Numbers: Their Tales, Types, and Treasures.

Chapter 1: Numbers and Counting

 

1.11.SYSTEMATIC ENUMERATION

The vocabulary of counting developed over many thousands of years in order to meet practical needs during the process of humankind's settling down. Expressing even large quantities by an exact number word became necessary to keep track of provisions or animals in a large herd, and for commerce. Hunters and gatherers had only a few number words, and then words like few and many. But if one only has a few counting words at one's disposal, such as names of body parts, one cannot count larger sets.

People need a set of counting words that cannot be exhausted—at least in principle, every natural number needs a unique name. Still, it should be easy to recite the sequence of number words in the correct order. In order to avoid a huge load on the human memory, the counting words have to be constructed in a systematic way based on simple logical repetition. Any such systematic method of creating number words is called a numeral system.

A basic idea for creating such a numeral system is the grouping of a large number of objects into manageable parts; for example, in such a way that each part can be counted with fingers. One can use this idea to count precisely and communicate the result, even at a stage of development where the language knows no number words at all, and even if the numbers involved are fairly large.

We do not know under what circumstances the first numeral system was developed. As a typical example, let us consider again the situation of the prehistoric shepherd who counted sheep by putting a pebble in a bag for every animal in his herd. With a growing number of sheep, the pile of pebbles might become unmanageable, and the method is not well suited for communication. So the shepherd follows a slightly different procedure. Let us assume he counts with his fingers, for example, by forming a fist and extending a finger for each animal coming through the gate. Once he has lifted all fingers of both hands, he puts aside a wooden stick and starts anew, counting the next group. So, for every group of 10 sheep, he would put a stick on a pile. For the final group, which probably does not amount to a full 10, he adds the corresponding number of pebbles. Eventually he will end up with 7 sticks and 9 pebbles, a handy and lightweight representation for a total of 79. He thus knows the number of sheep in his herd, probably without being able to name that number.

Once this process of grouping has started, it can be continued on a higher level. For counting larger quantities, people would take, for example, a bone for every 10 wooden sticks, and a big stone for every 10 bones. A collection of 1 big stone, 7 bones, 7 sticks, and 6 pebbles would thus symbolize the number 1776, as in figure 1.8. The progressive grouping is necessary to describe larger numbers. It is the basis for a systematic naming of numbers—a numeral system. It is not difficult to recognize our own numeral system in the shepherd's counting method: Just replace the word stick by the word tenbone by hundred, and stone by thousand.

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Figure 1.8: A natural representation of the number 1776.

We can also see how other base-n systems could have evolved. Had the shepherd used only the fingers of one hand, he probably would have counted in groups of five, and this would have been the origin of a numeral system with base-5—as it is, for example, in use in the Epi languages of oceanic island nation Vanuatu. Remnants of such a system are also visible in the system of roman numerals, where one has special symbols for five, fifty, and five hundred. Their symbols for the first numbers are I, II, III—easily recognizable as pictograms of fingers or counting sticks. The special symbol V, denoting five, represents a hand, and the letter X for ten obviously consists of two hands.

If the shepherd had used all his fingers and all his toes for basic counting, this would have been the origin of a system with base-20—as it is found in the Mayan culture and among the Celts in Iron Age Europe. In some European languages, the linguistic structure of the names of certain numbers still shows the Celtic heritage; for example, in French the word quatre-vingt for eighty means “four-twenty.” The Yan-tan-tethera (“one-two-three”) was a sheep-counting system in use in northern England until the Industrial Revolution. It was derived from an earlier Celtic language and used number words only for numbers up to twenty. For counting larger numbers, a shepherd would drop a pebble into his pocket every time he counted to twenty—that is, for each score. The word score, actually, comes from Old Norse, where it meant a notch on a tally stick. In a base-20 system, the twentieth notch was made larger, and this finally gave the meaning twenty to the word score.

If the shepherd had a system of tabbing with his thumb each of the three phalanges of the four opposing fingers, he would have created a system with base-12. A combination of phalanx counting on the right hand with finger counting on the left hand would lead to a system with base-60, as was used in ancient Mesopotamia. And in Toontown, the home of Bugs Bunny and Donald Duck, where all the cartoon characters have only four fingers on each hand, most probably a system with base-8 would have evolved.

How would the prehistoric ancestor of Donald Duck have represented the number 1776? To him, a stick would represent 8 items instead of 10. Consequently, he would replace 8 sticks by a bone (thus, a bone would represent 64 items), and a stone would represent 8 bones—or 8 × 64 = 512 items. It is not too difficult to figure out that he would need 3 stones, 3 bones, and 6 sticks, as in figure 1.9, because 1776 = 3×512 + 3×64 + 6×8.

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Figure 1.9: Representation of the number 1776 if we had only eight fingers.

What about other systems? A Sumerian using a base-60 system would have needed just 29 sticks and 36 pebbles (1776 = 29 × 60 + 36). On the other hand, with a small base—like 5—we need 14 stones, 1 bone, 1 pebble, and no sticks. And in order to be consistent, the 14 stones have to be represented with the help of the next higher category—say, pearls—where each pearl represents 5 stones (1776 = 2 × 625 + 4 × 125 + 25 + 1).