Multiplication of positive fractional numbers.

Question

I am reading this answer.
We know that multiplication of two positive rational number $\dfrac{a}{b}$ and $\dfrac{r}{s}$ respectivly is defined as follows:
$\dfrac{a}{b}\times\dfrac{r}{s} = \dfrac{ar}{bs}. \tag{0}$
The reason given for this definition is as follows:

If $\dfrac{a}{b}$ represents the solution to $bx=a\tag{1},$ and $\dfrac{r}{s}$ represents the solution to $sy=r\tag{2},$ their product will be the solution to $\tag{3}(bs)z=ar.$

What I don't get is how equation $1$ and $2$ imply equation $3$? I tried to multiply equation $1$ with equation $2$ and I got this:
$(bx)(sy)=ar \implies (\dfrac{b}{1}\cdot \dfrac{a}{b})(\dfrac{s}{1} \cdot \dfrac{r}{s})=ar\tag{4}$ If the commutative law also holds for the multiplication of rational numbers then equation $4$ will reduce to equation $3$ but since we are explaining the reason of the definition of multiplication of two positive rational number we can not presume commutative and associative law to hold good.

• So how the equation $3$ is obtained?

The author has explained the definition of addition of two positive rational number in a similar way, that is by assuming the commutative and associative law to hold good. So the same question is applicable to this explanation too.

• Why the commutative law and associative holds good in the multiplication of two positive rational numbers. Is it a consequence of definition$0$ or a presumption?

I have posted this as a comment on the linked questions but since the author is not active now I thought it would be more appropriate to ask this as a separate question.

P.S: I've reverted the question to it's original form and have asked the edit portion as a new question to get the answer.

Answer 1

You get $(bx)(sy)=ar$ alright, but rewriting it as $(\frac{b}{1}\cdot \frac{a}{b})(\frac{s}{1} \cdot \frac{r}{s})=ar$ is the wrong way to proceed.

Instead use the associative and commutative law on the left-hand side to get $$ (bs)(xy)=ar $$ In other words, if $x$ is a number such that $bx=a$ and $y$ is a number such that $sy=r$, then $xy$ will be a number that multiplied by $bs$ gives $ar$.

But we have chosen to use the notation $\frac{ar}{bs}$ for the number with that property!

Therefore it makes sense to require that the product of $\frac ab$ with $\frac rs$ should equal $\frac{ar}{bs}$.

since we are explaining the reason of the definition of multiplication of two positive rational number we can not presume commutative to hold good.

Yes we can -- at this place in the development we have decided that we want a way to calculate with fractions that will make the usual arithmetic laws, including the associative and commutative laws hold good. And we're exploring the consequences of that decision: Supposing that we find a way to achieve that goal, which properties will it have? The hope is that by assuming that the problem is already solved we'll be able to learn enough about the solution that we can actually write it down.

So far it is just hopeful thinking that we can ever get something that satisfies those rules, of course.

Once we have decided on the definition, of course we need to prove that the definition actually satisfies the associative and commutative laws. At that point we stop assuming that they do and view it as something to be proved.

Note that the reasoning leading up to the definition is not part of the formal development. It is meant to explain to the student why anyone would ever get the weird idea of writing down that particular definition, but mathematically it doesn't matter where we get our definitions from. It could be pulled out of a hat, or found in the left notes of a deceased genius, or dictated by a medium who claims to represent an etheric intelligence from a higher plane of existence. Mathematics doesn't care about that. All that matters is that once the definition is chosen, we can give proof that it has the properties we claim is has. And since the proof doesn't care about the reasoning you're quoting, all is actually well.