Analysis of Babylonian Mathematics

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As students, we are taught the basics about mathematics. What the core properties of addition, subtraction, multiplication and division mean. How they work, and if we are lucky, we go into a little history of these methods. For those of us who have learned history, we learned that the basis for modern mathematics came from the Greeks and their writings. While this is correct, to truly understand the historical aspect of mathematics and its origins, one must study a time before the Greeks, when math was a whole new language, and one we still today have not completely mastered. Perhaps the most interesting group to study is one of the first known civilizations, the Babylonians from Mesopotamia; the land between the Tigris and Euphrates Rivers in modern day Iraq. The Mesopotamian people are considered the founders of the first sophisticated, urban cities, and the founders of writing and keeping records. It was then that the idea of writing evolved as a means to record the most essentials of founding a city, mathematics. As a people who flourished from the land, it has been determined that the main uses for a mathematical language were utilitarian. It is believed that agriculture was invented in Mesopotamia, as the land between the rivers provided for much fertile ground (5). Because of this, research has found that the Babylonians made a number system to represent livestock, produce, and their basic way of life. According to Elanor Robson, they used “…small clay ‘tokens’ or counters’, made into various geometric...shapes” (2) . For them, each counter had both a qualitative and a quantitative meaning. So, there was actually not a one-to-one correspondence and this leads to the belief that the earliest mathematics by the Babylonians must not have been for counting and solving purposes, but instead for accounting and manufacturing purposes (2). From what we have discovered today, the Babylonians transcribed their work on many tablets written in what is today called Cuneiform notation (1). Because it was both a stable source and very plentiful, the Babylonians used clay to both build their buildings and transcribe their works. On these tablets, they kept record of the tables they created to solve linear equations, and used those tables to solve some of the most basic problems that they kept for a record. These tablets have become sources of pure satisfaction and amazement for researchers for generations. On these tablets, we have found that the Babylonians created their own number system, much like the one we use today, only of base 60 and of base 10. It is believed that this base 60 came to be because of how easily it worked with any numbers lower than it. For instance, 60 can be divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, and 30. Thus the Babylonians equated that number 60 to 0, much like number rings in ring theory (6). The break at 10 seems to show that the Babylonians did mathematics as a purely additive function, that is, the symbol for 40 is the symbol for 10 four times. In other words, the Babylonians used concepts like multiplication and division as faster forms of addition and subtraction, much like we do today. A table of the Babylonian base 60 is shown below. If one examines it, it is clear that their system was additive, in that the symbol for 10 is just duplicated over and over for multiples of 10. (4)

By creating a break at 60, the Mesopotamians displayed their knowledge of a place value system, and based on our research of ancient mathematics, they were actually one of only 4 cultures who understood the concept of place value in numerology (6). What is interesting is the Mesopotamians did not have a value for the number 0, and thus it seems they believed the notion of 0, like negative numbers, simply did not exist in the natural world (1). It is determined now that the notion of 0 likely came from the Indian mathematicians of the ancient worlds, who in turn likely got their notions of mathematics and number...
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