I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrational.

Eg. $\sqrt2 + 1$ can be expressed as a continuous fraction, and through looking at the fraction, it can be assumed $\sqrt2 + 1$ is irrational. I suggested that because of this, $\sqrt2$ is also irrational.

My professor said this is not always true, but I can't think of an example that suggests this.

If $x+1$ is irrational, is $x$ always irrational?

Actually, a better question is: if $x$ is irrational, is $x+n$ irrational, provided $n$ is a rational number?

^{1/2}+ 1 is irrational is enough to prove the irrationality of all numbers of the form 2^{1/2}+ x, x rational, but what about the rest of them? While it is true that `x+r` irrational implies `x` is irrational when `r` is irrational, you don't have another way of proving that `x+r` is irrational in the first place. – chepner Oct 28 '13 at 16:33