To Infinity, and Beyond:

A recent conversation reminded me that many of my lawyer friends don't know these really cool mathematical items, but enjoy them when they learn about them. I thought, then, that I'd quickly go through them; I'm no math maven these days, but I think I can get them right, substantively even if not with the level of formal precision that hard-core math people might prefer.

Here's the first question: Are there more positive integers (1, 2, 3, 4, 5, ...) or more integer squares (1, 4, 9, 16, 25, ...)? The answer: The two sets are of the same size.

Counterintuitive, some may say: After all, positive integers include all the squares, plus all the nonsquares, as well. In any interval from 1 to n, when n>1, there are more integers than squares; what's more, the ratio increases as n grows. In 1 to a million, for instance, there are a thousand times more integers than squares.

Yet this is quite true, under the standard mathematical definition of equal size. Let's define two sets as equal if there's a one-to-one mapping between them, so that each number in one set corresponds to precisely one number in the other. The mapping here is simple: Map any integer n to n squared, which is to say 1 to 1, 2 to 4, 3 to 9, 4 to 16, 5 to 25, and so on. Every number in the positive integers maps to precisely one square, and vice versa. The two sets are equal.

In fact, as one can easily prove through a similar approach, any two infinite subsets of the positive integers are equal. In fact, this may lead you to replace your old intuition (lots more integers than squares) with a new one (hey, both sets are infinite, and infinity = infinity). But not so fast! Two questions for you: Are there more positive integers or positive rational numbers (numbers that are the ratio of two integers, such as 2/7 or 549/100)? And are there more positive integers or real numbers? More on that soon.