The original question was: Why is it when we multiply fractions we multiply the numerators across and the denominators across? Whereas when we divide we don’t do the same? Who came up with these rules and why do they work the way they do?
Physicist: For all of these rules, start with addition and then extend and extend and extend, while doing the least damage possible. At first blush I thought this would be a short and straightforward question, but it really isn’t. There are wrong ways to construct the rules of arithmetic, but if you get technical there’s not necessarily a right way. That said, as far as basic arithmetic goes (no highfalutin calculus, or set theory, or anything) there’s really just one “right” way.
Rational numbers (“fractions”) have been in use for thousands of years and they, and their rules, have been independently invented at least dozens of times. More recently (19th century) a bunch of really A-type mathematicians got together to “put mathematics on a more rigorous footing“. Those mathematicians (there were many) are responsible for unpleasant statements like “there are exactly as many prime numbers as rational numbers“, and are a big reason behind why modern mathematicians always look just a little pained when they speak. It was these paragons of compulsivity that established how the rules are stated, and gave everything definite names like “associative”, “commutative”, etc.
Constructing the rational numbers and all of their behavior starts with an innocent statement; that “
Starting with integers, along with regular addition, subtraction, and multiplication (basic arithmetic tools), and the definition of “
Often 
For example: 
For example: 
Already we can say things like: 
For example: 
How do fractions multiply? It would be nice to know: 
For example: 
Now we’ve got the tools to define multiplication in general: 
For example: 
Now, again using nothing new, 
For example: 
It would be great if the addition and subtraction of fractions follow all of the same rules that hold for integers. As a mathematician, the way you make this happen is to declare that it’s true, and then check for inconsistencies. The rule most important for defining addition and subtraction of fractions is the “distributive law”, which says that “A(B+C) = AB+AC” (this doesn’t lead to any new inconsistencies).
First, 
For example:
But what about fractions with different denominators? The trick is to not have them:
For example:
Huzzah! We can multiply, add, and reduce fractions with ease and impunity!
But what about dividing fractions? Well, here we have to tread lightly and describe exactly what is meant by “dividing by a fraction”. So, in a perfectly reasonable extension of the original definition, “
But check this out: 
Another way to see this is to say
For example: 
Just a quick note on behalf of whoever grades your tests or homework: Please reduce fractions, 
For example:
What about subtraction and negative numbers? In general, in every case, without exception of any kind, whenever you see “-A” you can exchange it with “(-1)A” and exchange “B-A” with “B+(-1)A”. After that, treat “(-1)” just like any other number or variable*.
So (still using no new rules!), we can say
For example:
For those of you who’ve read this far, you can show using the same definitions and tricks above that:
Using the fact that 
Similarly, you can show that 
There are big issues with 
For example: If 
It’s also worth pointing out that when things are added in the denominator, there’s not much that can be done with it. So, if you’ve got something like 
Also, for those of you wondering (this is a bit technical), the rational numbers also inherit their positions in the number line very naturally. Using only the fact that “if A<B and C>0, then AC<BC” and the fact that we know how to order integers, you can figure out which rational number is bigger than which other rational numbers. Since 
For example:
![\begin{array}{ll}\frac{7}{4}\,[?]\,\frac{5}{3}\\\Rightarrow 3\cdot4\cdot\frac{7}{4}\,[?]\,\frac{5}{3}\cdot3\cdot4\\\Rightarrow 3\cdot7\,[?]\,5\cdot4\\\Rightarrow 21\,[?]\,20\\\Rightarrow 21>20\\\Rightarrow\frac{7}{4}>\frac{5}{3}\end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D%5Cfrac%7B7%7D%7B4%7D%5C%2C%5B%3F%5D%5C%2C%5Cfrac%7B5%7D%7B3%7D%5C%5C%5CRightarrow+3%5Ccdot4%5Ccdot%5Cfrac%7B7%7D%7B4%7D%5C%2C%5B%3F%5D%5C%2C%5Cfrac%7B5%7D%7B3%7D%5Ccdot3%5Ccdot4%5C%5C%5CRightarrow+3%5Ccdot7%5C%2C%5B%3F%5D%5C%2C5%5Ccdot4%5C%5C%5CRightarrow+21%5C%2C%5B%3F%5D%5C%2C20%5C%5C%5CRightarrow+21%3E20%5C%5C%5CRightarrow%5Cfrac%7B7%7D%7B4%7D%3E%5Cfrac%7B5%7D%7B3%7D%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0 "\begin{array}{ll}\frac{7}{4}\,[?]\,\frac{5}{3}\\\Rightarrow 3\cdot4\cdot\frac{7}{4}\,[?]\,\frac{5}{3}\cdot3\cdot4\\\Rightarrow 3\cdot7\,[?]\,5\cdot4\\\Rightarrow 21\,[?]\,20\\\Rightarrow 21>20\\\Rightarrow\frac{7}{4}>\frac{5}{3}\end{array}")
Even better, we can describe exactly where the rational numbers are!
For example: You can show that 
Just to get ahead of the most obvious follow up questions: There’s a lot of weird emphasis on how exactly mathematical notation is used in text. The first key to dealing with complicated text-based notation is: don’t. Writing equations using the symbols found only on a keyboard is something our unfortunate and sadly limited ancestors had to consider. If you’re reading this now, then you’re in a bigger and better-notated world.
In general, you can always write A/B = A(1/B), and A/B/C = A(1/B)(1/C), and so on. A little fancier: 
For example: 
That (way too long of a post) all said, if you’re reading this because you’re presently panic-studying the night before a test, and actually needed a short answer, memorize this:
“A/B is A times 1/B. 1/B times B is one. There’s nothing else to say about 1/B.”
and go to sleep sooner rather than later.
*If your math background is extensive enough to know some exceptions, then… be cool, you know what I mean.
7 Responses to Q: Where do the weird rules for rational numbers come from? (Dealing with fractions)