The following report will be based on the ancient Babylonian Numeration System. It has a great deal of mathematics involved and continually relates to modern everyday life experiences. The history behind this number system goes way back to the ancient times when it first originated in 1800 BC. (Babylonian Numbers, n.d)
Babylonian mathematics developed from Mesopotamia, the country between the Euphrates and Tigris, known as Iraq today. In the late nineteenth century archaeologists began digging in the ancient city of Mesopotamia, and thousands of clay tablets were discovered in evidence of ancient civilizations. There was some recognition of the relation to numbers and not until some thirty years ago, there was a greater understanding and appreciation of Babylonian mathematics. Now there are approximately 400 tablets and pieces of tablets resting in museums and collections in many countries. These have been copied, translated and comprehensively explained. An unbroken tablet is the size as big as the hand and usually made of unbaked clay. The writing on the tablet is called ‘cuneiform’; the signs are made of wedge shapes. Most of the tablets discovered date from about a couple of centuries at around 1700 B.C. The age of the tablet would have been established through the style of writing it had and from the mound area it was found (Aaboe, 1964).
The Babylonian Number system was based on a sexagesimal or base 60 numeric system. Much like the modern decimal system, Babylonians used a true place value system, where the digits on the left hand side held a greater value, but using base 60 not base 10. Essentially to represent the numbers 1-59, they used two distinct symbols a vertical wedge which represented a unit and a horizontal wedge which represented tens. These were combined and added to create bigger numbers, although the number 60 was represented by the same symbol as the number 1 (The Story of Mathematics, n.d.). The Babylonians were aiming for a positional system, where it made it much easier to read a whole number without being confused. Like the Egyptians, the Babylonians used an easy system for example, two ones represented two, and three ones represented three and so on. When they reached the number ten a new symbol was used, for example the symbol for ten and the symbol for one together represented eleven. The symbol for sixty turned out to be the same as the number one and this became quite confusing for example sixty is sixty and one which also looks like one and one, thus they decided to create this positional system (Edkins, 2006). Below is the table representing the numerical symbols used by the Babylonians and the corresponding modern number equivalents.
Babylonian Number Symbol (Easycalculation.com, n.d., Web)
Edkins (2006) states that the positional number system is created by columns; the numbers are specifically arranged. In today’s mathematics, we use a positional system with a base of 10 for instance; the furthest left hand column is the hundreds, then the tens following with the units. The Babylonians as mentioned used the same system but to the power of the 60s, this would be the opposite; so the left hand column was units, then multiples of 60, then 3600 etc. Edkins (2006) notes that “The representation of two has the two ones touching, while the representation of sixty one has a gap between them” (n.d.). This explains the confusion some may encounter whilst dealing with their number system. The development of the character for zero was also developed; it was much seen as a placeholder than a number in its own right. The lack of the zero was a drawback as one could not tell if it were 3600, 60 or 1, as there was not a decimal or symbol to represent this. After that development, the zero was only used in the middle of numbers, thus there was a great advantage to this positioning system as “You need only a limited number of symbols (the Babylonians only had two, plus their symbol for zero) and you can represent any whole number, however big” (Edkins, 2003).
Whibly and Scott (2003) note that the main idea of the Babylonian number system is that the first 59 digits were written with the combination of two digits. When we reached 60, the symbols were repeated to represent the number of sixties used; such as a single wedge denoted 60 and two single wedges 120, which is easy to comprehend. As previously stated sexagesimals also known as base 60 which are used today through semicolons and commas. Some of the problems associated with this number system are the lack of a sexagesmial point for example two wedges can represent 2 instead of 120 or the absence of the zero as beforehand acknowledged.
For the Babylonians, addition and subtraction were very much as it is for us today except that instead of the concept of carrying 10s the Babylonians carried 60s” (Whibly and Scott, 2003, p. 3). Their use of multiplication relied on the distributive law and the division was achieved by multiplying by reciprocals which were tables constructed by the Babylonians. These aided for calculations for regular sexagesimals numbers. The tablets discovered revealing that the Babylonians had a high level of computational mathematics ability for example, solving systems of linear equations. They also constructed tables of squares, for the integers 1 to 30. They used algebraic solutions through geometrical terms such as length and area and had an idea of the Pythagorean Theorem. The pie symbol was estimated at 3 by the Babylonians and thus estimated the circumference of a circle as 3 times the diameter (Whibly and Scott, 2003).
