The Babylonians lived in a region of Mesopotamia, which is now known as Iraq, as shown in the map below.
As with most ancient civilisations, a mathematical system developed as the bureaucratic need for a system to measure plots of land, tax individuals etc. arose due to the settlement of the civilisation.
There is evidence that from around 2600 BCE onwards the Babylonians produced multiplication and division tables, tables of squares, square and cube roots and worked on geometrical exercises and problems involving division. Furthermore, later tablets from 1800 to 1600 BCE show topics such as fractions, algebra and methods for solving linear, quadratic and cubic equations being tackled. The Babylonian mathematicians also produced a few approximations, including √2 which was accurate to five decimal places!
Unique Number System
The Babylonians used an advanced number system with a base of 60, rather than base 10, which is the base system in widespread use today. Counting physically in this base system was done using 12 knuckles on one hand and five fingers on the other. Unlike the other number systems used at the time by the Egyptians, Greeks and Romans, it was a true place-value system, where the digits in the left column represented larger powers of 60. The numbers 1-59 were given using two symbols that were combined in distinct ways, as shown below.
Base 60 was a wise choice of base system, as it has been conjectured that the advances the Babylonians made in mathematics were greatly facilitated by the fact that 60 has such a large number of factors; 60 is the smallest number to be divisible by all numbers from 1-6. The remnants of this number system can still be seen today. It was the Babylonians that divided the day into 24 hours, with 60 minutes in an hour and 60 seconds in a minute, a system still in use today.
Another great mathematical advance by the Babylonians was the concept of the number zero, which was represented by a circle character, something that had not been recognised by the Egyptians, Greeks or Romans. However, they are not necessarily credited with its discovery as it was used more as a placeholder, rather than being a number used in calculations.
Construction of Tables
One of the most astonishing aspects of Babylonian mathematics and their calculating skills was their construction of tables to facilitate their calculations. For example, two tablets that were found in Euphrates give squares of the numbers up to 59 and cubes of the numbers up to 32. To construct these tables Babylonians used formulae to make the calculations easier. For instance, to compute square numbers the following formula was used to make the multiplication easier:
ab = [(a + b)2 – (a – b)2]/4
On the Plimpton 322 clay tablet, which now resides in the British museum, the following is written:
“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.”
Due to this tablet, many claim that the Babylonians may have had an understanding on Pythagoras’ theorem before the Greeks. This claim is fortified by the fact that the Babylonian’s understanding of quadratics was extensive. However, there is a lot of controversy over this as due to the damage and age of the tablet, interpretations of the writings vary greatly.
Babylonians were not only fantastic at pure mathematics, but were very competent in astronomy and placed great value in its study due to the fact that they believed that if they understood what happened in the skies they would know what would happen on Earth as they were connected. They have recently been in the news over the deciphering of a clay tablet, by Ossendrijver, that reveals an early form of integral calculus to calculate the path of Jupiter, a technique which was thought to have been invented in Medieval Europe.
Although it was previously believed that the Babylonians did not use geometry for their astronomical calculations, this tablet undeniably shows that the Babylonians had geometric understanding, allowing them to develop a geometric technique to make arithmetic calculations.
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