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Babylonian Counting Systems

Sometime around 3000 BCE, the Babylonians developed a counting system based on two different bases, base 10 and base 60 (a sexagesimal system). There has been many opinions offered as to why they used a combined system, but there exists much reason to conclude their system exists in part to today's mathematics. 

The base 10 part of the Babylonian system had two symbols. The chart below explains the two symbols, one for a single counter and the other a 10 count. Each of these symbols could be grouped together to reach a count of up to 60, their second base system.

Above 60, the numbers represented powers of the base. In other words, if a scribe wished to represent numbers above 60, they would leave a space between the two groups of shapes. From the right, the columns were 60 to the zero power, 60 to the first, 60 squared, 60 cubed etc. An example of this is enmeshed in this paragraph. The calculation is 1*60≥+57*60≤+46*60+40 = 424,000. Common notation was to separate the columns by commas, as shown on the graphic. 

One problem came up with their system. There was no representation of zero in their culture. This was sometimes solved by leaving an internal space if it was missing a power. However, when it was on the end of a number, an appropriate word such as sixty or thirty-six hundred would be written to represent the missing power. Later their civilizations developed a symbol to represent an empty place. 

Later Babylonians used a system of sexagesimal fractions as well. Current systems we write show 0.125 equaling 1/10 + 2/100 + 5/1000 = 1/8. The Babylonian base 60 system fraction 0;7,30 represented 7/60 + 30/3600 which again written in our notation is 1/8.

It is widely felt this base 60 number system started with the ideas of the Sumerians and from the Akkadians, whose civilizations the Babylonians replaced. 

Most of the time this system was used for calculating. They used a different system felt to based on the Mesopotamian culture. It had a slightly different grouping procedure and symbols for the following numbers: 1,10,60,100,600,1000,3600. That system ultimately was modified to include pictographs and less numbers representing the quantity.

Earlier this section mentioned the origins of the base 60 system used by the Babylonians. A short mention of different theories may be of interest. The easy opinion is that they inherited the base of 60 from the Sumerians, whose civilization they replaced. Then we arrive at the question of why the Sumerians used base 60. It is felt that that the sexagesimal system originated with the Sumerians. Other historical mathematicians have asked such questions. Theon of Alexandria tried to answer this question in the fourth century AD and many historians of mathematics have offered an opinion since then without any coming up with a really convincing answer.

Theonís opinion was that 60 was the smallest number divisible by 1, 2, 3, 4, and 5, making it easily divisible. Obviously this is true but it may be too scholarly a reason. A base of 12 would seem a more likely candidate if this were the reason, yet no major civilization seems to have come up with that base. Some measures do involve 12, for example it occurs frequently in weights, money and length subdivisions. For example in old British measures there were twelve inches in a foot, twelve pennies in a shilling etc.

The Babylonian sexagesimal system is still in use today. It can be seen in time and angle measurements, units used in geometry and astronomical contexts that we see in our daily lives.

 

Additional information on this system may be found at:

Babylonian_numerals.html