Magnitude

A magnitude is a quantity with no restriction on how it can be divided into parts. [TODO I'm not a huge fan of this definition.]

Ratio

A ratio is an identification of two multitudes.

[TODO I mean that you identify them together, you equate one with the other. I do not mean that you say "ah this multitude is 5, and that multitude is 6." I'm not sure how to make this clear from the language.]

To take the ratio between two multitudes is to identify them together.

[TODO why can't I say it's the identification of a quantitative relationship between two multitudes? lol. Too vague?]

Just like in the case of natural numbers, it turns out that many of these identifications will be the same. [TODO ?]

I'll represent a multitude with the following notation. Suppose I identify $x$ As with $y$ Bs, where $x,y:\mathbb {N}$. Then I denote the ratio by $[x:y]_{A,B}$.

Examples

6 slices is the same as 1 pizza.

100 pennies is the same as 1 dollar.

1000 mm is the same as 1 meter.

36 inches is the same as 3 feet.

Facts about ratios

Suppose I identify 3 As with 5 Bs. Then if I have a multitude of As given by any multiple of 3, that's the same as having a multitude of Bs given by the same multiple of 5.

The general principle here is that

$$[x:y]_{A,B}=[nx:ny]_{A,B},{\text{ for any }}n:\mathbb {N} .$$

Now, suppose that I identify 3 As with 5 Bs, and I also identify 5 Bs with 2 Cs. Then clearly, I may also identify 3 As with 2 Cs.

The general principle here is that

$${\text{if }}[x:y]_{A,B}{\text{ and }}[y:z]_{B,C}{\text{ then }}[x:z]_{A,C}$$

[TODO this doesn't make sense....]