Zero - Charles Seife [58]
It takes a clever trick to do such an intricate calculation. Irregularly shaped objects can be very difficult to measure. For instance, imagine that you’ve got a stain on your wood floor. How much area does the stain take up? It’s not so obvious. If the stain were shaped like a circle, or like a square or a triangle, it would be easy to figure out; just take a ruler and measure its radius or its height and base. But there’s no formula for figuring out the area of an amoeba-shaped mess. However, there is another way.
Take a rectangular carpet and place it on top of the stain. If the carpet covers the stain entirely, we know that the stain is smaller than the carpet; if the carpet is one square foot, then the stain must take up less than one square foot. If we use smaller carpets, our approximation gets better and better. Perhaps the stain is covered by five carpets of size one-eighth square foot; we would then know that the stain takes up at most five-eighths of a square foot, which is less than our approximation with a one-square-foot carpet. As you make the carpets smaller and smaller, the covering gets better and better, and your total carpet area approaches the true size of the stain; in fact, you can define the size of the stain as the limit as your carpets approach zero size (Figure 43).
Let’s do the same thing with the rational numbers—but this time our carpets are sets of numbers. For instance, the number 2.5 is “covered” by a carpet that includes, for example, all the numbers between 2 and 3—a carpet of size 1. Using this sort of carpet to cover the rational numbers has some very odd consequences, as Cantor soon showed, thanks to his seating chart. That seating chart accounts for all the rational numbers—it assigns each of them a seat—so we can count them off one by one, in order, based on their seat number. Take the first rational number and imagine it on the number line. Let’s cover it with a carpet of size 1. Lots of other numbers are covered by that carpet, but we don’t have to worry about that. So long as the first number is covered, we are happy.
Figure 43: Covering a stain
Now take the second number. Cover it with a carpet of size ½. Take the third number and cover it with a carpet of size ¼, and so forth. Go on and on to infinity; since every rational number is on the seating chart, every rational number will eventually be covered by a carpet. What is the total size of the carpets? It’s our old friend, the Achilles sum. Adding up the size of the carpets, we see 1 + ½ + ¼ + 1/8 +…+ ½n goes to 2 as n goes to infinity. So we can cover the infinite cohorts of rational numbers in the number line with a set of carpets, and the total size of the carpets is 2. This means that the rational numbers take up less than two units of space.
Just as we did with the stain, let’s make the carpet sizes even smaller to get a better approximation of the size of the rationals. Instead of starting with a carpet of size 1, starting with a carpet of size ½ makes the total size of the carpets equal to 1; the rational numbers take up less than one unit of space, in total. If we start off with an initial carpet that has size 1/1000, all the carpets, in total, take up less than 1/500 unit of space; all the rational numbers take up less room than 1/500 unit. If we start with a carpet the size of half an atom, we can cover all the rational numbers on the number line with carpets that, in total, take up less room than an atom. Yet even those tiny carpets, all of which can fit in the span of an atom, cover all of the rational numbers (Figure 44).
We can get smaller and smaller—as small as we want. We can cover the rationals with carpets that, summed together, fit in the size of half an atom—or a neutron—or a quark—or as small as we can possibly imagine.
How big are the rational numbers, then? We defined size as a limit—the sum of the carpets as the individual sizes go to zero. Yet at the same time, we saw that as the carpets get smaller and smaller, the sum of the cover gets tinier and tinier—smaller than an