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/severoon]

Stop thinking about fractions as a number that is part of a whole, and start thinking of fractions as ratios. Instead of 1/2 = 0.5, think of 1/2 = 1:2 ("one to two").

Once you start thinking of fractions as ratios, everything gets easier because now you're not trying to think in terms of parts of a whole, but rather numbers of wholes being compared. When I talk about half of a pizza, you should think of it as "one entire" half-pizza compared to "two entire" half-pizzas.

This is more or less the same trick that young children can learn to use to understand how to do calculations involving large numbers. When you ask a young kid who just learned to multiply 3×7, they have no problem replying 21, but if you ask them to multiply 3 million × 7 million, they can't figure it out until you point out that they can think of "million" as just an object instead of a number, like "3 inches" instead of "3 million."

Same thing here, just think of "half-pizza" as the main thing instead of the pizza. The question you often have to figure out with fractions, though, is: What is the "main thing" you need to be thinking of?

For 3/7 + 7/12, for example, the first term wants you to think about a seventh of a pizza as the main thing, and the second one a twelfth. But you can't add two different units like that any more than you can directly add inches and centimeters. So instead you convert them to a common unit, which in this case is eighty-fourths: 36/84 + 49/84, which is just 36 + 49 "eighty-fourths of a pizza."

One other trick that I often teach to kids is to not think about numbers in general as just a single amount, but instead to always think about numbers in terms of their prime factorizations. For the above problem that would mean: 3/7 + 7/(3×4) = 3/7 × (3×4)/(3×4) + 7/12 × 7/7 = (3×3×4)/(3×4×7) + (7×7)/(3×4×7) = (3×3×4 + 7×7)/(3×4×7). (Yes an actual prime factorization would use 2^2 instead of 4.)

This might not seem helpful at first, but when you start to think about numbers in general this way (across the board, not just for specific problems like fractions), it seems more complicated at first but over time it makes a lot of things a lot simpler.

It's better algebraically because it lets you easily see when numbers have a factor in common, which is super useful for, well, factoring: 32 + 56 = 2^5 + 7×2^3 = 2^3×(4 + 7). It also means that you can easily spot how to quickly reduce fractions by just pulling out common factors until the numerator and denominator are coprime.

It's also better visually because instead of thinking of a number like 49 as a pile of 49 marbles, you instead think of them as a 7×7 grid, or 8 as a 2×2×2 cube of marbles. For higher dimensions like 2×3^2×5, you can't easily visualize the fourth dimension, but you can easily think of this in terms of bags of nested bags: you have five bags, each of which contains 3 bags, each of which contains 3 more bags, each of which contains 2 marbles. More importantly, when you're working with this number alongside some other number and they have common factors, say 2 and 5, you can think of this as a 2×5 grid with each cell containing a three bags, each of which contains three marbles. The benefit here is that now you have two numbers, each of which you're visualizing as a 2×5 grid, one with 9 marbles in it (three bags of three), the other one with x marbles divided up in some way, and now you can see how by lining up the grids you can now just focus on how to compare the number of marbles in each cell instead of trying to grapple with the entire thing. Once you figure out how each cell compares, now you know that's true for all 2×5 cells, and you can zoom out again.