subtraction of two irrational numbers to get a rational

Question

Say you have a number like $\pi$ or e. Is it possible to subtract another number from it and end up with a rational number? I mean I guess you could write an equation like $\pi-x=3$ But could there ever be a solution for x (that we could know and write out)?

Correction: Any number besides the irrational number itself. Damn math ppl are too quick..

Answer 1

If you take any real number $x\in\mathbb{R}$ you can show that the set of $y\in\mathbb{R}$ such that $x-y\in\mathbb{Q}$ is exactly $x+\mathbb{Q}$.

Let $y$ be in $\mathbb{R}$ with $x-y$ being rational then $y=x-(x-y)$ so that $y\in x+\mathbb{Q}$. On the other hand, if $y=x+q$ with $q\in\mathbb{Q}$ then $x-y=-q\in\mathbb{Q}$.

So this is kind of easy. However I think that it is not known wether $\pi+e$ is rational or not... What we know is that either $\pi e$ or $\pi+e$ (maybe both) is irrational...

Assume that both $\pi e$ and $\pi+e$ are rational then :

$P(x):=(x-\pi)(x-e)=x^2-(\pi+e)x+\pi e$ is a polynomial with rational coefficients. This implies that both $\pi$ and $e$ are algebraic numbers which cannot be true...

Answer 2

$\pi - \pi = 0$ which is rational.

Edit: $\pi - (\pi -1) = 1$ which is a difference of two different irrationals which is rational. How's that?

Answer 3

.223456789101112131415... - .123456789101112131415... = .1