Proof by Contradiction (Edexcel IAL Maths P4)

Q: How many marks are P4 proof by contradiction questions worth?

Proof by contradiction questions are typically 3 to 5 marks in Edexcel IAL P4 exams, usually appearing as an earlier short-answer question on the paper.

Q: Do I need to memorize the √2 and infinite primes proofs?

Yes! These two proofs are explicitly referenced in the P4 specification and are frequently tested directly, so you should be able to reproduce them perfectly under timed conditions.

Edexcel International A-Level Mathematics· WMA14 (2018 Specification Issue 3)· 15 min read

1. Structure of Proof by Contradiction★★☆☆☆⏱ 4 min

✓ Calculator OK

📘 Definition

Proof by Contradiction

A deductive proof method where you assume the negation (opposite) of the statement you want to prove is true. You follow logical steps to derive an impossible result (contradiction), which proves your initial assumption was false, so the original statement is true.

Example:

Proving there is no largest integer by assuming a largest integer N exists, then showing N+1 is larger, creating a contradiction.

All Edexcel P4 proof by contradiction questions follow a standard 4-step structure you should apply consistently to secure full method marks:

State the original statement you intend to prove.
Assume the negation (opposite) of the statement is true.
Use logical algebraic or numerical reasoning to derive a contradiction.
Conclude your initial assumption was false, so the original statement is true.

📐 Worked Example

Prove that there is no smallest positive rational number.

1
Step 1: State the original statement: There is no smallest positive rational number.
2
Step 2: Assume the negation is true: Let $a$ be the smallest positive rational number.
3
$b = \frac{a}{2}$
4
Since $a$ is non-zero and rational, $b$ is also positive and rational.
5
Step 3: Derive contradiction: $b < a$ , but we assumed $a$ was the smallest positive rational. This is impossible.
6
Step 4: Conclusion: Our initial assumption is false, so there is no smallest positive rational number.

Exam tip:

Always explicitly state the contradiction and your final conclusion, as these are explicitly allocated marks in Edexcel P4 mark schemes.

2. Mandatory Standard Proofs★★★☆☆⏱ 6 min

✓ Calculator OK

The Edexcel P4 specification explicitly requires you to memorize and reproduce two standard proofs: the irrationality of √2, and the infinity of primes. These are often tested directly, so learn them word-for-word.

📐 Worked Example

Prove that √2 is irrational.

1
Step 1: Original statement: √2 is an irrational number.
2
Step 2: Assume negation: √2 is rational, so it can be written as $2 = \frac{a}{b}$ where $a$ and $b$ are positive integers with no common factors (coprime, lowest terms).
3
$2 = \frac{a^2}{b^2} \implies a^2 = 2b^2$
4
$a^{2}$ is even, so $a$ must be even. Let $a = 2 k$ where $k$ is an integer.
5
$(2k)^2 = 2b^2 \implies 4k^2 = 2b^2 \implies b^2 = 2k^2$
6
$b^{2}$ is even, so $b$ is also even.
7
Step 3: Contradiction: We assumed $a$ and $b$ are coprime, but we have shown both are even (so they share a common factor of 2). This is impossible.
8
Step 4: Conclusion: Our initial assumption is false, so √2 is irrational.

📐 Worked Example

Prove that there are infinitely many prime numbers.

1
Step 1: Original statement: There are infinitely many prime numbers.
2
Step 2: Assume negation: There is a finite list of all primes: $p_{1}, p_{2}, p_{3}, ..., p_{n}$ .
3
$N = p_1 \times p_2 \times p_3 \times ... \times p_n + 1$
4
Consider the number $N$ constructed as above.
5
Step 3: Contradiction: When $N$ is divided by any prime $p_{i}$ on our list, it leaves a remainder of 1. So $N$ is either prime itself, or has a prime factor not on our original list. This contradicts the assumption our list contained all primes.
6
Step 4: Conclusion: Our initial assumption is false, so there are infinitely many prime numbers.

Exam tip:

For the √2 proof, you must explicitly state that $a$ and $b$ are coprime (in lowest terms) — this is required for the contradiction step to be valid, and omitting it will lose marks.

3. Unfamiliar Proof by Contradiction Problems★★★★☆⏱ 5 min

✓ Calculator OK

Most P4 exams include an unfamiliar proof by contradiction question, usually relating to number theory, algebra, or inequalities. The same 4-step structure applies, and you should look for contradictions involving parity (even/odd), divisibility, or impossible integer values.

