learningself-learningsoft-question
I want to find a starting point to learn "formal" math. While I've always been interested in math and am good at high school math I know hardly anything beyond that.
Some YouTube channels, like 3blue1brown, give you intuition for some phenomena I still don't know how I could start learning some rigorous math before going to university.
I feel like it depends on where you are headed. If you want to make mathematics you future profession, the way you take will be different from what say an engineer will take. For example, in my case I am engineering student and i got to study plenty of calculus, probability and much of the fancy stuff but by the end of the day i still felt my knowledge of maths to be unsatisfactory(that's why i am on his site by the way).
So on the way to achieving your goal, here is what i can tell depending on my experience.
If engineering is your way:
You have to work very much on problem solving. A possible way to approach the task, here it is. Start will "normal" calculus but now try to understand the concepts not just for computing answers but also try to understand what it means in real life. For example, say "limits". You must have studied those in high school. Understand carefully what it means. Try to find examples where this concept might fit. Here is an example: I am given a material whose 'flexibility' is modeled by a given function. And that function depends on temperature. Here limits may help you understand how the material behaves when the temperature tends towards a certain value. See ... Try to start thinking like that about concepts, not just solve some exercises - but don't get me wrong: exercises are of crucial importance in learning, but the difference between you and a maths software is that you must understand the why of every computation you are doing.
Now a possible road map:
I/ Calculus:
Integral transforms
Differential equations
II/ Algebra
From there you can go ahead and study other areas of interest mainly (i) Engineering optimization and numerical analysis (ii) Statistics and probability.
Those two because as an engineer the sooner you start producing results, the better off you are.
Starting with calculus is important because it has a lot of applications you can play with, it gives computational skills fast if you do exercises, has interesting concepts and forms the foundation of much mathematics engineers deal with.
Possible books:
So basically, it boils down to
If mathematics is your way:
Now if you want to make mathematics your profession, you will need a different frame of mind. First i am neither a professional mathematician nor have i reached a level where i can say that i am thinking like one. Yet that is my goal too. So i will share with you what i have learnt so far.
First, mathematicians, from i can tell so far, work differently from say physicists and engineers. When a you hit a theorem, don't go ahead and read the proof, first try to prove it yourself.
That will form the basis of the mathematician in you.
Here is the books i can advice to start with.
"How to prove it, A structured approach" by Daniel Velleman. Nice book for an introduction to proofs. I like the idea of givens and Goal.
"Book of proof" by Richard Hammack. Nice little book. You can either start with this one or Velleman. The thing i like with this one is that logic and set theory are separated in comparison with Velleman. - http://www.people.vcu.edu/~rhammack/BookOfProof/
Once you are grounded in Set theory ( not too much though, whatever is provided by the two previous will be enough ) and proofs, continue with these:
Always try to prove theorems before reading the proof. Every time you read a mathematics book, usually graduate level ( don't be concerned about these for now ), and they tell you that a certain amount of mathematical maturity is expected from the reader, what that simply means is that they expect you to be able to prove the theorems or at least follow the logical arguments.
Mathematical literature
Also I highly advice like others that you try to read about mathematics in the general sense. Some books you may start with, here they are.
You may not be able to follow, the proofs in the two first books but nonetheless, you will enjoy the ride!!!
So that's the best i can do for my level and I wish you good luck and success!
MIT has a website with some math courses online http://ocw.mit.edu/index.htm. They have the video lectures and homework sets available online. There are other schools as well. It really depends on what you want to learn about mathematics, and what you want to do with it. If you want to see it applied, I would suggest learning Linear Algebra pretty well as it is the basis(not a pun) for compression, wireless communication, coding, and a bunch of other cool stuff. If you want a more natural feel for the mathematics I would suggest looking at the first year physics course that is online. If you don't want to get bored, you can always pick random chapters from the book to look at and go back to fill in the gaps you need. As a starting point though, watching the physics 1, and single variable calculus lectures on MIT's OCW would be a good place to relearn some old and new math.
Best Answer
Historically, a lot of people have started with number theory or graph theory. They are fields with accessible concepts and interesting questions, and they'll get you into the practice of formal proofs.
My professors taught me that if I wanted to learn X theory, I should just walk down two floors to the science library and pull any book off the shelf called "Introductory X Theory", because they would all say pretty much the same thing. I must say that this rule has not let me down very often. You could do the same thing with Amazon. The cheapest used paperback book on a topic was reprinted in paperback because it was good enough to be read casually.