"GCD" of any two real numbers

Question

This isn't really a GCD question, because GCD is only defined for integers. I'm interested in the the existence of a common divisor of any two non-zero real numbers. In other words can you prove or disprove the following:

Given $x,y \neq 0\in \mathbb{R}, \exists \space g \space s.t. \space x/g \in \mathbb{Z}$ and $y/g \in \mathbb{Z}$.

(I hope my math is understandable, haven't done this in awhile). It's clearly possible for many numbers, including irrational ones (e.g. for multiples of $\pi$, $g = \pi$). Is it possible for all real numbers?

Answer 1

The following conditions are equivalent for nonzero reals $x,y$

1. There is a real $g$ such that $x/g$ and $y/g$ are integers
2. The quotient $x/y$ is rational

Proof:

$1 \implies 2$: Since quotient of integers is rational, your condition implies that

$(x/g) / (y/g) \in \mathbb{Q}$

after clearing $g$ in denominators

$x/y \in \mathbb{Q}$.

$2 \implies 1$:

If $x/y$ is rational: $x/y=p/q$ then define $g = y/q$ (or $g = x/p$), then $x/g = xq/y = p$ and $y/g = q$ are integers. QED

So any pair of reals with irrational quotient is a counterexample, for example $x=1$ and $y=\sqrt{2}$.

Real numbers $x,y$ with rational quotient are known as commensurable. This is how irrationality was formulated in the ancient times. It has been said that diagonal of a square is not commensurable with its side.

The Euclidean algorithm for finding GCD was originally formulated on segments (reals) - it found a common measure ($g$) given segments of length $x$ and $y$.

Answer 2

The statement is not true. Consider $x=1,y=\pi$. If $y/g=n\in\mathbb Z$, then $g=\pi/n$ so $x/g=n/\pi\notin \mathbb Z$.