How do people learn subjects so quickly? I'm pushing my

>>8472454
Not really. Brainletacademy will not teach you why addition is commutative or associative.

No one who does not study pure math even knows why it is that way. Everyone just memorizes it.

Engys and physicsfags here who talk about 'intuition' are talking out of their ass because they just memorized shit. Ask one of those faggots to prove commutativity of addition of constructive reals and they will give you a blank stare.

Don't swallow the intuition meme. No one knows the 'why' of even the most basic mathematics until they know set theory and formal logic.

>>8472454
You don't learn math by memorization. So, your biggest problem is that you're treating it like it's a history class.

Understand the intuition behind the concepts, and understand why problems are solved the way they are.

By memorizing only certain paths you can take to solve a problem, if the problem doesn't immediately make it clear that it can be solved by taking a similar path, you will struggle with solving it. You'd be doing yourself a favor by looking at problems more openly like "what is this question asking, what is the meaning of the solution, and how does that meaning follow from the question" rather than, "well what method that I've previously done can I apply here."

I didn't understand trig for shit until I understand how it relates to the circle, and now it comes second nature to me.

>>8472472
Book suggestions to understand why?

>>8472459
>Ask one of those faggots to prove commutativity of addition of constructive reals and they will give you a blank stare.

Memoryfag here, can't I just prove that with a pile of 4 rocks? I just shift the rocks from the left to the right and show that 1 rock and 3 rocks is 3 rocks and 1 rock and they're both 4 rocks.

When you say "you need to study pure math" I give a blank stare because it sounds like pretentious gibberish.

>>8472454
For high school stuff you just understand it. For college math, if you have the time you try to understand as much as possible, everything else you cram into your brain right before the exam, and hope that if you ever need it again you will get up to speed on the subject faster because you memorized it once.

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>>8472459
>prove an axiom

and you say we're the ones who dont understand math lmao

>>8472454
>and I'm making note cards in an attempt to memorize the mechanics that make it up better.
>memorize

Don't memorize, understand.

>>8472489
Congratulations, you proved that 3 + 1 = 1 + 3

That is not the same as proving that any addition of any pairs of numbers is commutative.

How do you know there is not some really big number, so big we will never count it, such that addition with it is not commutative?

I guess you better start gathering more rocks and prove commitativity for every single combination of numbers.

>>8472523
Commutativity of addition of real numbers is not axiom.

Commutativity of 'addition' of members of the arbitrary group G is an axiom.

It is best to get a book for the syllabus of your maths subject that you are struggling with and do constant questions until you can instinctively solve problems at a faster rate, what else helps is cds and dvds that correlate with the books, also don't trust khan academy he is dumb, I have just started my A-levels for Maths, Physics and History but have been able to use this constant repetition method to remember about half the syllabus in about 7 weeks. Knowledge is something that you have to work for and Khan academy won't help you with this.

>>8472525
But how should I whenever the properties are just given to me? :(

>>8472526
>How do you know there is not
That's a religion-tier argument. How do you know there is not an undiscovered flaw to your proof that any addition of pairs of constructive reals is commutative? Have you checked all possibilities, including the blatantly absurd ones? It takes a finite amount of time to prove a proof. It takes an infinite amount of time to disprove all potential disproofs. Until we actually try to add numbers that are so big we'll never use them and check the results, it's speculation. On faith of "it just works", proofs are right until they're proven wrong.

About the rocks, you don't need to know the number of rocks, only that you believe moving a rock from one pile to another is the same as removing a rock from the first pile and placing it in the second. Because once you have two separate piles of rocks, you can physically move them around. x+y and y+x is shorthand for literally picking up the x pile in a bag and placing it on the other side. Still, this doesn't prove it for negative numbers, which are pretty damn important.

>>8472532
What's the difference between a real number and members of an arbitrary group G? Is the official proof showing that two real numbers can be considered members of said group?

>>8472618
>How do you know there is not an undiscovered flaw to your proof that any addition of pairs of constructive reals is commutative?

Because I know formal logic and I know the axioms.

The reals are well established with axioms that are much more fundamental than "addition is commutative" and instead, through a lot of work, imply commutativity.

The more fundamental your axioms, the stronger your theory. We could just take commutativity as an axiom but why not just take every millenium problem as an axiom as well? That way we learn nothing about the structure of commutativity.

Why are things commutative? Why are some other things (like matrix multiplication) not commutative? We can't study those questions objectively if we just assume everything we want.

>>8472618
>What's the difference between a real number and members of an arbitrary group G? Is the official proof showing that two real numbers can be considered members of said group?

A group is an algebraic structure. As simple as I can, if I give you a collection of objects and a way to combine 2 objects so that it yields another object (the same way you combine 1 and 3 to yield 4 using addition) then you can do algebra.

In Algebra one of the more elementary proofs are proving that the traditional numbers (integers, rationals, reals, complex, etc.) form groups with addition. These things have to be proven.

As for the arbitrary group G, that is simply a way to explore the properties all groups share without relying on an specific (and probably non representative) one.

For example, the real numbers and the rational numbers are both groups, but these sets are really REALLY different.

So when you do algebra, instead of using real or rational numbers, you simply say:

Let G be a group and let a and b be elements of G...

and then you go with your proof. Every theorem you prove for G must be true for every thing you can show is a group.

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>>8472728
>looking up axioms on wikipedia
>mfw figuring out what an induction hypothesis is in relation to how the proof proves anything

Thanks for taking the time to reply. I appreciate it.

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>>8472728

The brainlets of /sci/ commends ye, Good Sire

May good fortune find thy path

I just want to be immortal already so I can pursue this at my own, snail's pace. Although I fear that that might actually take longer than the remaining lifespan of the universe, and so be completely pointless.

Alternatively, I'd like to be made immortal by contributing my couple of brain cells to a greater, hive-mind intelligence.

>>8472618
R