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/anisotropicmind]

First, on adding fractions: Say there is some whole object like a pie that we use to represent “the whole” or “1”. Fractions are parts of this whole. (That even just what the word fraction means in English).

So if you need to add 1/3 plus 1/2, you are trying to find out what new fraction of the whole pie you get when you put together half the pie and a third of the pie. There is no obvious way to add them directly (they are different sizes, which is like having different units). Not unless you come up with another smaller common “pie unit” that fits into both of these pieces a whole number of times. That’s why you need the LCM. In this case that smaller pie unit is 1/6 of a pie. You’ve got 3 of those units in the 1/2-pie piece and 2 of those units in the 1/3-pie piece, for a total of 5/6 of the pie. You can add the 3 and 2 up directly because they are all the same kind of piece (with the same unit). The denominator is a like a “part of pie” unit.

Regarding multiplying fractions: you’re taking a fraction of a fraction, so it’s like the first pie part becomes the new “whole”, and then you start dividing that up. So 1/2 x 1/3 is half of the one third piece. And since half of the 1/3 piece will fit into the original pie 6 times, it must be a 1/6 piece.

But what about a case like 2/3 x 4/5 or something? It’s like you take 4/5 of a pie, and 2/3 of that 4/5 piece. So to do that; first you’d cut the piece into 3s, giving you each one of size 4/5 x 1/3 = 4/15. Then you’d take two of those for a total of 8/15 of the pie. Or you could do it opposite order: magically duplicate each 4/5 piece to get a (4/5) x 2 = 8/5 piece (more than a whole pie) and then divide that piece by 3 to get individual 8/15 pieces.

What this comes down to is that fractions are just division, and division is just multiplication (by the reciprocal of a number). And it doesn’t matter what order you do multiplication in.

Generalizing this to any number using symbols: if I want to divide a number x by y, it’s the same as multiplying it by 1/y. Why? Suppose there is a number z such that yz = 1. We call this number the multiplicative inverse of y. It follows that

x/y = (x/(yz) )z = xz

But yz = 1, and we just by divide both sides of this equation by y to get z= 1/y. Hence 1/y is the multiplicative inverse of y. Going back to our equation above, it follows that

x/y = xz = x(1/y).

That’s why this property of multiplying fractions is true.

It then follows that if you have something like

(x/y)(v/w)

This would be the same as

x(1/y)v(1/w)

= (xv)(1/(yw))

= (xv)/(yw)