Say $x$ and $y$ are irrational numbers.

I know that $x + y$ could be rational, for example $\pi$ and $-\pi$. Also, x/y could be rational, for example pi and 1/pi.

But would $x + y + xy$ be irrational? I couldn't find an example where the expression could be rational. Could someone find an example where the expression is rational, or give a proof of how it has to be irrational?

  • 2
    $\begingroup$ $x=\sqrt{2}$, $y=-\sqrt{2}$ $\endgroup$ – Cornman Oct 11 '17 at 2:28
  • $\begingroup$ Thank you, now that I see it, I can't believe I didn't think of it haha. $\endgroup$ – IUissopretty Oct 11 '17 at 2:33
  • $\begingroup$ Please use MathJax to format your posts. $\endgroup$ – Chase Ryan Taylor Oct 11 '17 at 3:36

Note that $x+y+xy = (1+x)(1+y) - 1$. Hence, if $x+y+xy$ is irrational, then so is $(1+x)(1+y)$. Furthermore, $x$ and $y$ are assumed irrational, so $1+x,1+y$ are already irrational. Hence, $1+x,1+y$ are irrational numbers whose product is irrational.

This , as you know, need not happen all the time. For example, using the very common example $(\sqrt 2)^2 = 2$, one sees that $x = y = \sqrt 2 - 1$ would do the job here. Indeed, $x + y + xy = 1$ in this case.


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