# Thread: Rational & Irrational Numbers Problem

1. ## Rational & Irrational Numbers Problem

I came across this question in my exam, however I have exactly no clue on how to come across it.

''If the numbers
$\displaystyle e,\pi ,\pi^2, e^2,e\pi,$ are irrational,
prove that at most one of the numbers
$\displaystyle \pi+e, \pi-e, \pi^2-e^2, \pi^2+e^2$ is rational.''

I think the person who gets a solution to this deserves a prize

Suppose two are rational. Let's take $\displaystyle \pi + e$ and $\displaystyle \pi - e$.

Those can be expressed as ratios of integers.

$\displaystyle \pi + e = \frac{a}{b}$ and $\displaystyle \pi - e = \frac{c}{d}$

Solving simultaneously shows e is rational AND $\displaystyle \pi$ is rational, since:

1) Looking at denominators, they must be divisible by 2bd and
2) Looking at numerators, integers are closed over addition, subtraction, and multiplication.

I certainly have not covered the whole proof, but that seems a good piece of it.

3. seems like a good enough answer actually, well done -
thanks