📐 Worked Example

Prove that if $n$ is an integer and $n^{2}$ is odd, then $n$ is odd.

1
Step 1: Original statement: For all integers $n$ , if $n^{2}$ is odd then $n$ is odd.
2
Step 2: Assume negation: There exists an integer $n$ where $n^{2}$ is odd, but $n$ is even.
3
If $n$ is even, we can write $n = 2 k$ for integer $k$ .
4
$n^2 = (2k)^2 = 4k^2 = 2(2k^2)$
5
This means $n^{2}$ is even, which contradicts our assumption that $n^{2}$ is odd.
6
Step 3: Contradiction confirmed, so our initial assumption is false. Therefore the original statement is true.

Exam tip:

When tackling unfamiliar proofs, start by writing down the exact negation of the statement first. For 'if A then B' statements, the negation is 'A is true and B is false' — this is a common source of error.

4. Common Pitfalls

Wrong move:

Assuming the wrong negation of the original statement, e.g. negating 'all primes are odd' as 'all primes are even' instead of 'there exists at least one even prime'.

Why:

Incorrect negations lead to invalid proofs, as you are not contradicting the right statement, and you will lose all method marks.

Correct move:

For universal statements ('for all x, P(x) is true'), the negation is 'there exists at least one x where P(x) is false'. For conditional statements ('if A then B'), the negation is 'A is true and B is false'.

Wrong move:

Omitting the 'coprime' condition when proving √2 is irrational.

Why:

Without stating that a and b are in lowest terms, you cannot derive a contradiction when you find both are even, so the proof is incomplete.

Correct move:

Explicitly write 'where a and b are positive integers with no common factors (coprime)' when stating your assumption for the √2 proof.

Wrong move:

Failing to explicitly identify the contradiction and restate the conclusion at the end of the proof.

Why:

Edexcel mark schemes allocate 1-2 marks specifically for stating the contradiction and final conclusion, so omitting these costs easy marks.

Correct move:

End every proof with two clear sentences: first, state exactly what the contradiction is, second, state that the assumption is false so the original statement is true.

Wrong move:

Using circular reasoning in the proof, e.g. using the original statement you are trying to prove as part of your reasoning.

Why:

Circular reasoning is logically invalid, and will result in zero marks for the proof.

Correct move:

Only use known facts (e.g. even numbers are divisible by 2, primes have only 1 and themselves as factors) and your initial assumption in your reasoning steps.

Wrong move:

For the infinite primes proof, stating that N must be prime, instead of that N has a prime factor not on the original list.

Why:

N is not always prime (e.g. 2×3×5×7×11×13 + 1 = 30031 = 59×509, which is composite), so this is a factual error that invalidates the proof.

Correct move:

State that N is either prime itself, or has a prime factor not present on your original finite list of primes, both of which contradict the assumption that your list was complete.

5. Quick Reference Cheatsheet

Proof Type	Key Steps	Mandatory Marking Details
General Proof by Contradiction	State original statement Assume negation is true Derive contradiction Conclude assumption is false	Explicit contradiction statement, final conclusion matching the question
√2 is Irrational	Assume √2 = a/b (coprime integers) Show a² even → a even Substitute a=2k, show b even Contradiction (a,b share factor 2)	Explicitly state a and b are coprime (lowest terms)
Infinite Primes	Assume finite list of primes p₁ to pₙ Construct N = product of all primes + 1 Show N has prime factor not on list Contradiction	Note N may be prime or have a new prime factor, do not claim N is always prime

6. Frequently Asked

How many marks are P4 proof by contradiction questions worth?

Proof by contradiction questions are typically 3 to 5 marks in Edexcel IAL P4 exams, usually appearing as an earlier short-answer question on the paper.

Do I need to memorize the √2 and infinite primes proofs?

Yes! These two proofs are explicitly referenced in the P4 specification and are frequently tested directly, so you should be able to reproduce them perfectly under timed conditions.

Going deeper

What's Next

Now that you have mastered proof by contradiction for Edexcel IAL P4, you are ready to move on to the other core topics in the unit. Proof by contradiction is often combined with number theory and algebra concepts from earlier units, so revise those foundations if you struggled with any worked examples. Practice writing full proofs under timed conditions to ensure you do not omit mandatory marking points, and attempt all past P4 proof questions to familiarize yourself with the range of problems that can be asked. This skill will also be useful if you go on to study Further Mathematics, where more advanced proof methods are covered.