View Single Post
 2021-01-04, 01:15 #75 Dr Sardonicus     Feb 2017 Nowhere 32·5·101 Posts (1) OP seems to be confounding "fractions" (rational numbers) and "decimal fractions," i.e. fractions that can be expressed with a power-of-ten denominator. Not all rational numbers are decimal fractions. (2) OP also seems to think that invalidating a proof of A automatically proves ~A (not-A). It doesn't. (Here, A is "The square root of 2 is irrational.") OP, of course, did not invalidate the proof. What he actually did was (1). Expressing the statement that the (positive) square root of 2 is rational as an equation in positive integers p and q, (*) p2 = 2*q2 invites a Euclidean proof that the square root of 2 is not rational, because the equation is impossible. Euclid also proved a result now known as the Fundamental Theorem of Arithmetic, AKA unique factorization. The equation (*) violates the Fundamental Theorem, because the left side is divisible by 2 evenly many times, while the right side is divisible by 2 oddly many times.