simple multiplication question: multiplying radial fractions

Question

I'm having a bit of brain block right now.

In order to multiply any fractions you simply reduce to lowest form, then multiply num., then denom. When I tried with fractions in radian form it didn't work out so well.

consider: $(6 \pi/5)(2 \pi/3$)

If I want to keep the $\pi$ in the numerator I thought I should just multiply $6$ and $2$ to get $12$ for $12 \pi$. But I don't think that's a mathematically sound method of multiplication. So I figured I should just multiply the entire numerator and then divide out $\pi$: so that's $(6 \pi)(2 \pi)=118.4352528$ then divide out $\pi$ which gives me $37.69911184$ which isn't right because it should be an integer.I know I'm missing something very fundamental here. Any help appreciated.

Answer 1

In terms of multiplication, there is nothing special about $\pi$, you just multiply fractions in the usual way: $$\frac{6\pi}5\frac{2\pi}3=\frac{12\pi^2}{15}=\frac{4\pi^2}{5}\ .$$ It's not an integer, and there is no particular reason why it should be.