Does anyone know why fractions are reduced before multiplication, isn't just information loss?

@vidak 2 * 2 * 3 / 2 * 3 -> reduces because you can remove the factor of 2 on each side, but what I'm wondering is that, is there's a mathemathical reason for doing that? other than it saves the person doing the math time?

https://www.edweek.org/teaching-learning/approach-to-fractions-seen-as-key-shift-in-common-standards/2014/11
https://toinspiremath.com/is-simplifying-fractions-important/

@neauoire 4/6 is the same object as 2/3 and generally multiplying small numbers is easier

@hierarchon I think that getting the gcd() before doing the multiplication is moving more entropy around than multiplying the unreduced forms, that's where the question came from.

@neauoire I think it simplifies the intermediary numbers that you need to calculate. Which could be beneficial when working in cases like 32bits and you multiply two big numbers that might overflow in intermediary steps.

I don't think it's information loss because you could technically expand the fraction as well (1/2 -> 100/200) and the multiplication would come out the same.

Main reduction issues I had in school was when I did wrong reductions like try cancel 2 apple with 2 orange.

@neauoire not sure what you mean by "moving entropy around"

@hierarchon what I mean is: if I do cycles count on a recursive gcd() to get the reduced fraction, before doing the multiplication, it is using more resources than if I multiply the unreduced fractions.

@neauoire if you don't reduce then eventually multiplying by 2/2 will result in overflowing your numerator/denominator; I don't know if typical computer algebra systems reduce "eagerly" or not

@neauoire it's canonicalization of a fraction to an irreducible operation. that makes two fractions comparable.

@neauoire Regarding information loss, you are not supposed to know the two numbers. A fraction is an equivalence class.

I would take the 2d vector if I wanted to do that, which is easily converted to a fraction.

@neauoire reducing fractions is easy

@neauoire @vidak ironically my calculus teacher told us to avoid simplify fractions because he found it often lead to an increase in small mistakes, lmao.

As far as I can tell it's a bit of a mathematical style guide to reduce your fractions. Also an exercise in being able to see the equivalence between some fractions.

@neauoire @vidak but also things like the quotient rule and partial fraction decomposition would give you obscene fractions.

@lritter I wonder, like the needles of a clock for example, 3:15, there's value in differentiating 3/12 and 15/60, if I say 0.25, we are missing the information to know of what we are talking about. I wonder if they're really equal.

@th haha, that's very nice.

@neauoire

this is a really good question. I think the answers given from a computational perspective are not wrong.

But they ignore that the numbers in the numerator and denominator might come from a certain process and reducing (or multiplying them out) results in data loss.

Arithmetic is the trace of a computational process, deferring it keeps the history of it available to us.

@monkey1 I've spent the better part of the past two days asking myself why Conway would use reduced fractions and I haven't be able to turn up much.

I wish he was still alive so I could ask him! Otherwise, part of me feels like it might be a mistake?

@neauoire @lritter the thing is, mathematics are made up. so, in traditional algebra and such, 3/12 = 15/60, because that's a consequence of the axioms we started with, and how we defined fractions. nothing prevents us from coming up with alternative definition of a fraction-like object that has different properties 🤔 and as you said these might be useful in a given context, just like being able to reduce fractions is important in others

@neauoire @monkey1 i think I see what's happening. It looks like you've opened the Pandora's box that is number theory.

In day to day arithmetic, fraction reduction isn't required but is convenient because it removes common factors that just gets in the way of thinking about a problem. 16384/8191 is not equal to two, but if you're estimating something, it is sufficiently close to 16384/8192 or 2/1 that it won't make a difference. If you'd not known that 8192 was a common factor of your mental estimate, and this was able to mentally divide so easily, the mental math you'd need to jump through to arrive at that conclusion would be potentially impossible. I deliberately choose easy to reckon numbers here; but imagine if they weren't (or worse, based on transcendental numbers). For example, how would you compare 22/7 versus 355/113 without knowing what they abstractly represent ahead of time?

However, not all number systems are so convenient. There are number systems where multiplication and division follow different rules than for our daily use of numbers. I can't recall offhand what they're called now, but a quick search online will show some examples.

