How do people learn advanced math

I'm also interested. When learning to program I just found projects and learned how to write software for them. It didn't matter if it already existed, I just wrote my own implementation and became very good. Does something like this, with a guided tour starting from ~Calc 1 exist?

> How come some people can answer this so easily even though > I took college math and cannot? Why is the skill gap so great?

The maths you learned was not proof based, just knowing calq / linalg / diffeq's etc. is knowing how to use some tools required for STEM it's not really proper maths.

> How do people learn advanced math?

They do what you did but go a step further.

Most STEM undergraduates learn calculus, linear algebra, differential equations, maybe some laplace and fourier transforms.

The next step is to go right back to the beginning and learn to construct the lemmas and properties of numbers using logic and proofs. Most courses don't do this step. It's not a big deal unless you want to work in mathematics academia figuring out proofs to things.

> So I need to learn skills that will make me employable.

Mathematics is not what you should focus on. No one is going to start paying you to write proofs.

Have you done proof based maths before? I restarted my maths education from scratch awhile ago when realising my basic math foundations was shaky. I used basic mathmatics by Serge Lang which makes you prove things as you go. This got me used to proof by induction, and proof by contradiction for simple problems.

That might be bit basic though. Something like the book "how to prove it" or "book of proof" may help get you started.

Try this,seems to be quite detail and fairly reasoned: https://www.quantstart.com/articles/How-to-Learn-Advanced-Ma...

You start by writing your own examples. Remove all the abstract notation and expand out everything, distribute everything back in so you just have a series to be summed or integrated. Stare at it for k = 0, 1, 2, etc., like 5 choose 4, 5 choose 3...maybe you can prove this yourself. This is actually how you learn, trying to do it yourself and thinking for a few hours/days/weeks about this problem why the identity is true, and then getting all those 'How to prove it' style books filled with established techniques to see if your identity can be proved by them.

Try Terence Tao's book Analysis I https://learnaifromscratch.github.io/math.html you pick up the gist of this kind of math doing basic proofs of natural numbers. You see and do enough of these that you will get the first proof here, where the n+1 comes from, and how you can distribute things in/out of any summation notation. The second integral proof from math stackexchange the same applies, remove all the abstract notation and just write down a few series, distribute in the 1/2pi(i). Do this for a few examples and you can probably figure it out yourself.

In the above link there is resources for olympiad and putnam problem solving on YouTube, if you watch some of those you'll see how they train competitors to solve something like this problem. List every tiny idea or strategy you have to prove that binomial identity and pick one that seems the easiest, start there. Fill the variables with corner cases what happens. If it's geometry what happens if you 'stretch and pull' the geometric interpretation by some constant factor. Look at Pascals Array since this is a binomial identity. This is what they do in those seminars if you're interested and it's free for anybody to watch.

When all those proof books throw 'implies' and other logic at you, read through this: https://ncatlab.org/nlab/files/MartinLofOnTheMeaning96.pdf

A couple of things:

1) As others have pointed out, the example you use is not really calculus as such but about mathematical proofs. This might have not been covered in your college degree. You can pick this up if you want to, at least to the level required to follow the binomial example. It's interesting, its application in most jobs out there is limited though.

2) Math generally can get really, really hard. Insanely hard. It gets harder and harder for long, long time after it has reached the level that most mortals can understand.

2b) Fortunately (for most of us), the ability to earn money with math doesn't seem to increase significantly beyond the basic calculus, algebra, ... level.

If your motivation is to develop marketable skills since your crypto investments went belly up, then I think you're likely to find that math provides a poor return on your investment.

You're better off just learning to program and forgetting about all that higher level math, it's a tremendous amount of work, and it's not a path to riches.

With regard to that math stack exchange link- I heard an interesting anecdote once, that you don't really know a particular area of math until you study what comes after it.

So you don't know algebra until you study calculus, and you don't know calculus until you study analysis. The user who provided the solution in that post did his master's thesis in analytic number theory. A subject that may use relatively accessible tools, but is nevertheless notorious for its technicality.

Calculus is not "just calculus", it's a deep and enduring subject, you don't become fluent in it after a few courses. It's been more than ten years since I took calculus, and nearly ten years since I myself first taught it to undergraduates, and I'm still learning calculus.