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Babylonian Table of Squares


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Babylonian Numbers
Senkareh Table of Squares (Plate 18)

Senkareh Table of Squares (Plate 18). Here is an example of Babylonian mathematics, written in cuneiform. With this table of squares you can see how to put Base 60 put into practice.

http://www.gutenberg.org/files/16161/16161-h/16161-h.htm - The Seven Great Monarchies, G. Rawlinson
Three Main Areas of Difference From Our Numbers

Number of Symbols Used in Babylonian Math

Imagine how much easier it would be to learn arithmetic in the early years if all you had to do was learn to write a line like I and a triangle. That's basically all the ancient people of Mesopotamia had to do, although they varied them here and there, elongating, turning, etc.

They didn't have our pens and pencils, or paper for that matter. What they wrote with was a tool one would use in sculpture, since the medium was clay. Whether this is harder or easier to learn to handle than a pencil is a toss-up, but so far they're ahead in the ease department, with only two basic symbols to learn.

Base 60

The next step throws a wrench into the simplicity department. We use a Base 10, a concept that seems obvious since we have 10 digits. We actually have 20, but let's assume we're wearing sandals with protective toe coverings to keep off the sand in the desert, hot from the same sun that would bake the clay tablets and preserve them for us to find millennia later. The Babylonians used this Base 10, but only in part. In part they used Base 60, the same number we see all around us in minutes, seconds, and degrees of a triangle or circle. They were accomplished astronomers and so the number could have come from their observations of the heavens. Base 60 also has various useful factors in it that make it easy to calculate with. Still, having to learn Base 60 is intimidating.

In "Homage to Babylonia" [The Mathematical Gazette, Vol. 76, No. 475, "The Use of the History of Mathematics in the Teaching of Mathematics" (Mar., 1992), pp. 158-178], writer-teacher Nick Mackinnon says he uses Babylonian mathematics to teach 13-year-olds about bases other than 10. The Babylonian system uses base-60, meaning that instead of being decimal, it's sexagesimal.

The score is now 1:1 in the simplicity department.

Positional Notation

Both the Babylonian number system and ours rely on position to give value. The two systems do it differently, partly because their system lacked a zero. Learning the Babylonian left to right (high to low) positional system for one's first taste of basic arithmetic is probably no more difficult than learning our 2-directional one, where we have to remember the order of the decimal numbers -- increasing from the decimal, ones, tens, hundreds, and then fanning out in the other direction on the other side, no oneths column, just tenths, hundredths, thousandths, etc.

The tie remains.

I will go into the positions of the Babylonian system on further pages, but first there are some important number words to learn.

Babylonian Years

We talk about periods of years using decimal quantities. We have a decade for 10 years, a century for 100 years (10 decades) or 10X10=10 years squared, and a millennium for 1000 years (10 centuries) or 10X100=10 years cubed. I don't know of any higher term than that, but those are not the units the Babylonians used. Nick Mackinnon refers to a tablet from Senkareh (Larsa) from Sir Henry Rawlinson (1810-1895)* for the units the Babylonians used and not just for the years involved but also the quantities implied:

  1. soss
  2. ner
  3. sar.
A soss refers to a period of 60 years. The ner is a unit of 600 years, or one soss times 10 [while the Babylonian system is described as sexagesimal, it is also partly decimal] and the sar, a unit of 3600 years -- a soss squared.

Still no tie-breaker: It's not necessarily any easier to learn squared and cubed year terms derived from Latin than it is one-syllable Babylonian ones that don't involve cubing, but multiplication by 10.

What do you think? Would it have been harder to learn the number basics as a Babylonian school child or as a modern student in an English-speaking school?

*George Rawlinson (1812-1902), Henry's brother, shows a simplified transcribed table of squares in The Seven Great Monarchies of the Ancient Eastern World. The table appears to be astronomical, based on the categories of Babylonian years.
All photos come from this online scanned version of a 19th century edition of George Rawlinson's The Seven Great Monarchies Of The Ancient Eastern World.

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