Quick answer: To prove √2 is irrational, suppose it equals a fraction p/q in lowest terms and show both p and q must be even. That contradicts the fraction being in lowest terms.
Worked example
Prove that √2 cannot be a ratio of integers.
- Assume √2 = p/q, where q ≠ 0 and p,q have no common factor greater than 1.
- Square to get p² = 2q². Thus p² is even, which forces p even; write p = 2k.
- Then 4k² = 2q², so q² = 2k² and q is even.
- Both are divisible by 2: a contradiction.
Check the result
An odd integer 2n + 1 has odd square 4n² + 4n + 1, justifying why an even square has an even base.
Stuck on a question? Ask it here and get a step-by-step answer in seconds — pay only per answer, no subscription. Ask your doubt now →
Common mistake to avoid
The lowest-terms assumption is essential. Decimal approximations such as 1.414 do not prove irrationality.
Try it yourself
1. Is √9 irrational? 2. Why can the same contradiction method prove √3 irrational?
Answers and reasoning
1. No: √9 = 3. 2. If 3 divides p², primality forces 3 to divide p; substituting forces 3 to divide q, contradicting lowest terms.
Need help with your own attempt? Ask a study question. AI answers require an account and paid credits. Check important steps with your textbook or teacher.
Stuck on a question? Ask it here and get a step-by-step answer in seconds — pay only per answer, no subscription. Ask your doubt now →