fails. This is called a counter-example.

For example: "The square of any number is greater than the number." This is false. Counter-example: let the number be $0.5$ . Then $0.5^2 = 0.25$ , and $0.25 < 0.5$ . Since we found one case where the statement fails, the statement is disproved.

Good places to look for counter-examples:

$0$ (often breaks statements about multiplication or squaring)
Negative numbers (can flip inequalities or change signs unexpectedly)
Fractions/decimals between $0$ and $1$ (squaring makes them smaller, not bigger)

This is the opposite logic to a general proof: proving "always true" needs algebra covering every case, but proving "not always true" needs just one counter-example.

Key termscounter-exampledisprove

Think of it like this

Think of a counter-example like a single broken link in a chain — the whole claim of the statement being 'always true' snaps as soon as you find one case that fails.

Common mistake

Do not try to write algebra to 'prove' a false statement is false — one clear numerical counter-example is quicker and is what the mark scheme rewards.

Must Know

$\equiv$ means true for all values (identity); $=$ means true for specific value(s) (equation).
$n$ = any integer, $2n$ = any even number, $2n+1$ = any odd number.
$n$ , $n+1$ , $n+2$ represent three consecutive integers.
A full proof needs: define the letters, set up the expression, expand/simplify, and conclude with a sentence.
One number substituted in is never a valid general proof — only algebra with $n$ proves a statement for all integers.
To disprove a statement, find just ONE counter-example — try $0$ , negatives, or fractions/decimals first.

That's the notes covered.

Carry on to the next subtopic.