They’re distinct areas of math that have a substantial overlap: Galois theory, Galois cohomology, some other parts of algebraic number theory, some kinds of representation theory, and so on. Group theory studies a lot of things unrelated to number theory (classification of simple groups, general group cohomology and the related abstract nonsense, the word problem in groups, general representation theory, etc.), and vice versa (the entire field of analytic number theory, most notably).
No, I’d say Frylock’s right. There is some overlap between the subject; you’d find a discussion of modular arithmetic in an elementary textbook on number theory or on cyclic groups.
True, but it seems to me that thinking about the sort of thing that Frylock’s son was talking about would be most usefully studied as an application of group theory. That doesn’t mean that it would be a good idea for Frylock to give his son even an elementary textbook on group theory. But then, it wouldn’t be a good idea to give him an elementary textbook on number theory either. For the moment, give him books like those recommended in this thread. The hardest of them would be the collection of Martin Gardner’s articles. I would say that books that are considered to be popular accounts of “recreational mathematics” are generally the ones to look at. There’s no reason to restrict his son to any particular field of math at the moment. Let him read the popularized books on math, including Gardner’s, and let him decide on his own a few years from now whether he’s really interested in number theory, group theory, or any of the numerous other fields of math he’ll have read about.
I second this recommendation. It’s probably the most age-appropriate you’re going to get.
Some other, more “adult” recommendations, in case you think he’s ready for them, or if you want something for him to grow into, or if there are other people reading this thread who are looking for recommendations for themselves:
Number Story: From Counting to Cryptography by Peter Higgins
Several books by Calvin Clawson:
Mathematical Mysteries: The Beauty and Magic of Numbers
Mathematical Sorcery: Revealing the Secrets of Numbers
The Mathematical Traveler: Exploring the Grand History of Numbers
Excursions in Number Theory by C. Stanley Ogilvy
Recreations in the Theory of Numbers by Albert H. Beiler
A bit more wide-ranging and miscellaneous, but fun: Wonders of Numbers by Clifford Pickover
And the Teaching Company courses by Edward Burger:
Zero to Infinity: A History of Numbers
and Introduction to Number Theory
At that age I started watching Square One Television in the afternoons. Unfortunately I think it’s no longer on TV at all, but should you stumble across it…
Ghost from the Grand Banks has a lot on the Mandlebrot set.
Brian
Does he have a calculator? When I was young, I started messing around with repetitive operations. Like, add five, divide by eight, add five, divide by eight, add five, divide by eight… and noting how the result converged to a fixed answer.
By screwing around with this, I ended up independently figuring out Isaac Newton’s method of taking square roots!
Just mucking about with things is often a remarkable avenue to discovery.
Ah, yes, memories. My first exposure to the number e was from playing around on my calculator and realizing that it had a button to (for some reason) take logs to that base. I also figured out a repetitive pattern that gave me the Fibonacci sequence, and would sometimes kill time by factoring random numbers.
Is there any love for The Great Courses here? They’ve got one for Discrete Mathematics that seems to cover at least the beginnings of Number Theory, permutations, combinations, and other current topics. Even if it turned out to be too advanced for the kid, the OP might enjoy it just to keep a little ahead of the little rascal.
Maybe Simon Singh’s new book The Simpsons and Their Mathematical Secrets will be of interest to one of you. All of his books are great reading and he has a vast list of recommended maths books on his website.
I also love his youtube clip A Dramatic Demonstration of the Power of Mental Frames featuring Led Zeppelin in an an interesting way.
Several people have mentioned Martin Gardener. I was about your son’s age when I cut my teeth on his book Aha! Gotcha: Paradoxes to Puzzle and Delight. It’s actually about much more than paradoxes, and includes logic, number theory, set theory, and quite a bit more, in a wonderful and accessible style. I probably learned more from reading and rereading that book than from any other single book in my life, and it gave me a love of mathematics and logic that has stayed with me til today.
Yeah, I mentioned a couple in my post (#24). Haven’t seen the Discrete Math one, though.