# express an irrational as the sum of a rational and irrational number

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?

• 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. – lulu Nov 3 '15 at 10:02
• 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. – djna Nov 3 '15 at 11:59
• 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. – lulu Nov 3 '15 at 12:11

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