I am trying to better understand commensurability. Wikipedia says:
two non-zero real numbers a and b are said to be commensurable if $\frac{a}{b}$ is a rational number.
Richard Courant in Introduction to Calculus and Analysis says:
Two quantities whose ratio is a rational number are called commensurable because they can be expressed as integral multiples of a common unit.
First of all, can't we just "cheat" and say that the common unit is $1$? I'm not even sure if that's cheating or if that's what he actually means.
Furthermore, looking at the Wikipedia definition, since a "rational number" means that $a$ and $b$ must be integers, shouldn't the beginning of that sentence actually read, "two non-zero integers" (and maybe $a$ can also be $0$)? Or are there additional cases we want to allow to be commensurable?
Bottom line, I'm not sure what commensurability means other than "both numbers must be rational numbers," if that is indeed what it means.
For context, I'm reading this in the context of Courant demonstrating that irrational numbers exist. He's doing this by showing that some numbers exist which are not rational fractions (e.g. $\sqrt{2}$), but he equates that with being "incommensurable with the unit length":