LCM of irrational numbers

Question

So i read in a book that irrational and rational numbers do not have a common multiple and it said that lcm of irrational numbers is also only possible when both the irrational numbers have the same surd. I was wondering what this means.

Answer 1

https://www.quora.com/Does-the-L-C-M-of-two-irrational-numbers-exist With $5*\sqrt{2}$ and $3*\sqrt{2}$ their least common multiple is $15*\sqrt{2}$, because it's the smallest number that's an integer multiple of both.

However, they don't always have an LCM. Take $\sqrt{2}$ and $\sqrt{3}$. There is no number $L$ such that $\frac {L}{\sqrt{2}} $ and $\frac {L}{\sqrt{3}}$ are integers, otherwise their quotient, $\sqrt{\frac {2}{3}} $, would be rational. And it isn't.

Answer 2

Those are odd claims.

The first can, I think, be justified. Let's say $\alpha$ is an irrational number and $\frac ab$ is rational (with $a,b\in \mathbb Z$). Then it is certainly true that, for any non-zero integers $m,n$ we have $m\times \alpha$ is irrational and $n\times \frac ab$ is rational, so it is not possible for them to be equal.

But the second claim seems hard to follow, no matter what (standard) meaning you assign to "surd".

Originally, "surd" just mean "irrational". These days, it more often means an expression in radicals, such as $\sqrt 2$ or $\sqrt[3] 3$. However, numbers like $\pi$ and $2\pi$ clearly have common multiples so I'm not sure what meaning is intended.