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Using the fact that $\sqrt{2}$ is irrational, prove the following:

There exists real numbers $x$ and $y$, such that $x$ and $y$ are irrational, and $x+y$ is also irrational.

My attempt:

Let $x= \sqrt{2}$ and $y= \sqrt{2} + 1$.

Suppose $y= \sqrt{2} + 1$ is rational. Than $y= \sqrt{2} + 1 = \frac mn$, where $m,n$ are integers. So, $y= \sqrt{2} = \frac mn - 1$ which is rational. This contradicts the original statement that $\sqrt{2}$ is irrational. Thus, $y= \sqrt{2} + 1$ is irrational.

Now, $x+y= \sqrt{2}+ \sqrt{2} + 1 $ which equals $2\sqrt{2} + 1$.

Suppose $2\sqrt{2} + 1$ is rational. Then $2\sqrt{2} + 1 = \frac pq$ where $p,q$ are integers. So, $2\sqrt{2} = \frac pq - 1$ which equals $\sqrt{2}= \frac{p}{2q} + \frac 12 $ which is rational. This contradicts the original statement that $\sqrt{2}$ is irrational. Thus, $2\sqrt{2} + 1$ is irrational.

So, $x+y$ is irrational if $x$ and $y$ are irrational.

Does this proof make sense? Is there a simpler way?

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    $\begingroup$ Why not $\sqrt2$ and $2\sqrt2$, since nothing is asked about the product? $\endgroup$ – Parcly Taxel Oct 7 '16 at 11:26
  • $\begingroup$ because choosing that would just make $x+y$ = $3\sqrt(2)$ instead of the $2\sqrt(2)$ that I have, which doesn't really make a difference, right? $\endgroup$ – SeesSound Oct 7 '16 at 11:33
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Your argumentation is fine.

However, one has to be careful that the statement "So, $x+y$ is irrational if $x$ and $y$ are irrational." does not hold for all $x,y \in \mathbb{R}$ (but, of course, for your choice).

An easier choice should be $x = y = \sqrt{2}$.

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  • $\begingroup$ Assume that x and y are distinct terms. Considering that do you see a simpler choices for x and y that can be used? $\endgroup$ – SeesSound Oct 7 '16 at 11:28
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    $\begingroup$ The question does not ask for $x \ne y$. You can also take $y = 2 \, \sqrt{2}$ or, more generally, $y = q \, \sqrt{2}$ for any rational number $q$. $\endgroup$ – gerw Oct 7 '16 at 11:30
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    $\begingroup$ And just in case it isn't clear to the OP, a trivial example where $x+y$ is rational and both $x$ and $y$ are irrational, contradicting the conclusion, is $x=\sqrt2$, $y=-\sqrt2$, $x+y=0$. $\endgroup$ – hvd Oct 7 '16 at 11:34

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