- #1

Klion

- 14

- 0

Was thinking of maybe using product of primes somehow but.. hmm dunno

Anyway for sqrt(5) went like this (proof by contradiction)

Prove sqrt(5) is not rational.

Suppose sqrt(5) = a/b, where a,b E Z+ (suppose sqrt(5) is rational)

we'll also assume gcd(a,b) = 1 (otherwise just divide a,b by gcd)

sqrt(5) = a/b <=> 5 = a^2/b^2 or 5*b^2=a^2

so 5|a^2, but 5 is a prime so 5|a Ea' 5*a'=a so 5b^2=(5a)^2 or 5b^2=5^2*a^2 or b^2 = 5*a'^2 so 5|b^2 by prime lemma 5|b 5|a ^ b|b so 5|gcd(a,b)

so 5|1 which is nonsense therefore our initial assumption is incorrect and 5 is in fact irrational.

-Kli