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The Higher Arithmetic

How to count to a zillion without falling off the end of the number line

Brian Hayes

Numerical Eden

In their native habitat—which is not the digital computer—numbers are boundless and free-ranging. Along the real number line are infinitely many integers, or whole numbers. Between any two integers are infinitely many rational numbers, such as 3/2 and 5/4. Between any two rationals are infinitely many irrationals—numbers like √2 or π.

The reals are a Garden of Eden for doing arithmetic. Just follow a few simple rules—such as not dividing by zero—and these numbers will never lead you astray. They form a safe, closed universe. If you start with any set of real numbers, you can add and subtract and multiply all day long—and divide, too, except by zero—and at the end you’ll still have real numbers. There’s no risk of slipping through the cracks or going out of bounds.

Unfortunately, digital computers exist only outside the gates of Eden. Out here, arithmetic is a treacherous process. Even simple counting can get you in trouble. With computational numbers, adding 1 over and over eventually brings you to a largest number—something unknown in mathematics. If you try to press on beyond this limit, there’s no telling what will happen. The next number after the largest number might be the smallest number; or it might be something labeled ∞; or the machine might sound an alarm, or die in a puff of smoke.

This is a lawless territory. On the real number line, you can always rely on principles like the associative law: (a+b)+c=a+(b+c). In some versions of computer arithmetic, that law breaks down. (Try it with a=1030, b=–1030, c=1.) And when calculations include irrational numbers—well, irrationals just don’t exist in the digital world. They have to be approximated by rationals—the very thing they are defined not to be. As a result, mathematical identities such as (√2)2=2 are not to be trusted.

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