NUMBERS AND COUNTING
SYSTEMS: A SHORT HISTORY
Numbers are familiar to us from when we learn to talk. However, we do not learn to write numbers until later. There was a time when people did not write numbers at all. We have inherited number figures from the Persian Empire. They inherited much of their learning from the Greeks of the time of Alexander the Great. When Rome fell, the Persian Muslims kept learning more about numbers and astronomy. One of the only benefits of the Crusades was that this knowledge became available to the West. All of us are familiar with the numbers printed up above. They are what we call the decimal system, from the Latin word "deci" for ten. This seems like a logical system, for we have ten fingers and this system counts to ten and then counts by tens and ones and then by hundreds, tens and ones and so on. When we learn to add, we learn to keep the numbers in their columns of ten. My guess is that most people think that this decimal system, called "base 10" is some kind of numerical law. Let us look further. Here is an example of a system called "binary" which computers use.
I have put the familiar number first as a way to translate the binary system. Binary has two digits, not ten. These digits represent two states in the digital circuit. Zero is off and one is on. Think of a light switch going on and off. Or Morse Code, which is another binary system. Flick the light on and off and you have binary. Most of us are familiar with the word because we live with computers and some of the jargon associated with them. Few people know that binary is base two, from the Latin word for two, "bi." It would seem impossible for people to count in this system, but it is extremely easy for computers.
Another system with which we are familiar, but seems weird to us, is base twelve. We never use twelve digits, which is why it might seem strange to use it at all. However, we see clocks all the time that have twelve hours, and the old English system of measurement has twelve inches in a foot. We have twelve months in the year. Why? Sometimes kids ask this, but adults (unless they are scientists) rarely ask it. This answer is: because of Babylon. Yes, this system is from Babylon, the old, old Babylon of Hammurabi and the Bible. I shall explain more about this system in a moment. I want to show you one more system that is familiar and irritating to most people.
How many of you have had to learn something of the old Latin system of numbers? This system is in base five and ten, mixed. There are five or six digits and it quickly becomes almost as unreadable as the binary numbers. Yet people used it for over 1,000 years and continued to use it for another 1000 years. People still use it today. No one does calculation in these numbers, but for dates, volumes of books, and tombstones it is still very much in use. Although it is in a kind of base 5, note that the number 4 is not IIII, but IV or one from five. You see the same with the number nine, IX or one from ten. Fourteen is ten plus five less one. When people used this system in writing it seemed natural. It was not said in the way it was written, but with words that, to us, sound very normal.
Before people had pencil and paper, before they had adding machines, they had their hands. Here are two hands I have drawn. Ten fingers. Easy. Children add and subtract small numbers on their fingers. It is hard for some of them to "get" the written expression of numbers. How do you jump from three fingers to the number three? It may have three prongs, which is where it came from, three horizontal lines, but what of four? Eight? Nine? Children must learn the illogic of the system of writing. For all of them, counting on the fingers is much more natural. They don't think much about the words, since they learn the words with the fingers, but they do learn the "right" way to express the number 1, with the forefinger, or is it the thumb? In Europe it is the thumb. As you can see, even counting on the fingers is crazy.
This is where the Babylonians got their base 12. They did not count with the fingers opening for each number. The counted on one hand, using the thumb as the "counter" and the joints of the fingers as the numbers. Twelve joints. Three sets of four numbers or four sets of three numbers, depending on how you count. I did three sets of four numbers. The Babylonians were also attracted to the number 4. Four seasons, four directions, four castes, and four numbers times three was a magical number twelve.
Using the other hand as a counter, the Babylonians could easily count to 144. Twelve numbers with the forefinger. Twelve more with two fingers, first and middle. Twelve more with three fingers, twelve more with four fingers. Four times 12, or 48 numbers. Then twelve more numbers with the middle finger, twelve more with the ring finger and twelve with the pinky. 36 more numbers for a total of 84. Then twelve more numbers with the middle and ring finger, and twelve more with the ring and pinky, and twelve more with the first and ring and twelve more with the middle and pinky for another total of 48 more and finally, twelve more with the last three digits of the hand. 60 and 84 equals 144. We still have a name for it: one gross.
My next question may seem stupid, but no questions are stupid. The Europeans inherited some of the Babylonian counting systems, but did they inherit the words? This was not a European system. About the time of Babylon, Stonehenge was already abandoned. The people of the West Coast of Europe had traded heavily with Crete and Phoenicia, but they were equal to or more advanced that these societies in verbal math. They did not write anything down until very late, about the time of Greece. Then, they used an alphabet of twenty letters that were written as scratches or hatch marks in stone or wood. It looked like this:
All very well and good, but the fascinating part of this was that Ogham was a finger alphabet. Druids could flash signs across the room at each other using their finger alphabet. We don't use our fingers for words (unless we are deaf) but we use our fingers for counting. There is every reason to believe that the letters in Ogham were also numbers. This is the case in many cultures, the most famous of which is the Hebrew Kabbalah. Twenty letters, twenty numbers? But we only have ten fingers, so this seems unnatural.
Here is the Ogham alphabet on the hand. Look familiar? Look more closely, Four flights of five. Five fingers, four knuckles each. If you count using the finger combinations I described above, you can count to 240 or twelve finger positions and twenty numbers. 240 is a large number, sufficient for almost everything in the ancient world. But was it twenty? Let us look at our language for a clue. To this day, many European languages count the higher numbers in 20s. 80 is four twenties. The old word for twenty is "score." Most people know the passage "four score and seven years ago" or the Biblical reference: "three score and twelve" for the life of a man. In the old counting systems of the country folk of Britain and France, 11 was "one and ten" but 16 was "one and fifteen" and fifteen had its own word, not related to the words for five or ten. Twenty also was a unique word. Our own word, "twenty" is a contraction of "two tens." So here we are, base five again, like the Romans, but with four having its own word and not "five less one." After the number twenty there are no more unique words until 100 and then 1,000. So we count by fives and then by twenties and then by hundreds. The old English money system had 20 shillings in a pound. Under the influence of the Romans, some of the older system was abandoned, but I do believe that the people who lived in Europe before the Roman invasions, had a base five and twenty number system.
My next big question is: well counting is all well and good, but could they add?" The answer is yes, very easily.
Let's look at the hands again. Five fingers, ten fingers. No problem. But let's be creative. With a turn of one hand, we suddenly have fifteen. With another turn, we have twenty. Suddenly, this system is every bit as logical as the Babylonian.
Here I have illustrated some of the numbers in between. You can suddenly see why 16 would be "fifteen and one". 1 (sorry, if you use the thumb for 1) 6, 9 are normal to us. But look at 11. We turn one hand over and now start counting on the back, going to the back-front hand position for 15. Then, at 16, we start the next flight of five by turning the counting hand again and so on to twenty. This would suit the Druids, who could flash letters and numbers across the room at each other. Twenties would be flashed as the hands opening and closing. People still do this at auctions and when the noise is intense at markets.
Here is my illustration of an addition problem. 27 + 54. I want you to think of this problem as a verbal problem, not seeing the numbers in their decimal columns. Think of how the numbers look on the hands. Suddenly, you see three twenties. Then you see three palms, so three fives. then the remander fingers, which are two and four. Six is also five and one, so you have four fives, or twenty again. Four twenties and one. With practice, this could be extremely easy and fast, subtraction as well. There are many systems of using the hands for multiplication, even of large numbers. But this article was to introduce you to our own numbers and the possibilites of ancient systems without written figures. Many people I talk to about ancient math assume that math began with the Greeks because they wrote it down! This is as silly as saying that language began with the the people who first wrote it down.