I read Asimov on Numbers back in 4th grade and thoroughly enjoyed it, and probably even understood some of it.

My oldest kid (8) has started coming up at random moments with things like “Hey Daddy, did you know that the sums of the digits of consecutive multiples of eight decrease by one except after a sum of 1 it ‘decreases’ to 9?” (He didn’t put it that clearly, but anyway, it was what he was saying…) And he has a general fascination with numbers and infinity etc.

That made me think of Asimov on Numbers, but I don’t have that book anymore. I could get it for him, but for all I can remember of it it may simply be too much right now. (I’ll probably get it anyway, it can’t hurt to have a copy laying around for his perusal should he be curious…)

Meanwhile, are there good books for kids that go into things that might be thought of as “beginning number theory?” Primes, Fibonacci, infinities, rational and irrational and transcendental numbers, things like that?

The idea of Zero as a number was foreign to our ancestors (there’s no zero in Roman numerals). The idea arose in India, spread through the Arab world, and much later to Europe. It’s a fascinating account of something we take entirely for granted.

It looks like a fun book but, I’m sorry to say, the name turns me off. Jake knows he’s smart, but I try not to let him get a big head about it, focusing instead on the question of whether he’s a hard worker or not.

(I identified as a “smart kid” as a child, and I was also inexcusably lazy partly because of that–and it has led to many of the more regrettable aspects of my current life and habits. So it’s something that’s always on my radar.)

At eight, he’s a little too young for Martin Gardner’s collections of “Mathematical Games” columns. But keep them in mind for later! They’re fun, inspiring, witty, and educational in a peculiar “niche” sort of way. You might call them back alleys and country trails of mathematics.

On the other hand, he might be a good age for Abbott’s “Flatland.”

That’s too young to be given them. But if you happen to have them lying about the house, it’s a wonderful age for him to accidentally find them on his own. Even if he doesn’t understand a lick of them, they’re a wonderful illustration of just how much more there is to mathematics.

Even absent a book, an excellent thing for you to do, when he makes one of these observations, is to ask him why he thinks that is, or what causes it. That’s where real mathematics lies.

I also recommend that you buy your son a copy of The Colossal Book of Mathematics, an anthology of the best Mathematical Games articles from Scientific American by Martin Gardner. He probably won’t be ready for really understanding them for a couple of years, but who knows? Incidentally, the observation that your son came up with is not something from number theory. It can be proved with just high school algebra. In fact, it’s possible to roughly understand what’s going on without even knowing algebra. Tell him to figure out the pattern for the sum of the digits for consecutive multiples of 7, then of consecutive multiples of 11, then of consecutive multiples of 16, etc. Then ask him to figure out what the overall pattern is. Point out that these all involve treating the digits 1 to 9 as a circle. Let him think about how casting out nines and modular arithmetic fit into this.

I got"The Numbers Devil" for my godson when he was about 8. the Amazon citation say 11 up, but I’ve always regarded those as rather specious.
It’s got 4 color illustrations, so to me that says ‘8 year old’.

It is my understanding that any time you are studying the properties of integers, you are doing number theory. On this account, something’s being “from number theory” does not exclude it’s being “provable with just high school algebra.” In any case you’ll note I was careful to couch my post in terms of things being “on the way toward” or “along a path toward” and various other phrases to that effect.

Thank you for the book suggestion–I will probably buy it. Your suggestions of questions to ask him were good ones, but somewhat obvious. (Indeed–I started asking him to think about what happens with seven and it turned out he was way ahead of me.)

> It is my understanding that any time you are studying the properties of
> integers, you are doing number theory.

For what it’s worth, I would say this is isn’t correct. (And I’m a mathematician.) What you’re looking at is an aspect of group theory:

If you want to go beyond the questions I suggested, teach him about number bases. Look at base-eight arithmetic, for instance. Look at the sum of digits for successive multiples of various numbers in base-eight arithmetic. Look at this idea in other bases. What’s the generalization here?

Another thing that I remember specifically noticing in fourth grade:

8 times 8 equals 64.
9 times 6 equals 63.
10 times 6 equals 60.
11 times 5 equals 55.
12 times 4 equals 48.
And so on.

So the second answer is 1 less than the first, and the third answer is 4 less than the first, and the fourth answer is 9 less than the first, and the fifth answer is 16 less than the first, and so on. Hey, those are the square numbers. And this works when you start with 9 times 9 or 10 times 10 or whatever. Now this is easy to prove when you know algebra, but that wasn’t taught in my school until the freshman year of high school. Well, I taught myself algebra in sixth grade because I was tired of waiting, but nobody else learned it till freshman year. I showed this to my brother who was four years older than me. He essentially told me to quit bothering him.

Note that there’s also another pattern in the above:

64 - 63 = 1
63 - 60 = 3
60 - 55 = 5
55 - 48 = 7

Neat, I thought. Those are just the odd numbers. I wondered how all this connected together. Maybe this might interest your son.

I actually thought group theory was part of number theory. But on poking around, it seems there’s a historical connection of some kind but not a connection of the “X is a sub-topic of Y” kind.

Anyway: My kid likes to think about patterns in numbers a lot.

So now I’m wondering: What’s number theory then? I have always read definitions that amount to saying it’s the study of whole numbers and their relationships. A brief google session would seem to bear this out. Is this a case where the popularizations are oversimplifying?

I think he may be old enough to understand parts of One, Two, Three … Infinity by George Gamow. The first two chapters are about numbers and rest is mostly physics, chemistry, and biology.

It reads like a story and has a way of immersing into a topic. Afterwards, he initiates discussion on a given topic - like, few weeks ago, we talked about the concept of function f(x) and inverse function.

My mom just gave my daughter this book last night (daughter is also 8) and while I haven’t read through it yet, it seems pretty interesting. One of the neat ways we can look at numbers is to look at how other people looked at numbers, and counted and calculated, and that seems to be the focus of this book, with both intellectual and hands on projects to explore. The Secret Life of Math: Discover How (And Why) Numbers Have Survived From The Cave Dwellers To Us!

It’s a little ! heavy, and the cover illustrations a bit babyish, but the interior content seems solid.

If he were a little older, I would insist that you buy him Math Girls. Don’t be scared by the title and cover: it’s not a girl-y book. Heck, go ahead and get it for yourself, and pass it along to him when you think he’s ready for it.