Maybe, for Conway's game, he's using a number system deliberately which suffers from daily use?

@vertigo @monkey1 I think reducing is akin to stripping types off of a program, if we reduce 3:15pm, or 3/12 and 15/60, to 0.25 and 0.25, the only information that we lose really is that which specified what we are talking about. Somehow feels more like type theory than number theory.

@capital @vidak I ran diagnostics during multiplying against reduced by gcd() and unreduced fractions, the amount of memory moved around in that process of reduction is A LOT more than just multiplying longer names, it makes me doubt that it's a practical thing to make students do.

@imyxh in computers(I think?) in this case.
https://wiki.xxiivv.com/site/fractran.html

@neauoire I’m not sure I follow how it’s information loss, since it’s just a representation change. In humans, it’s done this way because generally humans find it easier to simplify fractions when numerator and denominator are small.

If you don’t simplify first, you’re more likely to introduce errors and compute more slowly; and you’re less likely to be able to solve the problem in your head.

@calcifer my thinking goes something along the lines of datatypes. For example, if we reduce 3:15pm, which is really 3/12 and 15/60, to 0.25 and 0.25, what we lose in the translation is the size of units which specified what we are talking about.

@neauoire in fact, a lot of the way I was taught basic maths in school involved rewriting terms to make them easier to think about

And part of my programming education Back When was understanding when certain tactics that are better for humans are worse for computers — something I think you’re deeply familiar with based on the work of yours I’ve seen.

@neauoire @lritter it's like when you normalize a vector to keep just its direction to get a versor - sometimes you need the norm, sometimes you don't...

@neauoire @monkey1 That would be an invalid reduction though, I think. The hour and minute specifies a single quantity with each in a unique base. It really is specifying 3*60+15 (or, going the other way, 3+(15/60)). So, either 195 minutes or 3.25 hours, respectively. The type remains thanks to dimensional analysis, and in fact, is crucial to determining which simplified form is correct in a given context.

@neauoire what you want is a tuple. fractions (i.e. numbers in ℚ) are just tuples equipped with an equality relation allowing for reducing the fractions (so reduction is allowed for them to be equal). a tuple without this equality relation will "preserve the information" as you seem to want

@neauoire @lritter They are equal but you’re throwing out your units. The short hand indicates 0.25 of 12 hours have passed. The long hand indicates 0.25 of 1 hour has passed. The fractions are equal, but what they’re fractions *of* is different.

@neauoire @capital @vidak to me, it seems that reducing the fraction before multiplication is mostly done to keep the amount of energy required for future computation with the result from getting out of hand. it is probably rarer that we would only ever care about the raw value, using it for no further computation?

i.e. the reduced and unreduced artifacts are not equally computationally useful

@opfez I keep my information as tuples behind the scene, but I was wondering more why do we think that fractions have this mechanism baked in, and does it even have this mechanism, that's more what I was curious about, but from what I can tell from the answers on the thread and what I could find online, it's more that it's just tradition to do so, than for any kind of mathematical soundness.

@neauoire ah, yeah shorting fractions isn't necessary, it's just that by their definition, a fraction (np)/(nq) = (mp)/(mq) for any n, m. this allows shortening, but doesn't require it, like you note. if you're ever curious, i'd recommend taking a look at the constructions of our numbers (ℕ, ℤ, ℚ, and most importantly ℝ) at some point! you already know ℕ from the peano axioms but it gives a lot of deep insight into a lot of mathematics rather quickly :)

@opfez I will! One weird unicode letter at a time

@neauoire

Where is the quoted text from?

@monkey1 This is from this page: https://wiki.xxiivv.com/site/fractran.html

@xavier @neauoire @lritter afaik mathematics aren't really made up. They exist in nature. We just try to prove everything and create abstract constructs to help with that.

@lw64 @neauoire @lritter it's very much a philosophical debate, but what I mean is: they are made up in the sense that they're a tool we construct to solve problems. common theories of maths are what they are because they're useful to physics and other applied sciences. and it's common to create other "mathematical theories" (with different axioms and definitions) that are better adapted to a specific problem.

of course, the fact that the same maths can solve problems in many different contexts could be an argument toward saying that maths is real 😅