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 $y$ :
 * $\exists m, n \in \Z: m x > n, n y > m$

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 $n$, $n p < m q, n p = m q, n p > m q$ according as $n r < m s, n r = m s, n r > m s$ respectively.