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The Higher Arithmetic

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

Brian Hayes


The kind of computer arithmetic that comes closest to the mathematical ideal is calculation with integers and rationals of arbitrary size, limited only by the machine’s memory capacity. In this “bignum” arithmetic, an integer is stored as a long sequence of bits, filling up as much space as needed. A rational number is a pair of such integers, interpreted as a numerator and a denominator.

A few primitive computers from the vacuum-tube era had built-in hardware for doing arithmetic on integers of arbitrary size, but our sophisticated modern machines have lost that capability, and so the process has to be orchestrated by software. Adding two integers proceeds piece by piece, starting with the least-significant bits and working right to left, much as a paper-and-pencil algorithm sums pairs of digits one at a time, propagating any carries to the next column. The usual practice is to break up the sequence of bits into blocks the size of a machine register—typically 32 or 64 bits. Algorithms for multiplication and division follow similar principles; operations on rationals require the further step of reducing a fraction to lowest terms.

Looking beyond integers and rationals, there have even been efforts to include irrational numbers in exact computations. Of course there’s no hope of expressing the complete value of pi or √2 in a finite machine, but a program can calculate the values incrementally, supplying digits as they are needed—a strategy known as lazy computing. For example, the assertion pi < 3.1414 could be tested—and shown to be false—by generating the first five decimal digits of pi. Another approach is to treat irrational numbers as unevaluated units, which are carried through the computation from start to finish as symbols; thus the circumference of a circle of unit radius would be given simply as 2pi.

The great virtue of bignum arithmetic is exactness. If the machine ever gives an answer, it will be the right answer (barring bugs and hardware failures). But there’s a price to pay: You may get no answer at all. The program could run out of memory, or it could take so long that it exhausts human patience or the human lifespan.

For some computations, exactness is crucial, and bignum arithmetic is the only suitable choice. If you want to search for million-digit primes, you have to look at every last digit. Similarly, the security module in a web browser must work with the exact value of a cryptographic key.

For many other kinds of computations, however, exactness is neither needed nor helpful. Using exact rational arithmetic to calculate the interest on a mortgage loan yields an unwieldy fraction accurate to hundreds of decimal places, but knowing the answer to the nearest penny would suffice. In many cases the inputs to a computation come from physical measurements accurate to no more than a few significant digits; lavishing exact calculations on these measurements cannot make them any more accurate.

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