How can I deepen my knowledge in Mathematics without getting a degree in Mathematics?

Question

I'm an undergraduate student, I'm pursuing a Bachelor Degree in Computer Engineering. I had some exams about Mathematics (calculus 1/2/3, probability and statistics, algorithms and data structures, complex functional analysis).

While I don't want to become a researcher/professor (I would like to be a Software Engineer) I also love Mathematics, and I feel like I always want to know more.

Question: Which is the best way of studying on my own? Should I get some textbooks? Aren't textbooks suitable for those who are going to have an exam?

Actually I don't know which part I would like to deepen the most, but I'm really amazed by the Riemann hypothesis. What would it take to understand a wrong proof?

Answer 1

Not all math textbooks are designed for self-study, but a decent number of them are. You often won't find yourself with the answer key, but resources like Math StackExchange are always full of people willing to help folks who show a little enthusiasm and legwork.

If you had asked this question twenty years ago, I would have said that your best friend would be Dover Press, because they reprint great math textbooks in paperback in both popular and esoteric subjects and sell them at a significant discount. While this is still true, you will no doubt also find that you can get even more popular textbooks in PDF form for free on the web. That hits me a little bit in the morality, but there it is.

I think it's a great idea. As a pure mathematician who spent quite a while as a software engineer, your perspective will be a useful part of your team. Whether it is through coursework or self-study, I'd recommend that you get some discrete math under your belt like combinatorics and graph theory, because those modes of thought will aid in software design much more than the analysis side of math.

Answer 2

There is no right answer to your question, but I can highly recommend the book Concrete Mathematics

by Ronald Graham, Donald Knuth, and Oren Patashnik, first published in 1989, is a textbook that is widely used in computer-science departments as a substantive but light-hearted treatment of the analysis of algorithms.

Perhaps you are already familiar with this book. In the topics it covers I think it is an excellent book for self-study.

If you are interested in graph theory I can recommend Introduction to Graph Theory by Richard J. Trudeau available as a Dover Book.

I highly recommend the Schaum's Outline Series of books on many mathematics topics. They have lots of problems and solutions. Solving problems is a good way to get familiar with a subject.

Answer 3

I’ll recommend youtube - if you like visual learning. Here are some channels I find interesting: Numberphile, (also related channel computerphile), 3blue1brown, blackpenredpen, mathlogger, flammable maths, Dr Peyam

For Riemann zeta MrYouMath is absolutely great. However, don’t expect to solve it after watching his videos. ;) 3 blue1brown and numberphile have introductory videos on this subject.

As you get interested in specific topics you will find wealth of videos from various professors.

I recently came across:

Rob Shone when I needed deeper understanding of sequence of functions.

nptelhrd videos by Prof Venkata Balaji are wonderful for complex analysis (not all professors on this channel are as good); Prof S H Kulkarni is my second favorite on this channel.

Prof Steven Miller has great lectures on complex analysis.

You will find many more as you start marking the videos you like and dislike. YouTube algorithm will suggest similar videos.

Here is a playlist of some of the videos I have found useful and you may find interesting: https://www.youtube.com/playlist?list=PLRlT_RP_FVhnw0RzXdrZhki7F-c7AU6Uc