Resources for learning mathematics for intelligent people?

Question

Could people recommend resources to help my wife learn more complicated mathematics? She had a really terrible maths education, and while she essentially OK with every day maths she keeps wanting to know more about topics that would be college and university level.

She often wants to dig much deeper into a subject than basic texts allow, but there are some fundamentals that she has never been taught, which means there is quite a lot of going back to basics needed.

I can find a lot of resources for remedial mathematics that are aimed at basic numeracy, but the questions she wants to ask are things like: What is set theory(which led us into questions on what numbers mean, and Peano axioms), How does cryptography work? What are imaginary numbers? Various stats problems.

I've got a maths and comp sci degree so I usually know what the answer is, but there is such a void of knowledge between us, we can spend hours getting deeper and deeper trying to resolve a side issue from the main question, and it can get frustrating for both of us. :)

General resources would be great, or advice on how best to approach it. Specific resources about Crypto, number/set theory, and Statistics would also be appreciated.

Answer 1

I suggest some fun books such as "Mathematics and the imagination" by Kasner and Newman; "Geometry and the Imagination" by Hilbert; "Flatland" by Abbott; "How to lie with statistics"; and instead of cryptography, coding theory from "From Error-Correcting Codes through Sphere Packings to Simple Groups" by T.M. Thompson, as it is a fascinating story, even if you grasp only bits of it. Books by Tobias Dantzig about Numbers. "Zero: the history of a dangerous idea" by Charles Seife.

Good luck and enjoyment!

Answer 2

Bourbaki's Algebra and Set Theory. The structure of the books makes knowledge outside the series irrelevant, and they clean past misconceptions and handwaving out of the mind. They are very dense and challenging, though. While the books are not considered good introductions by many, I personally used them to start learning mathematics, and while under the same teaching circumstances as the OP, having the student read Algebra was the only way I was able to teach any real facility, in spite of many exercizes. The first chapter of Algebra can take a while to sink in, but in spite of protests it does not require reading Set Theory first, only knowledge of basic set operations and notation and of the product of sets. However, after a definition has been read, illustrations will make sense of it, and the illustrations will themselves make sense, because the difficulty comes from having to realize that the definition of the thing is exactly what it takes for something to be that thing, and that the definition is so open that you can make things that satisfy it at will.

After getting used to Algebra, Rotman - "An introduction to the theory of groups" is number theory-minded, and its historical expositions and diagrams add depth and explain, to the same degree Bourbaki doesn't, what it is you're doing. Note however, that neither of these introduce Cayley graphs early, which if she's geometrically minded you'll want to show her how to make as soon as she has a handle on the ideas (you wouldn't want her to rely on them entirely, as they depend on the group axioms).

These lay considerable conceptual groundwork: Between Bourbaki's Algebra and Rotman, she would have enough familiarity with axiomatic constructions and the interest in solutions to polynomial equations that you could introduce the complex numbers axiomatically, by showing how the algebra of the reals adjoined with a token solution to $x^2+1$ is defined completely just by assuming certain axioms on multiplication and addition that apply to the real numbers also apply to the new algebra, the ring axioms. I think that does its job to such satisfaction that it justifies the whole language. It would also be a shortcut from the early chapters of Rotman to the idea of the Galois group, which comes up somewhat late in the book.

Answer 3

I would suggest going to Khan Academy. They have many videos on math with a wide variety of skill levels.