LCM of irrationals

Question

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?

Answer 1

I would write about an l.c.m. of two irrational numbers only if either (1) I explicitly say what I mean by the term rather than assuming the reader knows, or (2) I'm writing in some context in which it is appropriate to expect the reader to think about what the term ought to mean.

The smallest number that can be written as $n\Big(4\sqrt2\Big)=m\Big(6\sqrt2\Big)$ where $n$ and $m$ are positive integers is $12\sqrt2$, so one could say that is the l.c.m. of $4\sqrt2$ and $6\sqrt2$. One can write a sensible definition of l.c.m. so that the question of what is $\operatorname{lcm}\left(4\sqrt2,6\sqrt2\right)$ makes sense, but I would heed what the paragraph above says.

Note, however, that in that sense, $\operatorname{lcm}(a,b)$ will exist only if $a/b$ is rational and positive.

Answer 2

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 $\xi/\eta$.