Why the square root of two is not a fraction

Question

The number 1.4142... squared is 2, or very nearly. Could some fraction p/q, with whole numbers p and q, square to exactly 2?

Prove that it cannot. Then explain what the same argument says about the square root of 3, and why it fails for the square root of 4.

Accepted Answer

No fraction squares to 2. The proof is by contradiction. Suppose it did. Write the fraction in lowest terms as p/q, so p and q have no common factor, and suppose (p/q) squared = 2. Then p squared = 2 q squared. p is even. The right-hand side is even, so p squared is even. The square of an odd number is odd, so p must be even: p = 2k for some whole number k. q is even too. Substitute: (2k) squared = 4 k squared = 2 q squared, so q squared = 2 k squared. By the same reasoning, q is even. Contradiction. Both p and q are even, so they share the factor 2. But we chose p/q in lowest terms. The assu…

Why the square root of two is not a fraction

Your answer

Discussion

Learn the concepts

Related questions