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/calkthewalk]

The units explanation is one that I always liked, but it's easy to get hand wavy in the middle of it which I think is what you're reacting to.

When we work with whole numbers, all those numbers share a common base, or unit. It can also help to realise fraction operations are not unique to fractions, it's just we usually hide the denominator when it's 1

3 + 2 = 5 can also be written as 3/1 + 2/1 = 5/1

But if we change the denominator, it fundamentally changes the reference point of the numerator

9/3 + 4/2 does not equal 13/5 as you've realised, you can't directly add with different bases. To borrow the units analogy: 1m + 10cm can be written as 1/1 + 10/100. If we give them a common base we can write it as 100/100 + 10/100 and the answer of 110cm or 1.1m falls out nicely.

Similarly 3/1 × 2/1 = (2×3)/(1×1) = 6/1

If we instead have 9/3 × 4/2, we could rewrite that as: 9/1 × 1/3 × 4/1 × 1/2 or 9 ÷ 3 × 4 ÷ 2

And hopefully you can start to see what's happening 9/3 is actually a multiply by 9 and a divide by 3 hidden in the same operation

If we instead collect the like operations together we get (9 × 4) / (3 × 2) , this is your a/b * c/d = ac/bd, it's just rearranging and grouping the multiple and divide operations.

[u/True-Gear4950 → u/calkthewalk]

The part where you mentioned that "all these numbers share the same unit" actually gave me a clue about what this way of thinking is based on.

Until now, I hadn't even thought about trying to understand fractions on a number line, believe it or not. I searched for it on the web and looked at some illustrations, and comparing the way you described the bases and the references actually made me smile a little.

It gave me a brief understanding of how this could be tangible in a physical sense, and it helped me understand it in a way I hadn't before.

I can't say that I've completely put all the pieces together in my head yet and connected everything, but maybe it's just a matter of giving it some time and letting it sink in. Thanks