Prove that √3 is irrational.

1. 1 · Write what is given and what is to be found. To prove: √3 is irrational.
2. 2 · Write the theorem. If a prime p divides a², then p divides a, where a is a positive integer.
3. 3 · Assume the opposite. Assume √3 is rational. Then √3 = a/b, where a and b are co-prime integers and b ≠ 0.
4. 4 · Square and use the theorem. √3 b = a, so 3b² = a². So 3 divides a², and hence 3 divides a. Write a = 3c.
5. 5 · Substitute. 3b² = 9c², so b² = 3c². So 3 divides b², and hence 3 divides b.
6. 6 · Reach the contradiction. 3 divides both a and b. This contradicts the fact that a and b are co-prime.
7. 7 · Check and box the answer. So our assumption is wrong. Therefore √3 is irrational.

Scoring tip: Write each 'divides' step on its own line and name the theorem each time. Missing the co-prime line costs a mark.