Non-trivial examples of irrational numbers $x, y$ such that $x - y$ is rational.

Question

If we factor the additive group $\mathbb R$ by $\mathbb Q$ we will have the quotient group $\mathbb R / \mathbb Q$ where $\mathbb Q$ is the identity element. And I try to find non-trivial examples of irrational numbers which are in the same coset (i.e $x - y \in \mathbb Q$). Yet I am not able to come with one.

What i mean when i say "trivial" is like $x = y$ or $x = \sqrt{2}$ and $y = \sqrt{2} - 2$.

$x-y$ being rational means $x=y+q$ for some rational $q$ so... what do you mean by trivial precisely? — Simply Beautiful Art, Dec 17 '19 at 16:08
The problem is that $y$ will be a rational number plus $x,$ and so if we think of rational numbers as "trivial adjustments", $y$ and $x$ will be trivially different. Or, if you have in mind a rational number that you don't consider to be trivial, then just add it to $\sqrt 2$ to get an example. — Dave L. Renfro, Dec 17 '19 at 16:09
@Peter Foreman: I was going to search for an answer of mine where some examples like this show up after making my initial comment, but you beat me to it! (3 minutes later) Ah, here it is --- my answer to sum of irrational numbers - are there nontrivial examples?. — Dave L. Renfro, Dec 17 '19 at 16:10
The intent of the question is probably an utterly unexpected case, like as $\pi-e$, if it actually would turn out to be rational (which we cannot rule out at the moment). — Peter, Dec 17 '19 at 16:17

Arthur · Accepted Answer · 2019-12-17T17:11:39.643

5

There aren't going to be any "non-trivial" examples. Whatever we try, we will have $y= x-q$ for some rational number $q$. We might use two different ways of describing $x$ to camouflage this fact (e.g. $x=\sqrt2, y=\frac{\sqrt8}{2}-2$ is a simple example), or we can cleverly hide $q$ (an example given in the comments above), but ultimately we can't change this fact.

edited Dec 17 '19 at 17:11

answered Dec 17 '19 at 16:09

Arthur

204,511

Marios Gretsas · Answer 2 · 2019-12-17T16:14:09.357

4

You can find many examples also using appropriately the identity $\cos^2{x}+\sin^2{x}=1$.

Or the identity $\ln{\frac{x}{y}}=\ln{x}-\ln{y}$

For instance take $x=2e^3$ and $2e^2$

edited Dec 17 '19 at 16:14

answered Dec 17 '19 at 16:08

Marios Gretsas

22,350

Non-trivial examples of irrational numbers $x, y$ such that $x - y$ is rational.

2 Answers2