The number system with which we are most familiar is the decimal (base-10) system, but over time our ancestors have experimented with a wide range of alternatives, including duo-decimal (base-12), vigesimal (base-20), and sexagesimal (base-60) …

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The Decimal Number System
The number system we use on a day-to-day basis is the decimal system, which is based on ten digits: zero through nine. The name decimal comes from the Latin decem, meaning “ten,” while the symbols we use to represent these digits arrived in Europe around the thirteenth century from the Arabs who, in turn, acquired them from the Hindus. As the decimal system is based on ten digits, it is said to be base-10 or radix-10, where the term radix comes from the Latin word meaning “root.”

Outside of specialized requirements such as computing, base-10 numbering systems have been adopted almost universally. This is almost certainly due to the fact that we happen to have ten fingers (including our thumbs). If Mother Nature had decreed six fingers on each hand, we would probably be using a base-twelve numbering system. In fact this isn’t as far-fetched as it may at first seem. The term tetrapod refers to an animal that has four limbs, along with hips and shoulders and fingers and toes. In the mid-1980s, paleontologists discovered Acanthostega who, at approximately 350 million years old, is the most primitive tetrapod known – so primitive in fact, that these creatures still lived exclusively in water and had not yet ventured onto land.

The First Tetrapod
The decimal system with which we are fated is a place-value system, which means that the value of a particular digit depends both on the digit itself and on its position within the number. For example, a four in the right-hand column simply means four – in the next column it means forty – one more column over means four-hundred – then four thousand, and so on.

Unfortunately, although base-ten systems are anatomically convenient, they have few other advantages to recommend them. In fact, depending on your point of view, almost any other base (with the possible exception of nine) would be as good as, or better than, base-ten. This is because, for many arithmetic operations, the use of a base that is wholly divisible by many numbers, especially the smaller values, conveys certain advantages. The number ten is only wholly divisible by one, two and five, so, given the choice, we might prefer a base-twelve system on the basis that twelve is wholly divisible by one, two, three, four, and six. By comparison, for their own esoteric purposes, some mathematicians would ideally prefer a system with a prime number as a base: for example, seven or eleven.

The Ancient Egyptians
Number systems with bases other than ten have appeared throughout history. For example, the ancient Egyptians experimented with duo-decimal (base-12) systems in which they counted finger-joints instead of fingers. Each of our fingers has three joints (at least they do in our branch of the family), so if you use your thumb to point to the joints of the other fingers on the same hand, you can count one-two-three on the first finger, four-five-six on the next, and so on up to twelve on your little finger.

If a similar technique is used with both hands, you can represent values from one through twenty-four. This explains why the ancient Egyptians divided their days into twenty-four periods, which is, in turn, why we have twenty-four hours in a day. Strangely enough, an Egyptian hour was only approximately equal to one of our hours. This was because the Egyptians liked things to be nice and tidy, so they decided to have twelve hours of daylight and twelve hours of nighttime. Unfortunately, as the amount of daylight varies throughout the year, they were obliged to adjust the lengths of their hours according to the seasons.

One of the methods used by the Egyptians to measure time was the water clock, or Clepsydra, which consisted of a container of water with a small hole in the bottom through which the water escaped. Units of time were marked on the side of the container, and the length of the units corresponding to day and night could be adjusted by varying the distance between the markings or by modifying the shape of the container (by having the top wider than the bottom, for example). (The term “Clepsydra” is derived from the Greek klepto, meaning “thief,” and hydro, meaning “water.” Thus, Clepsydra literally means “water thief.”)

In addition to their base-twelve system, the Egyptians also experimented with a sort-of-base-ten system. In this system, the numbers 1 through 9 were drawn using the appropriate number of vertical lines; 10 was represented by a circle; 100 was a coiled rope; 1,000 a lotus blossom; 10,000 a pointing finger; 100,000 a tadpole; and 1,000,000 a picture of a man with his arms spread wide in amazement. Thus, in order to represent a number like 2,327,685, they would have been obliged to use pictures of two amazed men, three tadpoles, two pointing fingers, seven lotus blossoms, six coiled ropes, eight circles, and five vertical lines. It requires very few attempts to divide tadpoles and lotus blossoms by pointing fingers and coiled ropes to appreciate why this scheme didn’t exactly take the world by storm.

Actually, it’s easy for us to rest on our laurels and smugly criticize ideas of the past with the benefit of hindsight (the one exact science), but the Egyptians were certainly not alone. As an example, we might consider Roman numerals, in which I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000, and so forth. Now try to multiply CCLXV by XXXVIII as quickly as you can. In fact, Roman numerals were used extensively in England until the middle of the 17th century, and are still used to some extent to this day; for example, the copyright notice on films and television programs often indicates the year in Roman numerals!

The Ancient Babylonians
Previously we noted that, for many arithmetic operations, the use of a number system whose base is wholly divisible by many numbers – especially the smaller values – conveys certain advantages. And so we come to the Babylonians, who were famous for their astrological observations and calculations, and who used a sexagesimal (base-60) numbering system. Although sixty may appear to be a large value to have as a base, it does offer some interesting features advantages. Sixty is the smallest number that can be wholly divided by one, two, three, four, five, and six ... and it can also be divided by ten, fifteen, twenty, and thirty. In addition to using base sixty, the Babylonians also made use of six and ten as sub-bases.

