Babylonian Table of Squares

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Babylonian Numbers 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.

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.

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.
soss
ner
soss
sar
soss

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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The Numbers of Babylonian Mathematics Cuneiform Table of Squares. http://www.gutenberg.org/files/16161/16161-h/16161-h.htm - The Seven Great Monarchies, G. Rawlinson

Since we grew up with a different system, Babylonian numbers are confusing.

At least the numbers run from high on the left to low on the right, like our Arabic system, but the rest will probably seem unfamiliar. The symbol for a one is a wedge or Y-shaped form. Unfortunately, the Y also represents a 50. There are a few separate symbols (all based on the wedge and the line), but all other numbers are formed from them.

Remember the form of writing is cuneiform or wedge-shaped. Because of the tool used to draw the lines, there is a limited variety. The wedge may or may not have a tail, drawn by pulling the cuneiform-writing stylus along the clay after imprinting the part triangle form.

The 10, described as an arrowhead, looks like a bit like < stretched out.

Three rows of up to 3 small 1s (written like Ys with some shortened tails) or 10s (a 10 is written like <) appear clustered together. The top row is filled in first, then the second, and then the third. See next page.

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1 Row, 2 Rows, and 3 Rows Table of Squares. http://www.gutenberg.org/files/16161/16161-h/16161-h.htm - The Seven Great Monarchies, G. Rawlinson

There are three sets of cuneiform number clusters highlighted in the illustration above.

Right now, we're not concerned with their value, but with demonstrating how you would see (or write) anywhere from 4 to 9 of the same number grouped together. Three go in a row. If there is a fourth, fifth, or sixth, it goes below. If there is a seventh, eighth, or ninth, you need a third row.

The following pages continue with instructions on performing calculations with the Babylonian cuneiform.

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The Table of Squares Senkareh Table of Squares in Cuneiform. http://www.gutenberg.org/files/16161/16161-h/16161-h.htm - The Seven Great Monarchies, G. Rawlinson

From what you've read above about the soss -- which you'll remember is the Babylonian for 60 years, the wedge and the arrowhead -- which are descriptive names for cuneiform marks, see if you can figure out how these computations work. One side of the dash-like mark is the number and the other is the square. Try it as a group. If you can't figure it out, look at the next step.

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How to Decode the Table of Squares Arabic Conversion of the Cuneiform Table of Squares. http://www.gutenberg.org/files/16161/16161-h/16161-h.htm - The Seven Great Monarchies, G. Rawlinson

Can you figure it out now? Give it a chance.

...

There are 4 clear columns on the left side followed by a dash-like sign and 3 columns on the right. Looking at the left side, the equivalent of the 1s column is actually the 2 columns closest to the "dash" (inner columns). The other 2, outer columns are counted together as the 60s column.
• The 4-<s = 40
• The 3-Ys=3.
• 40+3=43.
• The only problem here is that there is another number after them. This means they are not units (the ones' place). The 43 is not 43-ones but 43-60s, since it's the sexagesimal (base-60) system and it's in the soss column as the lower table indicates.
• Multiply 43 by 60 to get 2580.
• Add the next number (2-<s and 1-Y-wedge = 21).
• You now have 2601.
• That's the square of 51.

The next row has 45 in the soss column, so you multiply 45 by 60 (or 2700), and then add the 4 from the units column, so you have 2704. The square root of 2704 is 52.

Can you figure out why the last number = 3600 (60 squared)? Hint: Why isn't it 3000?