I am having a bit of trouble understanding the concept of an incommensurable unit.
From what I have gathered so far, it is simply a magnitude that cannot be expressed as the ratio of two natural numbers (i.e. $\sqrt{2}$)
However, in the text I am currently reading
“In describing the concept of a real number, Newton in his “General Arithmetic” wrote: “by number we mean not so much a collection of units, as an abstract ratio of a certain quantity to another quantity taken as the unit.” This number (ratio) may be integral, rational, or if the given magnitude is incommensurable with the unit, irrational.”
Excerpt From: A. D. Aleksandrov. “Mathematics.”
This seems to say that an incommensurable magnitude CAN be expressed as a ratio of "abstract units." Would this imply natural numbers? Maybe my understanding is completely off.
Thank you!
I'm not sure what you mean by "an incommensurable unit" or "an incommensurable magnitude".
A number can't be "incommensurable". "Incommensurable" means not "commensurable", which in turns means (I'm not a native english speaker, so I won't embarass myself with guesses on the etymology) something like "measurable with the same tool". Technically "two non-zero real numbers a and b are said to be commensurable if a/b is a rational number." (wikipedia). If $b$ is $1$, the condition is that $a$ must be a rational number.
$\sqrt{2}$ is incommensurable with $1$ (or indeed any rational number), but it is commensurable with $\sqrt{8}$ (which is also incommensurable with $1$).