Definition:Ratio

Definition
Let $x$ and $y$ be quantities which have the same dimensions.

Let $\dfrac x y = \dfrac a b$ for two numbers $a$ and $b$.

Then the ratio of $x$ to $y$ is defined as:
 * $x : y = a : b$

It explicitly specifies how many times the first number contains the second.



Existence of Ratio
A ratio exists between two quantities $x$ and $y$ iff:
 * $\exists m, n \in \Z: m x > n, n y > m$

This condition is known as the Archimedean property.



Equality of Ratios
Let $p, q, r, s$ be quantities.

Then $p:q = r:s$ if, for any positive integers $m$ and $n$, $np < mq, np = mq, np > mq$ according as $nrms$ respectively.