Computers—the machines we think with

by D. S. Halacy

Using units in this manner to express a sum is called the unitary system. It is the concept that gives rise to the “if all the dollars spent in this country since such and such a time were laid end to end—” analogies. Put to practice, this might indeed have a salutary effect, but long ago man learned that it was not practical to stick to a one-for-one representation.

How long it was before we stumbled onto the fact that we had a “handy” counting system attached to our wrists is not positively known, but we eventually adopted the decimal system. In some places the jump from one to ten was not made completely. The Pueblo Indians, for instance, double up one fist each time a sum of five is reached. Thus the doubled fist and two fingers on the other hand signifies seven. In the mathematician’s language, this is a modulo-5 system. The decimal system is modulo-10; in other words we start over each time after reaching 10.

Besides the word digit in our vocabulary to tie fingers and numbers, the Roman numerals V and X are graphic representations of one hand with thumb widespread, and two hands crossed, respectively. A point worth remembering is that the decimal system was chosen arbitrarily because we happen to have ten digits. There is no divine arithmetical significance in the number 10; in fact mathematicians would prefer 12, since it can be divided more ways.

The ancient Mayans, feeling that if 10 were ten times as good as 1, then surely 20 would be twice the improvement of the decimal system. So they pulled off their boots and added toes to fingers for a modulo-20 number system. Their word for 20, then, is the same as that for “the whole man” for very good reason. Other races adopted even larger base systems, the base of 60 being an example.

If we look to natural reasons for the development of number systems, we might decide that the binary, or two-valued system, did not attain much prominence in naïve civilizations because there are so few one-legged, two-toed animals! Only when man built himself a machine uniquely suited to two-valued mathematics did the binary system come into its own.

Numbers are merely conventions, rigorous conventions to be sure with no semantic vagueness. God did not ordain that we use the decimal system, as evidenced in the large number of other systems that work just fine. Some abacuses use the biquinary system, and there are septal, octal, and sexagesimal systems. We can even express numbers in an ABC or XYZ notation. So a broad choice was available for the computer designer when he began to look about for the most efficient system for his new machine.

Considering only the question of a radix, or base, which will permit the fewest elements to represent the desired numbers, mathematicians can show us that a base of not 10, or 12, or any other whole number is most efficient, but the fraction 2.71828. The ideal model is not found in man, then, since man does not seem to have 2.71828 of anything. However, the strange-looking number does happen to be the base of the system of natural logarithms.

Now a system of mathematics based on 2.71828 might make the most efficient use of the components of the computer, but it would play hob with other factors, including the men who must work with such a weird set of numbers. As is often done, a compromise was made between ideal and practical choices. Since the computer with the most potential seems to be the electronic computer, and since its operation hinges on the opening and closing of simple or sophisticated switches, a two-valued mathematical system, the binary system, was chosen. It wasn’t far from the ideal 2.71828, and there was another even more powerful reason for the choice. Logic is based on a yes-no, true-false system. Here, then, was the best of all possible number systems: the lowly, apparently far-from-sophisticated binary notation. As one writer exclaimed sadly, a concept which had been hailed as a monument to monotheism ended up in the bowels of a robot!

The Binary System

It is believed from ancient writings that the Chinese were aware of the binary or two-valued system of numbers as early as 3000 B.C. However, this fact was lost as the years progressed, and Leibnitz thought that he had discovered binary himself almost 5,000 years later. In an odd twist, Leibnitz apprised his friend Grimaldi, the Jesuit president of the Tribunal of Mathematics in China, of the religious significance of binary 1 and 0 as an argument against Buddhism!