Shuttleworth (2010) notes that this mathematics was developed from the early days of the Sumerians to the fall of Babylon in 539 BC; their contribution was the development of the cuneiform script. The Sumerians also use a base 60 system, which is the reason why we still divide a circle into 360 degrees, count hours, minutes and seconds. This sexagesimal system was used for weights and measures, astronomy, and for the development of mathematical functions (Shuttleworth, 2010). This system also allowed the Babylonians to use fractions and quarters, it found its way into Greece and centuries later, is still used today; where the decimal system was developed. Due to their agricultural base, the Babylonians based the sexagesimal system on astronomy, and the need to develop accurate calenders, to mark the turning of seasons and predict the best time for planting which was extremely important in their culture. The Babylonians believed that there were 360 days in a year, and this formed the basis of their numerical system; they divided this into degrees and this represented the daily movement of the sun around the sky. They then transferred this into measuring circles by dividing degrees in minutes. Our entire system of astronomy, geometry, and dividing the day into hours, minutes and seconds hails from this period of history (Shuttleworth, 2010).
Moreover, the Babylonians also constructed tables which were very similar to the multiplication tables we use today. If it had not been for the Babylonians, our modern understanding of mathematics and its concept would have not been anywhere near advanced or coherent. They were behind the development of “The calendar, units of measurement including length, volume, and weight, the 360 degree circle, knowledge of lunar eclipses, square roots, and exponents” (Arithmetic.com, 2011). Edkins (2006) states that sixty is seen as a very great number for a base as there are many factors in it for example factors of 10 and 60 (2, 3, 4, 5 etc.). In today’s modern world it is believed that the Babylonians have handed down their base 60 through generations and generations as explained above. It was the building blocks of today’s current mathematical systems and the telling of time (n.d.). Bolneni (2010) notes “Time can be written in the form of a sexagesimal fraction. For example, 6 hours, 15 minutes, and 26 seconds are just the sexagesimal fraction 6, 15/60, 26/3600” (Bolneni, 2010, p. 2). Bolneni states that the Babylonians made a great deal of advancements in the various fields within mathematics today and this led to other civilizations creating new number systems and expanding upon the discovered ones.
In conclusion, the high amount of Babylonian tablets discovered, approximately five hundred, we have been able to gain a much better understanding and a deeper appreciation of the mathematical concepts which evolved from the Babylonians. (Zara, 2008) This has been a vast contribution to the mathematics that exists today. Some of the areas are particularly; the positional number system base 60 (sexagesimal), the Pythagorean, calculations of equations, fractions and the square numbers. Present scholars and students in today’s society are indebted to the foundation they have laid on mathematics.
Aaboe, A. (1998). Chapter 1: Babylonian Mathematics. In Episodes from the early history of mathematics (1st ed., p. 5). Washington, DC: Mathematical Association of America.
Arithmetic.com (n.d.). Babylonian Math History. Arithmetic.com. Retrieved September 15, 2012, from http://www.arithmetic.com/math/history/babylonian.php
Bolneni, P. (2010). Mathematical Intuition. Massachusetts Academy of Math and Science, 0(1), 2. Retrieved from http://www.scientiareview.org/pdfs/113.pdf.
Easycalculation.com (n.d.). Babylonian Numerals, Ancient Numbers. Free Online Math Calculator and Converter. Retrieved from http://easycalculation.com/funny/numerals/babylonian.php
Edkins, J. (2006). Babylonian Numbers. gwydir.demon.co.uk. Retrieved September 15, 2012, from http://gwydir.demon.co.uk/jo/numbers/babylon/index.htm
Shuttleworth, M. (2010). Babylonian Mathematics and Numerals - developed in Mesopotamia. Experiment-resources.com. Retrieved September 15, 2012, from http://www.experiment-resources.com/babylonian-mathematics.html
Storyofmathematics.com (n.d.). Sumerian/Babylonian Mathematics The Story of Mathematics. storyofmathematics.com. Retrieved September 15, 2012, from http://www.storyofmathematics.com/sumerian.html
Whibly, S., & Scott, P. (2003). Babylonian Mathematics. Web.ebscohost.com. Retrieved September 15, 2012, from http://web.ebscohost.com.ezproxy1.acu.edu.au/ehost/pdfviewer/pdfviewer?vid=5&hid=21&sid=a4f0a7c6-7486-4228-9ebb-173fa527ae8b%40sessionmgr113
Zara, T. (2008). A Brief Study of Some Aspects of Babylonian Mathematics. Babylonian Mathematics 1. Retrieved from http://digitalcommons.liberty.edu/cgi/viewcontent.cgi?article=1050&context=honors&sei-redir=1#search=%22Robertson%2C%202000%20babylonians%22.