Where do I start learning Higher Mathematics? [closed]

Solution 1:

Khan academy might be a good starting point.

On youtube you can watch hundreds of their short videos on subjects like algebra, calculus, number theory, probability, linear algebra, geometry, trigonometry etc.

When you feel like you master that material you can move on to MIT OpenCourseWare on youtube.

Gilbert Strang's Linear algebra.

David Jerison's Single variable calculus.

Denis Auroux's Multivariable calculus.

Tom Leighton's Mathematics for Computer Science.

They also have courses on probability, statistics, differential equations etc.

Other personal favourites, which i actually liked more than MIT, are these

Discrete Mathematics. Arsdigita University. Instructor: Shai Simonson

The Fourier Transforms and its Applications. Standford. Professor Brad Osgood

Probability. Harvard

Probability Primer. Mathematicalmonk's channel

General topology from the very basics, including set theory, techniques for proofs

Graph theory by Sarada Herke

Short course on writing proofs in mathematics by Sidney Morris

At the same time you are following this online material i recommend buying books and solve a lot of problems. There is no better way of learning mathematics than solving problems. I would buy books which have solution manuals.

I would also recommend to start to learn to program in R, matlab, mathematica, maple, python or whatever environment you like best. This will become a very useful skill when you reach higher mathematics.

Solution 2:

I think you're in a good position to benefit from Keith Devlin's course (or just read the book) on mathematical thinking: https://www.coursera.org/course/maththink. Rather than dive into advanced mathematics directly, he takes time to reflect on what it means to "think like a mathematician" and develops some necessary logical prerequisites. The course/book concludes with a taste of different kinds of advanced mathematics. Here's the table of contents so you can see what I mean:

1. Introductory material
2. Analysis of language – the logical combinators
3. Analysis of language – implication
4. Analysis of language – equivalence
5. Analysis of language – quantifiers
6. Working with quantifiers
7. Proofs
8. Proofs involving quantifiers
9. Elements of number theory
   10. Beginning real analysis

Solution 3:

Well well, this is a broad question and you might get widely different answers, so take mine with a grain of salt!

I believe the first thing you should do before attempting to study any more mathematics is solidify your knowledge of basic logic. Study first-order logic, learn to write proofs of theorems in this language. This should not take long (only a general overview is required), but it will give you essential foundations.

You can then move on to other topics. In my first year at university we had a course which was using "A Concise Introduction to Pure Mathematics" by Martin Liebeck. I recall it was a pretty decent book, and it has some number theory in it! There is also a chapter on logic, although I would advice you look at external sources, too.