In the last lecture we saw how different societies invented different number systems. One big achievement of a number system is that it allows you to give names to numbers and invent symbols to write them down. Without a number system you would have to give every number its own name and invent thousands of words and symbols.
The decimal system, to take a familiar example, needs words only for the numbers 1 to 10, then 100, then 1000 and so on, and uses these words to describe all other numbers: "1263" in words is "one thousand two hundred and sixty-three". (In a strict sense this should be "one thousand two hundred six ten three". In the English language, as in many other languages, multiples of 10 less than 100 allude to the strict principle but have their own proper names.)
The ease of use of a number system depends very much on the size of its base. The Sumerian number system, the first known number system that existed in written form, used the base 60. Inventing and remembering names for sixty numbers is easier than inventing thousands of names, but it is still a challenge.
The problem is worse when it comes to writing. When people first started writing numbers down they made kerbs in a piece of wood or pressed lines into a lump of clay. The number 4 is then written as IIII. A new symbol, for example two angled kerbs (V), can then be used to indicate the next power of the base. If the number base is small, for example 5, we write V for 5 and represent the number 9 as VIIII. If the number base is large, for example 20, the first opportunity to use the new symbol would occur at 20, and the number 9 would have to be written as IIIIIIIII.
We know from our test in the previous lecture that any quantity larger than 4 can only be grasped through counting, so IIIIIIIII is not a very useful representation of the number 9. The solution is the introduction of an auxiliary number base.
The city of Sumer in Mesopotamia developed its number system well before its script, which it invented around 3000 BC. Its number system used the main base 60 and the auxiliary base 10. A Sumerian number is thus structured into units like this, starting with the smallest unit:
| 1 | 1 |
| 10 | 10 |
| 60 | 60 |
| 600 | 10*60 |
| 3,600 | 602 |
| 36,000 | 10*602 |
| 216,000 | 603 |
| 2,160,000 | 10*603 |
| 12,960,000 | 604 |
Sumerian writing began as a pictorial script, and numbers were originally depicted on clay tablets as images of the corresponding pebbles. The introduction of the stylus did not change that, but the original impressions now had to be simulated through narrow stylus impressions. This made the representation of some symbols more complicated but did not prompt a movement away from pictorial script, and the number symbols introduced around 3000 BC remained in use for some 1500 years.
Introducing the auxiliary base 10 is only a partial solution to the problem of comprehensible number representation. In a sexagesimal system the maximum required number of tens is five (in the numbers 50 - 59). This is just acceptable as a quantity that can be grasped without counting. But the "small change" (if we want to call it that) can still go up to 9, and the number 59 still requires five tens and nine ones.
Having nine ones can be avoided by using not 10 but 5 as an auxiliary base. But that does not solve the problem in a sexagesimal system, because we would then need up to 11 fives to represent the numbers before we reach 60. This inability of the sexagesimal system to write numbers without the need for counting is one of the reasons why it came out of fashion - Mesopotamia eventually adopted the much simpler number system of Egypt.
Like all number systems of early civilizations, the Sumerian number system developed as an absolute number system. The only such system still in use is the Roman number system. The Roman numbers are much easier to read than Sumerian numbers because they use the decimal system familiar to us, and because they are not written in pictorial form but use letters of the alphabet. The well-known Roman numerals are
| I | V | X | L | C | D | M |
| 1 | 5 | 10 | 50 | 100 | 500 | 1,000 |
Note the use of the auxiliary base 5: A Roman number never requires the repeat of more than 4 identical symbols, which makes reading Roman numbers relatively easy.
In an absolute value system the value of a numeral is independent of its place in a number. The Roman numeral C always has the value 100, whether it occurs in third and fourth place (counted from right to left) of CCXI (211) or in second, fourth and fifth place of MDCCXCI (1791). In place-value number systems, on the other hand, the value of a numeral depends on its place in a number: the numeral 1 is worth 1 at the end of the number 211 but 10 in the middle, and in 1791 it is worth 1 at the end but 1000 at the front.
The next lecture will discuss why the invention of the place-value number system is one of the greatest achievements of the human mind. For thousands of years humans did not know it and had to work with absolute number systems. An example from the American continent comes from the Aztec civilization, which flourished between 1300 and 1500 AD. Like all other South and Central American civilizations the Aztec number system used the main base 20. But unlike the Maya civilization they did not use an auxiliary base, which made their numbers rather cumbersome. Aztec written numbers therefore invariably involved counting.
The absolute value system that formed the basis of European number development was the system invented in Egypt. Egyptian numbers used the decimal system without an auxiliary base. The smaller base made Egyptian numbers somewhat easier to read than Aztec numbers. The fact that the pictographs for Egyptian numerals all relate to features of the Nile valley are a clear indication that the Egyptian number system was an independent local invention. It developed at about the same time as the Sumerian number system.
A severe problem of absolute value systems is the difficulty to represent fractions. The Egyptians were very advanced in this regard, but their system was still quite restrictive. Egyptian fractions were always fractions of the unit 1. More complicated fractions had to be represented as the sum of simple fractions: 13/40 was written as 1/5 (+) 1/8.
An intriguing feature of the Egyptian number system was the "Eye of Horus" to depict fractions developed from powers of 2: 1/2, 1/4, 1/8, 1/16, 1/32 and 1/64. The origin of this fractional number system with base 2 is religious; an Egyptian epic reports how the god Horus is killed and cut into pieces by his brother Seth. The re-assembly of the eye is one of the central events of the epic. The use of Horus' eye as a means to write fractions is another reminder that in the Egyptian civilization science was practiced by the priests and closely linked to religion.
In this lecture we established the groundwork for the first great step in the development of science, the place-value number system, which will be the topic of the next lecture. The main points of this lecture are:
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