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.

  • 3
    $\begingroup$ Suppose $a=\pi$ and $b=-\frac {\pi}2$. $\endgroup$ – lulu May 8 at 22:30

This is false. Take $a=100-2\sqrt{2}$ and $b=\sqrt{2}$.

  • $\begingroup$ I am so sorry..I fixed the question $\endgroup$ – Learner May 8 at 22:32
  • $\begingroup$ You have just changed the domain of $k$, then also we have $a+2b=0\in\mathbb{R}$ with $a=-2\sqrt{2},b=\sqrt{2}$ $\endgroup$ – tarit goswami May 8 at 22:33
  • $\begingroup$ What is $\{a,b\}$ here? $\endgroup$ – Mark May 8 at 22:35
  • $\begingroup$ are Positive numbers. $\endgroup$ – Learner May 8 at 22:36
  • $\begingroup$ You mean both $a,b$ are positive? $\endgroup$ – Mark May 8 at 22:37

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