Irrationality of $ 1/a + 1/b$

Question

I have thought about this and was wondering if anyone could provide an example of real numbers $a$ and $b$ such that $a + b$ is rational but $1/a + 1/b$ is irrational or prove the statement false.

Answer 1

Let $a=\sqrt{2}$ and $b=1-\sqrt{2}$ obviously their sum is $1$, but if you check what their inverses are, they don't sum to a rational number. Almost every example in this form will work. I say almost every because here might be some highly contrived examples, but if you pick any irrational you happen to like, it will work with high probability (probably probability $1$ actually).

Answer 2

Let $a = x $ be irrational.

Let $b = 1 -x$ so $a + b = 1$.

$\frac 1 x + \frac 1 {1 - x} = \frac {(1-x) + x}{x(1-x)} = \frac 1 {x(1- x)}$

So we just need to find an irrational $x$ such that $x(1-x)$ is irrational.

The list is endless. $\sqrt{n};n$ not a perfect square, is a good choice as $\sqrt{n}(1 - \sqrt{n}) = \sqrt{n} - n$ is irrational. Any non algebraic irrational such as $\pi$ or $e$ is good as $x(1-x) \in \mathbb Q \implies x$ is algebraic.

In fact, it takes a bit of work to find an irrational $x$ where $x(1-x)$ is not irrational.