Rational + irrational = always irrational? 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?
Suppose $x$ is irrational and $x+\dfrac pq=\dfrac mn$ then, $x=\dfrac mn-\dfrac pq=\dfrac{mq-np}{nq}$ so, $x$ would then be rational. :)
This follows essentially from the fact that as additive abelian groups $\mathbb Q$ is a normal subgroup of $\mathbb R$, which is true since both groups are abelian and the latter contains the former. Hence we have a quotient homomorphism $\varphi: \mathbb R \rightarrow \mathbb R / \mathbb Q$. $r \in \mathbb R$ is rational iff $\varphi (r) = 0_{\mathbb R / \mathbb Q}$, where $0_{\mathbb R / \mathbb Q}$ is the identity element of $\mathbb R / \mathbb Q$.

Consider then for $x \in \mathbb R - \mathbb Q$, $r \in \mathbb Q$, that
\begin{align*}
 \varphi (x+r)&amp;= \varphi(x) + \varphi(r) \\
&amp;= \varphi(x) + 0_{\mathbb R / \mathbb Q} \\
&amp;= \varphi(x) \\
&amp; \ne 0_{\mathbb R / \mathbb Q} \text{ by assumption that }x \not\in \mathbb Q,
\end{align*}

so $x+r \not \in \mathbb Q$.

Of course, the above is just reinterpreting the above elementary proofs in a more general context, but this lets us apply the same line of reasoning to a wide variety of things including modular arithmetic, rotations/reflections of geometric objects, Rubik's cube moves, matrix multiplication, permutations of sets of numbers, exotic differentiable structures on spheres, ...