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I am an engineer. Hence, my knowledge of math is limited to applied math and primarily calculus. I want to study math on my own and get a good understanding of various concepts. How should I go about it?

11 Answers
Ben Carriel
Ben Carriel, I have a degree in pure mathematics
24.1k Views
I want to stress that there is a huge difference between engineering math and math as mathematicians know it. I'll define math as "the thing that mathematicians do" and assume that is what you are trying to learn. I don't mean to insult you, but the background you described means that you probably know very little math, if any.

That being said, it's never too late to start! Contrary to what others have been recommending, I do not recommend starting out with baby Rudin, especially if you have never written a proof before. I would say your best bets are as follows:

Absolute Beginner

By this I mean, you have little, if any, experience writing proofs. I do not recommend many of those "learn proof techniques" books because they often make math seem like a bag of ad hoc tricks. I would recommend:
I think that these books provide a introduction to mathematical thinking "in context" and should help you decide if math really is for you.

Beginner

At this stage you have either done one or more of the books above. Maybe you have some experience writing proofs from an engineering mathematics class, or physics. At this stage you have a solid understanding of precalculus fundamentals. Contrary to what is usually taught in schools, calculus is not necessarily the only next step. I would recommend:
I think that the first two options here are absolutely fundamental, but it is a damn shame that a lot of math students never study discrete math. Also discrete math is fairly accessible without a lot of background, and you can find yourself working on interesting problems fairly quickly.

Core Undergraduate

This level would be for you if you have completed the beginner stage, or what is typical of an affiliated math major at a research university. There are three "core subjects" that math degrees universally require. They are:
You don't have to grind through all of the above, each of those books has a wealth of wisdom. You should aim to do one or two from each group.

Other Undergraduate

This is where things start to really diverge. Honestly, with solid foundations in each of the three above areas you could start tackling the graduate level texts in those subjects. By this point, you should have enough "mathematical maturity" to pick up any undergraduate level book and get going with little resistance. For more variety, I'll give you my favorite books in a bunch of other areas that an undergrad might be interested in:
The above should not be considered exhaustive. By this point, you are on the level of most math major upperclassmen.

After going through the above you'll be ready to start doing some real mathematics. Pick up a graduate level text, or a research paper in the field of your choice and get cracking. Have fun!

Edit** Added links and fixed some typos.
Daniel McLaury
Daniel McLaury, [math]P[A \wedge B] \neq P[A] P[B][/math]
3.7k Views • Most Viewed Writer in Pure Mathematics
Math is a huge area, and even professional mathematicians generally only know one particular piece of it.  So you have to choose what direction you want to move in and then make a decision.

Probably the single best multiplier in any field is really understanding linear and multilinear algebra well (including those basic parts of functional analysis that often get lumped into "linear algebra").  You might try Axler's "Linear Algebra Done Right" for an introductory approach from an algebraic perspective, followed by the relevant sections from Aluffi's "Algebra: Chapter 0" (which I prefer to other texts like Hungerford or Lang for this sort of thing) to get the more advanced algebraic perspective.  Then you'd also want to learn about the various matrix tricks like Sylvester matrices and such, and if you want to learn about using matrices to do calculations with data then you probably want to read one of Strang's books and also parts of Golub and van Loan.
Karthik Hemmanur
Karthik Hemmanur, full stack Product Manager, gym-rat, and a Spartan for life!
1.9k Views
I'm a comp sci/ math major and I can very much relate to what you are feeling. In most of my higher level math classes it doesn't help at all if I go in with my analytical first mindset. Be open to the idea of learning concepts that might not necessarily have engineering applications right away, and be patient with it. I started loving abstract algebra only towards the end of my semester. It's a process, I hope you enjoy it.

These are some of the books I've seen widely used at undergrad level.

Linear Algebra Robert Messer
Abstract Algebra Thomas Hungerford
Combinatorics and graph theory Mossinghoff
Bill Bell
Bill Bell, Graduate studies so long ago that they are mere diaphanous memories.
3.2k Views
You might consider watching some of the videos by Professor N L Wildberger on youtube at njwildberger. He doesn't discuss everything that you might ultimately want but he does offer interesting approaches to a variety of topics.
Ryan Howe
Ryan Howe
2k Views
In addressing your question, I believe it is necessary that you are not mislead. Such as here...

"I want to stress that there is a huge difference between engineering math and math as mathematicians know it. I'll define math as "the thing that mathematicians do" and assume that is what you are trying to learn. I don't mean to insult you, but the background you described means that you probably know very little math, if any."

This is first untrue and secondly completely unnecessary to mention. I will not begin to concern myself with that definition of math. I will not list off books, as it has been done and I believe you are above that.

This boils down to what a few have said, developing ones own capability for mathematical discourse and thought and maintaining ones own motivation, direction and goals over time.

Your knowledge insofar can help but really the first, as it has been called "Real", and hence Pure, math studies you must begin are on Mathematical Proofs at an introductory level. There are a million different ones, and they will all cover: Set Theory, Basic Logic, and then Meaning of Mathematical Statements -> Meanings of Proofs -> Methods of Proofs. The purpose is that you begin to frame your thoughts and problems as mathematical objects that have definite purpose and meaning so they can be manipulated together to form more complex ideas.

Disregarding the various branches created by the set theory used and logic then followed,  that in essence covers "what mathematicians do" rather fully (foundations). All other math really falls into three categories that sometimes mesh:  Geometry (Topology), Algebra, Combinatorics, Applied and Analysis. Most students then naturally go to study the beginnings of analysis and algebra, (real -> complex) and abstract and linear algebra (both at the same time or either or) respectively.

As you start, you should begin to develop habits in asking yourself why structures exist or are defined are they are and if so what others ones are similar or could be. The most immediate is, once you begin to learn about Groups, you'll think about arithmetic and then immediately you understand without truly knowing, what a Field is, or specifically the F(C) the field of complex numbers which you as an engineer use...everywhere.

The point really is that being a mathematician is not about, as I once thought, lauding all the great men before and their contributions or discriminating against what are simply different uses of math, but really choosing to question what and how you already understand mathematics and as you continue to apply those concepts to view the world and its issues in terms that relate to either a completely new mathematical construct you must determine and create or (by relating to proof) calling on historical use of the techniques to choose an accurate model to create a construct to logically describe a system of manipulating that object and laying claims against or for it.

In many ways whole branches of mathematics that are tremendously important (Abstract Algebra and then, god help me, Galois Theory, and followed by Representation Theory) are instances where a set of assumptions on existing framework or mathematical object are either loosened or tightened and explored. e.g. Abstract Algebra is not about simply the algebra of just real or complex numbers but the general concept of a binary operation on a set and a set of assumptions.

well, I went on a bit..but you should see, mathematicians don't wake up and say "I think I'll solve this great problem" rather really it is in beginning to consider they could declare that, they find somewhat which has opportunity. This festers in their mind, my mind, for years sometimes, and then it clicks.
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