What strategies should one use to prove that some expression

>>8573006
proof by contradiction for the reals and rationals

what do you mean by 'expression', a set?

>>8573011
Let's say one needs to prove that the square root of n^2+1 doesn't belong to the set of natural numbers. How does one go about doing this?

What I did was a proof by contradiction: first assume that it belongs to the set of natural numbers, then assume that this implies that this implies that this expression must be either pair or impair, which produces a contradiction.

I think I didn't do it correctly.

>>8573025
if sqrt(n^2+1)=k is a natural then n^2+1=k^2

so n^2-k^2=1

so (n-k)(n+k)=1

so n-k and n+k have to both equal 1 or -1

so n-k=n+k

so -k=k which is a contradiction to being a natural number

>>8573110 (You)
woops, should obviously be:


if sqrt(n^2+1)=k is a natural then n^2+1=k^2

so k^2-n^2=1

so (k-n)(k+n)=1

so k-n and k+n have to both equal 1 or -1

so k-n=k+n

so -n=n

so n=0

so the only solution is when n=0, k=1