# Prove that: If $a+b$ and $b$ are irrational, then $a+kb$ is irrational.

How can I prove that,

If $$a+b$$ is irrational and $$b$$ is irrational, then $$a+2b$$ irrational; where $$a,b>0$$

More generally, if $$a+b$$ and $$b$$ are irrational, then $$a+kb$$ is irrational; where $$k\in\mathbb{Z^{+}}$$ and $$a,b>0$$

I know, for every $$k\in\mathbb{Z^{+}}$$, $$k×b$$ is irrational. But I'm stuck here. I can't continue.

• Suppose $a=\pi$ and $b=-\frac {\pi}2$. – lulu May 8 at 22:30

This is false. Take $$a=100-2\sqrt{2}$$ and $$b=\sqrt{2}$$.
• You have just changed the domain of $k$, then also we have $a+2b=0\in\mathbb{R}$ with $a=-2\sqrt{2},b=\sqrt{2}$ – tarit goswami May 8 at 22:33
• What is $\{a,b\}$ here? – Mark May 8 at 22:35
• You mean both $a,b$ are positive? – Mark May 8 at 22:37