The Babylonians’ sexagesimal system, which first appeared around 1900 to 1800 BC, is also credited with being the first known place-value number system, in which the value of a particular digit depends on both the digit itself and its position within the number. This was an extremely important development, because – prior to place-value systems – people were obliged to use different symbols to represent different powers of a base. As was illustrated by the Egyptian and Roman systems discussed above, having unique symbols for ten, one-hundred, one thousand, and so forth makes even rudimentary calculations very difficult to perform.

(Although the Babylonians’ sexagesimal system may seem a tad unwieldy to us, one cannot help but feel that it was an improvement on the Sumerians who came before them. The Sumerians had three distinct counting systems to keep track of land, produce, and animals, and they used a completely different set of symbols for each system!)

The Concept of Zero and Negative Numbers
Interestingly enough, the idea of numbers like one, two, and three developed a long time before the concept of zero. This was largely because the requirement for a number “zero” was less than obvious in the context of the calculations that early men and women were trying to perform. For example, suppose that a young man’s father had instructed him to stroll up to the top field to count their herd of goats and, on arriving, the lad discovered the gate wide open and no goats to be seen. First, his task on the counting front had effectively been done for him. Second, on returning to his aged parent, he probably wouldn’t feel the need to say: “Oh revered one, I regret to inform you that the result of my calculations lead me to believe that we are the proud possessors of zero goats.” Instead, he would be far more inclined to proclaim something along the lines of: “Father, some drongo left the gate open and all of our goats have wandered off.”

In the case of the original Babylonian system, a zero was simply represented by a space. Imagine if, in our decimal system, instead of writing "104" (one-hundred-and-four) we were to write "1 4" (one-space-four). It’s easy to see how this can lead to a certain amount of confusion, especially when there are multiple zeros next to each other and one is writing on tablets of damp clay in the middle of a thunderstorm.

After more than 1,500 years of potentially inaccurate calculations, the Babylonians finally began to use a special sign for zero. Many historians believe that this sign, which first appeared around 300 BC, was one of the most significant inventions in the history of mathematics. However, the Babylonians only used their symbol as a placeholder and they didn't have the concept of zero as an actual value.

In fact, the use of zero as an actual value, along with the concept of negative numbers, first appeared in India around 600 AD. Although negative numbers appear reasonably obvious to us today, they were not well understood until modern times. As recently as the eighteenth century, the great Swiss mathematician Leonhard Euler (pronounced “Oiler” in America) believed that negative numbers were greater than infinity, and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless!

Aztecs, Eskimos, and Indian Merchants
Other cultures, such as the Aztecs, developed vigesimal (base-20) systems because they counted using both fingers and toes. The Ainu of Japan and the Eskimos of Greenland are two of the peoples who make use of vigesimal systems to the present day.

Another system that is relatively easy to understand is quinary (base-5), which uses five digits: 0, 1, 2, 3, and 4. This system is particularly interesting, in that a quinary finger-counting scheme is still in use today by Indian merchants near Bombay. This allows them to perform calculations on one hand while serving their customers with the other.

Jobs Abound for Time-Travelers
To this day, we bear the legacies of almost every number system our ancestors experimented with. From the duo-decimal systems we have twenty-four hours in a day, twelve inches in a foot, and special words such as dozen (meaning 12) and gross (meaning 12 x 12 = 144). Similarly, the Chinese have twelve hours in a day (each equal to two of our hours) and twenty-four seasons in a year (each approximately equal to two of our weeks). From the Babylonians’ sexagesimal system we have sixty seconds in a minute, sixty minutes in an hour, and 360 degrees in a circle, where 360 degrees is derived from the product of the Babylonian's main base (sixty) and their sub-base (six); that is, 60 x 6 = 360. And from the vigesimal systems we have special words like score (meaning 20), as in Lincoln's famous Getttysburg Address, in which he proclaimed: "Four score and seven years ago...”

Because we’re extremely familiar with using numbers, we tend to forget the tremendous amounts of mental effort that have been expended to raise us to our present level of understanding. In the days of yore when few people knew how to count, anyone who was capable of performing relatively rudimentary mathematical operations could easily achieve a position of power. For example, if you could predict an eclipse (especially one that actually came to pass) you were obviously someone to be reckoned with. Similarly, if you were a warrior chieftain, it would be advantageous to know how many fighting men and women you had at your command, and the person who could provide you with this information would obviously rank highly on your summer-solstice card list. (You wouldn’t have a Christmas card list, because the concept of Christmas cards wasn’t invented until 1843.) So should you ever be presented with the opportunity to travel back through time, you can bask in the glow of the knowledge that there are numerous job opportunities awaiting your arrival.

Note: The material presented here was abstracted and condensed from The History of Calculators, Computers, and Other Stuff document provided on the CD-ROM accompanying our book How Computers Do Math (ISBN: 0471732788).