Definition:Ratio

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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.


In the words of Euclid:

A ratio is a sort of relation in respect of size between two magnitudes of the same kind.

(The Elements: Book $\text{V}$: Definition $3$)


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.


In the words of Euclid:

Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

(The Elements: Book $\text{V}$: Definition $4$)


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 $nr<ms, nr=ms, nr>ms$ respectively.


In the words of Euclid:

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

(The Elements: Book $\text{V}$: Definition $5$)