can't remember "the distributive property"? ez if you understand it as geometry

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basically you have to understand that until the 1600s or so the concept of a number as an abstract quantity was basically nonexistent. numbers are lengths. addition is sticking one length on to the end of another length. multiplication is the production of an area from two lengths.

treating numbers as abstract entities is powerful but it makes mathematical operations way harder for a beginner to understand

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i think it's actually very hard to understand anything in terms of pure abstraction, at least for me. i formally studied linear algebra and it made absolutely no sense to me until i started using it practically for game development.

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@dankwraith split the ab rectangle diagonally to produce a right triangle, align 4 of those at the corners of the a+b square and now you have 2ab + c^2 so you've also geometrically proven the Pythagorean theorem

@dankwraith does this mean there's still hope for me when it comes to understanding quaternions

@dankwraith learning Aristotle and the categories helped me understand math way more than the years of taking classes in it had

I live in pure abstractions, but the issue you end up with if that's how your brain works, is if you OPERATE in full-abstraction mode, you tend to lose track of things.
I straight up do not have the attention span to do linal in a purely abstract way, despite that being how my brain works, and not having THAT low of an attention span!
I cannot imagine what kind of people actually manage to do it "naturally".

@dankwraith I majored in Math and all linear algebra ever was to me was a set of theorems to do proofs with. I did really well in the multivariable sequence but I really cannot say that I learned Linear Algebra.

@dankwraith linear algebra IMO should be taught in full thinking with vectors as lists of numbers, and matrices as ways of transforming space. the geometric intuition helps everything.

the abstract notion of a vector space is immensely powerful and opens up so much, but at the same time it should be introduced at the end, in which they say "hey, we've been working with lists of numbers the entire time, what if we didn't do that"

@dankwraith like so many things like the span and the determinant and arbitrary choice of basis and eigen-everything make so much more sense when matrices are derived from thinking about space, not the other way around

@dankwraith this makes a lot of sense to me, i didn't understand integrals at all until i finally could see in my head, like, the slices adding together

@dankwraith totally agreed, seeing how calculus relates to physics and geometry is the only thing that made me a) not hate calculus and b) finally pass the goddamn class

@dankwraith 100% I had the same experience! I literally only passed math classes because of pity from my teachers. Yet when I got a job and just said 'yes' to everything even if I didn't know it, I found it was pretty easy to learn almost anything with a real problem to apply it to. I went through my whole grade school years thinking I was really just stupid with math only to find out they didn't teach it in a way that made sense to me. I also found out late in life that I was dyslexic.

@dankwraith i have said this in the past but this isn't entirely true, since book V establishes a theory of proportion and with it introduces the idea of ratios in a way that is compatible with incommensurable magnitudes (explored in book X)

@parenthetical true! but i would argue that at this point the student has already worked through four entire books of problems that treat numbers as magnitudes rather than throwing them in the deep end with "a number is uhhh, well, i'ts just a thing you see"

@dankwraith numbers (meaning positive whole numbers) actually aren't dealt with until books VII and VIII, where euclid confusingly doesn't establish that ratios between numbers are a subset of ratios between magnitudes and proceeds to re-prove a bunch of the propositions he proved of general magnitudes for the special case where everything is numbers

@parenthetical i need to crack open euclid again, i clearly have something mixed up in my noggin

@dankwraith which is important because proportions can be compounded indefinitely (let A have a certain ratio to B, and B a different ratio to C, and C a different ratio to D) whereas if you're merely enclosing areas and volumes you run out of dimensions to multiply in very quickly

@dankwraith and assuming you don't consider geometry the worst thing to exist

@dankwraith i used to be in a mathlete type competition when i was a kid and they taught us algebra via geometry there, unlike at school. and when i was in school my classmates asked me to help them with math, and i explained it with geometry. suddenly math was a lot easier for everyone

@dankwraith Why is this the first time I see this?? :blobfoxsurprised: That's awesome!

@dankwraith Can't go wrong with the good ol' distribution property.

@dankwraith I wish I could have had you for my maths teacher in school...

@dankwraith hot fuck this looks like it could actually be used for things

k12 geometrical models for multiplication 

@dankwraith fourth grade uses a version of this for two digit by two digit multiplication. by fifth grade, we ditch that for lattice because the partial products get too large in three digit by four digit multiplication. There's way too much storage of numbers in working memory using the standard algorithm.

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