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Simple question, apologies. This is from some sample high school math questions, target is age 16 pupils. I don't think any great sophistication is expected.

$$ P + Q = \sqrt {5}. $$

$P$ is a rational number and $Q$ is an irrational number. Give possible values of $P$ and $Q$.

I can think of the trivial $$ P = 0, Q = \sqrt {5}. $$ I'm not even sure that the pupil would be expected to come up with $$ P = 2, Q = (\sqrt {5} - 2). $$ I don't think $Q$ could then be simplified to another named irrational. Any ideas what an expected answer might be?

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    $\begingroup$ Generalizing your own example, $P$ could be any rational! If, say, $P=\frac mn$ then $Q=\sqrt 5 - \frac mn$. One should argue that such a $Q$ is necessarily irrational, but this is straightforward. $\endgroup$ – lulu Nov 3 '15 at 10:02
  • $\begingroup$ Just so. My puzzle is what the target student was expected to give as an answer. I wondered if there was some value of P that can yield a simpler value of Q. $\endgroup$ – djna Nov 3 '15 at 11:59
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    $\begingroup$ Well, I'd have thought that what I wrote was the intended answer. For what it's worth $-1+2\phi=\sqrt 5$ where $\phi$ is the Golden Ratio. But I don't think that simplifying $Q$ was the point of the exercise. $\endgroup$ – lulu Nov 3 '15 at 12:11
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Let $a,b\in\mathbb{Q}$ with $b\neq 0$ and let $P=\frac{a}{b}$. Then we have $Q=\sqrt 5 - P$ such that

$$ P+Q=\sqrt 5. $$

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