How will you prove that the square root of 2 is irrational?
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Answer Posted / Pramod Kumar Mishra
To prove that the square root of 2 is irrational, you can use Euclid's method for finding the infinitude of primes. Assume that sqrt(2) is rational and express it as a fraction in lowest terms: sqrt(2) = m/n where m and n are integers with no common factors other than 1. Squaring both sides, we get 2 = m^2/n^2, which implies m^2 is even (since 2 is not divisible by an odd number squared). Therefore, m must be even, meaning it can be written as 2k for some integer k. Replacing m with 2k in the equation, we get (2k)^2 = 4k^2 = 2n^2. This shows that n^2 is an even multiple of 2, and thus n must also be even. However, if both m and n are even, they share a common factor of 2, which contradicts our initial assumption that m and n have no common factors other than 1. Therefore, sqrt(2) cannot be rational.
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