Proof with irrational numbers
Objective. Use contradiction and exact forms in accessible surd proofs.
- 1. Learn
- 2. Worked example
- 3. Practice
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- 5. Continue
Learn: the key idea
Proof with irrational numbers uses properties of surds, such as that a rational number can never equal an irrational one, often via proof by contradiction. Assume the opposite of what you want to prove is true, follow logical algebraic steps, and show this assumption leads to something impossible, such as a surd equalling a fraction. A classic accessible example proves √2 is irrational by assuming √2 = a/b in lowest terms, then showing both a and b must be even, contradicting 'lowest terms'. For GCSE, simpler contradictions like assuming 3 + √5 is rational leading to √5 being rational are expected. Always state clearly where the contradiction arises.
Worked example
Prove that 4 + √3 is irrational, given that √3 is irrational.
- 1Assume for contradiction that 4 + √3 is rational, equal to r
- 2Then √3 = r − 4, which is rational since r and 4 are rational
- 3This contradicts the given fact that √3 is irrational
- 4Therefore 4 + √3 must be irrational
Answer: 4 + √3 is irrational by contradiction
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Start practisingCommon mistake. A common mistake is trying to prove irrationality using decimal approximations instead of exact algebraic contradiction, which can never give a rigorous general proof.
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