TITLE: LCM of irrationals
QUESTION [7 upvotes]: So, I was recently asked by a friend about the lcm of two irrational numbers.
As far as I know, mathematically speaking, lcm is generally defined only for positive integers (and sometimes extended to negative integers and also even rationals).
But I have never heard of it being extended to irrationals. But my friend argues that lcm of irrationals is defined (in general).
He gives me an example: $\textrm{lcm}(e,2e)=2e$ where $e$ is the irrational Napier's constant.
I think he is wrong. I still want to hear opinions from others on this. What do you guys think, fellow MSE users?

REPLY [4 votes]: The ancient Greeks knew about this. Finding a least common (integer) multiple of two positive numbers is essentially the same problem as finding a greatest common divisor (if $\zeta$ is a g.c.d  of $\xi$ and $\eta$, then $(\xi\eta)/\zeta$ is an l.c.m.). The Greeks understood that greatest common divisors exist for some $\xi$ and $\eta$ but not for all $\xi$ and $\eta$. They called $\xi$ and $\eta$ commensurable if they have a greatest common divisor: we would say that $\xi$ and $\eta$ are commensurable if $\xi/\eta$ is rational.
Euclid's algorithm works perfectly well for two arbitrary positive real numbers $\xi$ and $\eta$. From a modern point of view, you carry out the arithmetic in the ring $\mathbb{Z}[\xi, \eta]$. If the algorithm terminates, it finds the greatest common divisor: the largest $\zeta$ such that $\xi$ and $\eta$ are both integer multiples of $\zeta$.
What ought to be much better known is that the Greeks studied the case when the algorithm does not terminate: Euclid's Proposition X.2 gives non-termination of the algorithm as a necessary and sufficient condition for the irrationality of $\xi/\eta$.