Page 176 - HOW TO PROVE IT: A Structured Approach, Second Edition
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162 Proofs
√
√ 2
Proof. Either 2 is rational or it’s irrational.
√
√ 2 √
Case 1. 2 is rational. Let a = b = 2. Then a and b are irrational,
√ √ 2
b
and a = 2 , which we are assuming in this case is rational.
√ √
√ 2 √ 2 √
Case 2. 2 is irrational. Let a = 2 and b = 2. Then a is
irrational by assumption, and we know that b is also irrational. Also,
√ √ √ √
√ √ ( 2· 2) √
b 2 2 2
a = 2 = 2 = ( 2) = 2, which is rational.
