Zero - Charles Seife [49]
But how do you prove that two feet is actually the limit of the race? I ask you to challenge me. Give me a tiny distance, no matter how small, and I will tell you when both Achilles and the tortoise are less than that tiny distance away from the limit.
As an example, let’s say that you challenge me with a distance of one-thousandth of a foot. Well, a few calculations later, I would tell you that after the 11th step, Achilles is 977 millionths of a foot away from the two-foot mark, while the tortoise is half that distance away; I have met your challenge with 23 millionths of a foot to spare. What if you challenged me with a distance of one-billionth of a foot? After 31 steps, Achilles is 931 trillionths of a foot away from the target—69 trillionths closer than you needed—while the tortoise, again, is half that distance away. No matter how you challenge me, I can meet that challenge by telling you a time when Achilles is closer to the mark than you require. This shows that, indeed, Achilles is getting arbitrarily close to the two-foot mark as the race progresses: two feet is the limit of the race.
Now, instead of thinking of the race as a sum of infinite parts, think of it as a limit of finite sub-races. For instance, in the first race Achilles runs to the one-foot mark. Achilles has run
1
1 foot in all. In the next race Achilles does the first two parts—first running 1 foot, and then a half foot. In total, Achilles has run
1 + ½
1.5 feet in all. The third race takes him as far as
1 + ½ + ¼
1.75 feet, all told. Each of these sub-races is finite and well-defined; we never encounter an infinity.
What d’Alembert did informally—and what the Frenchman Augustin Cauchy, the Czech Bernhard Bolzano, and the German Karl Weierstrass would later formalize—was to rewrite the infinite sum
1 + ½ + ¼ + 1/8 +…+ ½n +…
as the expression
limit (as n goes to ?) of 1 + ½ + ¼ + 1/8 +…+ ½n
It’s a very subtle change in notation, but it makes all the difference in the world.
When you have an infinity in an expression, or when you divide by zero, all the mathematical operations—even those as simple as addition, subtraction, multiplication, and division—go out the window. Nothing makes sense any longer. So when you deal with an infinite number of terms in a series, even the + sign doesn’t seem so straightforward. That is why the infinite sum of +1 and -1 we saw at the beginning of the chapter seems to equal 0 and 1 at the same time.
However, by putting this limit sign in front of a series, you separate the process from the goal. In this way you avoid manipulating infinities and zeros. Just as Achilles’ sub-races are each finite, each partial sum in a limit is finite. You can add them, divide them, square them; you can do whatever you want. The rules of mathematics still work, since everything is finite. Then, after all your manipulations are complete, you take the limit: you extrapolate and figure out where the expression is headed.
Sometimes that limit doesn’t exist. For instance, the infinite sum of +1 and -1 does not have a limit. The value of the partial sums flips back and forth between 1 and 0; it’s not really heading to a predictable destination. But with Achilles’ race, the partial sums go from 1 to 1.5 to 1.75 to 1.875 to 1.9375 and so forth; they get closer and closer to two. The sums have a destination—a limit.
The same thing goes for taking the derivative. Instead of dividing by zero as Newton and Leibniz did, modern mathematicians divide by a number that they let approach zero. They do the division—perfectly legally, since there are no zeros—then they take the limit. The dirty tricks of making squared infinitesimals disappear and then dividing