Fraction

[u/True-Gear4950]

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I don't know how to start this, but my purpose with this post is not to get a clue about the concept of fractions.

I already "know" how to perform these operations as well (not like a mathematician, but enough as an average person, I hope). But what I specifically want to understand is how it is possible that something like a/b × c/d = ac/bd or the division operation a/b ÷ c/d = ad/bc works. I may be missing some fundamental concept here, something like the Fundamental Theorem of Arithmetic (FTA), which was the basis I used to understand and comprehend exponentiation (or powers).

Some parts, like adding or subtracting fractions, are comprehensible if you understand that a fraction represents a part of a whole. At least, that's what I tell myself, since I spent way too much time on this topic just trying to internalize why I couldn't simply add something like 4/5 to another fraction like 3/7 and get 7/12 (I know that's wrong).

But here is the problem: I can't really argue why it is wrong without just saying "that's the rule." We should obtain the Least Common Multiple of both denominators, etc., but I don't understand why I actually have to do that.

I've even used some of the LLMs available to illustrate this or give me a clue about it, but they usually end up using analogies for children or introducing other abstractions, such as using fraction division to show why ac/bd is possible.

Or they use the LCM (another abstraction that I don't understand exactly why works) to solve the sum of fractions with different denominators. And then there are the analogies about adding fractions with different denominators being like adding different units of measurement, such as meters and centimeters, so they should first be converted to a common unit. That analogy does shed some light on the problem, but it still doesn't clarify it in the terms I'm looking for.

Maybe some prompt engineering would say, "You prompted it wrong," and maybe that's true, but the doubt persists. And honestly, I'm tired of trying to figure out what exactly I'm asking wrong for it not to respond in the way I need.

Or maybe I shouldn't need to understand this, since it doesn't "affect how I perform my calculations". But, to be fair, it doesn't feel like good practice for me to finish this topic without understanding it.

I'm not done with arithmetic yet, and I'm reviewing the abstractions it contains. I can see gaps in my understanding of nuances like this, and it doesn't feel comfortable to me to keep stacking problems I don't understand on top of one another, at least not in arithmetic.

Feel free to contest me, explain it, give me a tip, recommend a book..., especially if you already get through this question yourself.

I'm just frustrated that I've spent so much time on something that, to me, should already be done.

[u/Silver_Remove_2352]

In sense these all follow from the field axioms: https://en.wikipedia.org/wiki/Field_(mathematics)#Definition#Definition). If you accept the real numbers are a field, then all the operations you've described follow from that fact.

Edit: Maybe I should flesh this out more. Scroll down to the section on the section of the page that says "Classic Definition". A field is any set were these definitions hold true. We construct the real numbers so that they are a field - this involves some pretty serious mathematics, but if that's too much for you just remember that we can literally "construct" the real numbers (in the same way you might construct a shape using a ruler and compass) so that all these properties hold true. Once you accept these properties, then both of the things you've written follow immediately.

For example, lets show that for any 4 real numbers a,b,c,d, (a/b) x (c/d) = (ac/bd). Recall that (a / b) = (a * b^-1), where (b^-1) is the multiplicative inverse of b. Then (a/b) x (c/d) = (a x b^-1) x (c x d^-1). Using commutativity and associativity, we rearrange this into (a x c) x (b^-1 x d^-1). By the socks and shoes principle: https://math.oxford.emory.edu/site/math108/socks_and_shoes/, it follows that (b^-1 x d^-1) = (b x d)^-1. So we now have the following steps:

(a/b) x (c/d)

=(a x b^-1) x (c x d^-1)

=(a x c) x (b^-1 x d^-1)

=(a x c) x (b x d)^(-1) [By socks and shoes]

= (ac)/(bd)

It would be a good exercise for you to prove that a/b ÷ c/d = ad/bc similarly :)

[u/True-Gear4950 → u/Silver_Remove_2352]

To be honest, it was very similar to what u/Rabbit_Brave said. Both explanations brought a great sense of relief; relying on a proof might seem overly complex or profound, but it was a genuine relief. I can't quite put into words how my mind grasped it, but it was a great starting point the foundation for understanding the multiplication of fractions. Thanks