Practice question · Put in order
Arrange the steps of the classical proof that the square root of two is irrational.
- Note that both being even contradicts the assumption of lowest terms
- Conclude that p must be even, and write it as twice some number
- Substitute back and find that q must be even as well
- Assume it equals a ratio of whole numbers in lowest terms
- Square both sides to get p squared equals two q squared
Hints
- The proof is an indirect one: it assumes what it means to refute.
- The contradiction is with the stipulation that the ratio was in lowest terms.
Show the answer
- Assume it equals a ratio of whole numbers in lowest terms
- Square both sides to get p squared equals two q squared
- Conclude that p must be even, and write it as twice some number
- Substitute back and find that q must be even as well
- Note that both being even contradicts the assumption of lowest terms
Why
This is reductio ad absurdum applied to arithmetic, and the proof survives essentially unchanged after 2500 years. The contradiction is precise: both numbers turn out even, which the lowest-terms assumption forbade.
Practise Pythagoras and Early Pythagoreanism
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