Ratios of Fractions in Lowest Terms

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Theorem

Let $a, b, c, d \in \Z_{>0}$ be positive integers.

Let $\dfrac a b$ be in canonical form.

Let $\dfrac a b = \dfrac c d$.


Then:

$a \mathrel \backslash c$

and:

$b \mathrel \backslash d$

where $\backslash$ denotes divisibility.


In the words of Euclid:

The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.

(The Elements: Book $\text{VII}$: Proposition $20$)


Proof

Let $CD, EF$ be the least (natural) numbers of those which have the same ratio with $A, B$.

We need to show that $CD$ measures $A$ the same number of times that $EF$ measures $B$.

Euclid-VII-20.png

Suppose $CD$ is an aliquant part of $A$.

Then from Proposition $13$ of Book $\text{VII} $: Proportional Numbers are Proportional Alternately and Book $\text{VII}$ Definition $20$: Proportional, $EF$ is also the same aliquant part of $B$ that $CD$ is of $A$.

Therefore as many aliquant parts of $A$ as there are in $CD$, so many aliquant parts of $B$ are there also in $EF$.

Let $CD$ be divided into the aliquant parts of $A$, namely $CG, GD$ and $EF$ into the aliquant parts of $B$, namely $EH, HF$.

Thus the multitude of $CG, GD$ will be equal to the multitude of $EH, HF$.

We have that the numbers $CG, GD$ are equal to one another, and the numbers $EH, HF$ are also equal to one another,

Therefore $CG : EH = GD : HF$.

So from Proposition $12$ of Book $\text{VII} $: Ratios of Numbers is Distributive over Addition, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.

Therefore $CG : EH = CD : EF$.

Therefore $CG, EH$ are in the same ratio with $CD, EF$ being less than they.

This is impossible, for by hypothesis $CD, EF$ are the least numbers of those which have the same ratio with them.

So $CD$ is not an aliquant part of $A$.

Therefore from Proposition $4$ of Book $\text{VII} $: Natural Number Divisor or Multiple of Divisor of Another, $CD$ is an aliquot part of $A$.

Also, from Proposition $13$ of Book $\text{VII} $: Proportional Numbers are Proportional Alternately and Book $\text{VII}$ Definition $20$: Proportional, $EF$ is the same aliquot part of $B$ that $CD$ is of $A$.

Therefore $CD$ measures $A$ the same number of times that $EF$ measures $B$.

$\blacksquare$


Historical Note

This theorem is Proposition $20$ of Book $\text{VII}$ of Euclid's The Elements.


Sources