Computing with real arithmetic.

Computers and calculators generally have a fixed, limited number of slots in which to stuff the digits that make up a given number. This presents a problem when a computation involves a number that has more digits than the number of slots available. To overcome that difficulty, computer scientists in the 1950s developed the floating-point system of arithmetic, expressing each number in two parts. One set of digits gives the rounded-off number, and the other set represents, in effect, the position of the decimal point.

Although widely applied, the floating-point system still causes problems, especially when calculations using operations such as multiplication or subtraction yield answers that are extremely large or extremely small, falling outside the range of numbers that the system can represent. At that point, the system fails, and the computer produces the wrong answer or no answer at all.

To overcome this flaw, mathematician Peter R. Turner of the U.S. Naval Academy in Annapolis, Md., and others have been studying an alternative, logarithm-based method of representing numbers on a computer. For a given number, the idea is to find natural logarithms repeatedly until the result lies between 1 and 0. Thus, the original number is represented by the number of times the logarithm has been determined (the level) and the final logarithm (the index). The computer then uses this particular combination of digits in its calculations.

This "symmetric level-index" scheme avoids many of the problems the floating-point system encounters with numbers close to zero or approaching infinity. But it is slower, and if built into a computer, it requires more complicated circuitry than equivalent floating-point schemes. Nonetheless, says Turner, "I would maintain that useful information attained more slowsly is better than getting no answer at all quickly."

Similar criticisms initially greeted the floating-point system, which was slower than whole-number arithmetic. Yet that method flourished because it allowed researchers to develop efficient ways of solving problems they couldn't tackle before. The same may yet happen for symmetric level-index arithmetic. Although individual operations are slower, simpler programs could result.