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 Powers and Fractions   Notes About Squares   Decimal Expansions   Repeat Length Again   Number Maze   Primes   Divisibility   Intrinsic Nature of Primes   Other Topics   Other Topics (2)

Numbers

It's a funny thing … I like mathematics, but I also like numbers. Probably you already know the difference, but let me explain it anyway, since a lot of people confuse the two.

I saw the confusion a lot last year, when I was being a physics tutor. We'd be working on some simple problem, maybe trying to figure out how long it takes an object moving at 2 m/s to go around a circle of radius 10 m, and the student would do something like this.

The total distance is the circumference of the circle, which I know is 2 pi r, or—punching on calculator—62.8 m. Then, I know how distance and velocity are related, that's d = vt, so then t = d/v, which gives—back to the calculator—31.4 s.

So, you can imagine that if I'd tried to explain that the problem was about mathematics, not numbers, the student would have said, what do you mean, it was numbers all the way through.

The way I see it, the real answer to the problem is t = 2 pi r / v. It's true that one can plug in the numbers and get another number, but that is much less interesting … or perhaps I should say, even less interesting.

By the way, I've never been able to get any of my students to stop thinking in terms of numbers … maybe it's one of those things that can't be explained.

So, anyway, now you can see the difference I was referring to: mathematics is about abstract, symbolic calculation, while numbers are as concrete as can be, like dividing 62.8 by 2.

Numbers do show up in mathematics, but typically only simple numbers. The integers from 0 to 4 are common, as is 6; 5 turns up once in a while, 7 rarely. The powers of 2 are relatively common. Of course you also see functions of these numbers … negatives, reciprocals, square roots, and the like, even an occasional non-reciprocal fraction like 2/3. Then there's pi, and e, and the like, which are symbolic numbers … do they count?

Then, there are some branches of mathematics that deal with numbers, but none of them really captures what it is that I like.

Arithmetic is close, since it's about individual numbers, but it's too uniform, too general-purpose; it doesn't say anything about some numbers being more interesting than others. As far as arithmetic is concerned, there's no difference between 128 + 256 and 126 + 258.

Number theory is close, too, but in the opposite direction. It says a lot about the patterns that make numbers interesting, but it tends to lose the individual numbers in the process.

If you'll permit me an analogy … learning arithmetic is like learning to walk; learning number theory is like studying maps; and playing with numbers is like hiking. It is enjoyable in an entirely different way.

Before I move on, I wanted to mention a couple of books. I have a book on number theory, Introduction to Number Theory, that's appealing partly because it's so amazingly dense. And, believe it or not, I also have a book on arithmetic, How to Calculate Quickly, which I just noticed was originally called The Art of Calculation.

Speaking of calculation, I just love the image it conjures up, of a time before computers, of accountants in green visors adding up numbers. It makes me think of the 1880s, for whatever reason … of Kafka, or maybe Brazil, or Memoirs Found in a Bathtub, or, instead, of The Difference Engine.

Anyway, by now I hope you're beginning to have some idea what I meant when I said that I like numbers. I like actual numbers, mostly integers; I like how they interrelate and form patterns. I'm no Ramanujan, to tell you what's interesting about 1729, but I can tell you what's interesting about 27, or 37. And, that's what this category is for: collecting thoughts about numbers that are interesting to me.