# Resources for learning mathematics for intelligent people?

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.

-
Seriously though what numbers mean, peano axioms, you're going in the wrong direction. Get her a book on elementary algebra/trig and tell her to do all the problems, intuition and context is built up from the inside out – cuckmaster5000 Jun 21 '13 at 17:17
My advice: be sure your wife first understands thoroughly high school mathematics. If it is in Europe, Israel, etc., level A-B high school mathematics can be fine. From what I've heard, "usual" H.S. maths. in USA is of very low level, so perhaps try to enhance it a little. When this is done, then the way into more advanced mathematics will be smoother. – DonAntonio Jun 21 '13 at 17:18
Perhaps something like this? amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/… (I have no financial interests) – Bitwise Jun 21 '13 at 17:41
The title seems to suggest that the usual resources for learning mathematics (textbooks, articles, Wikipedia) are for stupid people. – Rahul Jun 21 '13 at 18:31
@RahulNarain My thought exactly. Where can I find a source to learn mathematics for "dumb" people? – Pedro Tamaroff Jun 21 '13 at 18:40

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!

-

Here's my advice: Work through each of the texts listed below. By "work through", I mean understand all proofs and do most of the exercises. This will bring her up to speed with roughly the average junior math major at the average US college. Note that prerequisites parenthesis.

1. Simmons, Precalculus Mathematics in a Nutshell
2. Axler, Precalculus: A Prelude to Calculus (1)
3. Stewart, Calculus: Early Transcendentals (2)
4. Velleman, How to Prove It: A Structured Approach (2)
5. Axler, Linear Algebra Done Right (4)
6. Niven, Elementary Number Theory (4)
7. Herstein, Abstract Algebra (4)
8. Apostol, Mathematical Analysis (3,4)
-

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.

-
-1: This is just about the worst possible answer I can imagine. The OP is asking about resources for someone to break into university level mathematics. If such a person could profitably read Bourbaki there would be no reason to ask such a question: she could simply read university-level books on set theory, cryptography and so forth. – Pete L. Clark Jun 21 '13 at 18:57
And the part about reading Bourbaki "in any order" is even worse: I (a research mathematician 10 years past my PhD) do own several volumes of Bourbaki. And there is great stuff in there for me -- but it is incredibly annoying the way they write as though the reader has the other $50$ or so volumes spread out on his desk, even when there is no need to do so. – Pete L. Clark Jun 21 '13 at 18:58
@PeteL.Clark Sorry, but that hasn't been my experience of the required expertise for those books. Maybe not in any order, but in the order you'd expect things to be limited to - topology, algebra, and set theory, the first three volumes, should be independent, as they are, while topological vector spaces should probably depend on topology and algebra. – Loki Clock Jun 21 '13 at 22:51
"Maybe not in any order." Then why did you say "in any order"? I'll give an example: the results on completions in commutative algebra use results from General Topology. Since completion is a partially topological concept it seems inevitable to use some notions from topology. But many times the justification in a step in a proof is given by quoting a result from General Topology by number...without even stating the result! Thus it is not enough for me to know general topology: I need to own or have ready access to General Topology. This makes Bourbaki very hard to read in places. – Pete L. Clark Jun 22 '13 at 0:25
@Loki: Wait, so you recommended to someone to read Bourbaki, in any order, but in no order have you actually read the majority of Bourbaki? Oy vey. Restricting to recommending the algebra book makes your answer much less ridiculous -- it merely becomes a recommendation that I don't agree with. But recommending the set theory is borderline irresponsible: unlike most of the other Bourbaki books which at least the experts can respect and often enjoy, the set theory text has a very poor reputation, especially among researchers in set theory. – Pete L. Clark Jun 24 '13 at 17:16

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

-
Thanks, it definitely looks like a good way to fill in the missing bits of her education. – Matt Waddilove Jun 25 '13 at 8:29