Can we calculate LCM of irrational numbers. Specifically for $\pi$

Question

Irrational numbers can not have LCM. But in one of the books I read and LCM for thr triplet $( \pi/2 , \pi , 3\pi/2)$ was calculated and the answer was $3\pi$. If we can't find LCM for irrational numbers and pi is one,who is the result possible for the problem I read.

Answer 1

Suppose for simplicity that the numbers are positive. Then one can define $\gcd(a,b)$ of real numbers provided the Eclidean algorithm for determining GCD stops atfer a finite number of steps. This is the case if $a$ and $b$ are commensurable.$\def\lcm{\operatorname{lcm}}$

This means $a$ and $b$ have a "common measure", i.e. a finite length that fits into either value without remainder.

Put differently, the quotient $a/b$ must be rational.

$$\text{Euclidean algorithm stops for } a,b \quad\iff\quad \frac ab\in\Bbb Q $$

You can then introduce LCM as $$\begin{align} \lcm(a,b) &= \frac{a\cdot b} {\gcd(a,b)} \tag 1 \\ \lcm(a,b_1,\ldots,b_n) &= \lcm(a, \lcm(b_1,\ldots,b_n)) && \text{ for } n\geqslant 2 \tag 2 \end{align}$$

As an aside, this also relates to regular continued fraction expansion of real numbers: The expansion of $r\neq 0$ has fintely many terms iff $r\in \Bbb Q$, i.e. iff $r$ and $1$ are commensurable.

In your specific case,

$$\gcd(\pi/2, \pi, 3\pi/2)=\frac \pi2$$ $$\lcm(\pi/2, \pi, 3\pi/3)=3\pi$$

Answer 2

Yes, we can calculate the LCM of irrational numbers if and only if the ratios between all the pairs of irrational numbers are rational. Basically, LCM is the least common positive integral multiple of the given numbers. In your case, (pi/2)/(pi)=1/2,(pi/2)/(3pi/2)=1/3 and (pi)/(3pi/2)=2/3 are all rational and 3pi is the least common positive integral multiple of (pi/2),pi and (3pi/2).