Babylonian Mathematics
Created | Updated Jan 28, 2002
Mathematics is a pure thing. It is not tainted by the subjective beliefs and values of man. It is the language of the Universe. Numbers are the interpretation of reality. They hold a concept fundamental to science and human progress. Succeeding civilizations carried on the legacy of Babylonian math, and brought new ideas to this elemental science.
History of the Numbering System
Babylonia was located in what is now Iraq. The Sumerians were the original inhabitants, happily creating things such as writing, the wheel and a postal service. The less technological Akkadians invaded Sumeria, and after a time the Sumerians revolted and were subsequently invaded by the Babylonians around 2000 BC. They set up a society much like the Sumerians, and used the number system of their predecessor: a base 60, which had 59 different characters as opposed to our ten. This system is called the sexagesimal system. Their numbering scheme closely simulates ours for the first 59 numerals. There are two basic numerals, one and 10. A table of the numerals is shown below in Figure 1. The base 60 is still used for modern time-keeping. Ever wonder about the 60 minutes in an hour or 60 seconds in a minute? Thank Nebuchadnezzer and his mathematically inclined countrymen.
Cuneiform
Cuneiform originally had no place-holder. 402 could only be distinguished from 42 by the context. The concept of a place-holder had not been developed. It appeared on tablets that date to 400 BC as two wedges. Other tablets show three hooks. A place-holder was never used at the end of a number, only between digits. Babylonia used Sumerian writing, called cuneiform. It is a pictographic alphabet using wedge-shaped characters. A wedge-shaped stick was pressed into wet clay, and has provided archeologists with a durable record of the ancient writings.
Familiar Mathematical Concepts
The Babylonians also developed important mathematical concepts such as the quadratic formula, the Pythagorean theorem and the natural base e. Their mathematics was algebra based, unlike the Greeks, who focused on geometry. Their algebra arose from geometry, however, and the words they used to represent variables were the words for length, breadth, diagonal and volume. The Pythagorean theorem. Here is an example of the theorem taken from a clay tablet:
4 is the length and 5 the diagonal. What is the breadth ?
Its size is not known.
4 times 4 is 16.
5 times 5 is 25.
You take 16 from 25 and there remains 9.
What times what shall I take in order to get 9 ?
3 times 3 is 9.
3 is the breadth.
This tablet is dated to well before Pythagoras's granddaddy was a twinkle in his ancestor's eye. The necessity of such a formula is evident, so it is unsurprising that it was developed. The quadratic formula was also cultivated by these agricultural buffs. The fractions that often result from this formula were not a concern; the sexagesimal system allowed for many fractions to be whole numbers.
The Numbering System
The numbering system is very fascinating. To express numbers in sexagesimal notation, it is common to put a comma between the numbers. 1, 1; for example, is two. This is where the confusion can arise. 1, 1 can also indicate 61. The single wedge is both one and 60. Originally, the wedge indicating 60 was slightly different from the one indicating one. Eventually this distinction dissolved and Babylonians relied solely on the context to determine which number is being expressed. This is much like how one would say that a movie ticket is four fifty, and how an airline ticket is four fifty. The distinction is clear. A space used as a place holder came into use, as mentioned above, an actual character was used as a place holder. No character ever meant zero, however. What use was zero goats? The positional system was the difference between the Sumerians and the Babylonians. This was a great accomplishment.
Babylonian Influence
The influence of the Babylonians is unquestionable. The concepts and structured mathematics are still in use today. Whenever a clock is read, the legacy of Babylon beats on. But the influence is not limited to time-keeping. The Babylonians had a rich algebra, and their use of tables is still carried on today. The lessons for me provided by learning about Babylonian math is rather fundamental. It is like learning a second language. You find out about all these rules you have never heard of, and realize they apply to your native tongue. I now have brand new insights into the decimal system we use. For instance, our numbering scheme implies the following:
19843 implies 1 x 10^4 + 9 x 10^3 + 8 x 10^2 + 4 x 10 + 3
Who would have ever seen 19843 and even thought anything like that? Not me, that's for sure. That is a pretty big glimpse into how numbers work. It feels like everything changes now, that I understand a few things more than before. The value of studying is actually demonstrated also. That is a big leap for me also. I personally find ancient civilizations very interesting. I was excited to learn about something deeply historical. Modern history is engaging, but stories and legends from over 2000 years ago still apply today. That is an amazing statement about humanity. It seems to imply that everyone is essentially the same in some fashion, regardless of our technology or clothing or language.
