What Is Proof by Contradiction?
Proof by contradiction is a powerful logical technique where you work backwards to prove something is true. Instead of directly showing that a statement is correct, you assume it's false, then follow the logic step by step. Eventually, you'll hit something impossible — and that impossibility means your original assumption was wrong. Therefore, the statement must be true.
The Core Idea
Assume the opposite → Follow logic → Find contradiction → Original statement is true
Why Is This Useful?
Sometimes, directly proving something is hard. For example, proving that √5 is irrational is much easier if you assume it is rational and show why that leads to nonsense. It's like a detective ruling out suspects by showing each one has an alibi — when all the false options are eliminated, the truth becomes clear.
Think of it like this:
If someone says "It's raining outside," you could look out the window and observe. But you could also say: "If it's NOT raining, the ground would be dry. But the ground is wet. So it must be raining." That's proof by contradiction — you assumed the opposite and found it doesn't match reality.
How to Use Proof by Contradiction
The Method (Step by Step)
Every proof by contradiction follows the same basic structure:
- State your assumption clearly. Start with "Assume [opposite of what you want to prove]"
- Use that assumption. Write it in a useful form (e.g., as a fraction a/b for rational numbers)
- Apply logical steps. Use algebra, factorisation, properties of numbers, etc.
- Find the contradiction. Show that your assumption leads to something impossible
- Conclude. State clearly: "This contradicts our assumption, so the original statement must be true"
Marking Tip:
Examiners want to see your assumption stated clearly at the start, your working shown step by step, and the contradiction explicitly identified. Don't skip steps or assume the reader knows what you mean.
Key Techniques You'll Need
1. Rational Numbers
A rational number can be written as a/b where:
- a and b are integers (whole numbers)
- b ≠ 0 (can't divide by zero)
- a and b have no common factors (the fraction is in "lowest terms")
For example, 3/4, 2/5, 7/1 are all rational. If you can't write a number this way, it's irrational.
2. Prime Factorisation
Every composite number (a number greater than 1 that isn't prime) can be written as a product of primes. For example:
28 = 2 × 2 × 7 = 2² × 7
This is useful because if a² is a multiple of a prime p, then a must also be a multiple of that prime. This logical link is key to many contradiction proofs.
Key Insight:
If a² is even, then a is even. If a² is divisible by 5, then a is divisible by 5. Use this pattern in proofs about irrational numbers.