VI:Babylonian`s Wedge Numerals

VI. Babylonian’s Wedge Numerals 
      [Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri.] 

To search for counting system and numerals of another famous civilization, called “Babylonian” we are to go back to a period of earliest human occupation in the Lower Paleolithic period (best known as Old Stone Age) 2.6 million years ago and extending up to the 7^th century A.D. during which this civilisation flourished in a region, called Mesopotamia - the land of two rivers, Tigris and Euphrates roughly corresponding to the modern-day Iraq, Syria and Kuwait including regions along Turkish - Syria and Iraq-Iran border.

Widely considered to be one of the cradles of civilisation, Bronze Age Mesopotamia included Sumer, the Akkadian, Babylonian and Assyrian empires, all native to the territory of modern-day Iraq.

Of these, Sumer was the first ancient urban civilisation in the historical region of Southern Mesopotamia, modern-day Southern Iraq during the Chalcolithic (Copper Age) and Early Bronze ages and arguably the first civilisation in the world.

The indigenous Sumerians and Akkadians including Assyrians and Babylonians dominated Mesopotamia from the beginning of the written history in about 3100 BC to the fall of Babylon in 539 BC conquered by the first Persian Empire. Then it fell to Alexander the Great in 332 BC and subsequently became the part of Seleucid Empire founded by the infantry general Seleucus of Alexander the Great.

Sumer is the first of these civilisations and named that way by the Akkadians. Later the Akkadians took over Sumer and established Babylonian civilisation. Much later, as has already been mentioned, Greeks came and called the region Mesopotamia. None of these words were coined by the people of the region themselves. The modern word Iraq thus is used as the word Sumerians used for one of their important cities Uruq. 

Sumer, a region of historic Mesopotamia and present-day Iraq was the birth place of many important ingredients of civilisation, like writing, invention of the wheel, agriculture, the arch, the plow and many other innovations.

Sumerians developed the earliest known writing system – a pictographic writing system known as cuneiform script using wedge( ) shaped characteristics inscribed on clay tablets baked in the sun or in the fire. These tablets provide us much knowledge about the ancient Sumerian and Babylonian mathematics.

The Sumerian mathematics initially developed from practical needs. Possibly as early as the 6 millennium BC when their civilization settled and they developed agriculture, they needed idea of mathematics for the measurement of plots of land and levy tax on citizens. The Sumerians and Babylonians needed to describe quite large numbers, astronomical figures in our modern language, as they attempted to study and chart the night sky and to develop their lunar Calendar

In an attempt to make description of larger number easier, perhaps they were the first people to assign symbols. They moved from using separate tokens or symbols to represent, say, sheaves of wheat, jar of a liquid, like oil etc. to the more abstract use of symbol for specific numbers of anything. Starting as early as the 4^th millennium BC, they developed clay tokens to represent number. For example a small clay cone to represent one, a clay ball for ten and a large cone for sixty. During the third millennium BC these objects were replaced by cuneiform equivalents so that the numbers could be written with the same stylus that was being used for the words in the text. From some evidences it appears that a rudimentary model of the abacus was probably in use in Sumer from as early as 2700 to 2300 BC.

The unique feature of the Sumerian and Babylonian numeric system is the use of Sexagesimal or 60 as the base. Perhaps they performed counting physically by using twelve knuckles on one hand and the five fingers on the other hand.

It has been thought that wonderful advances of Mathematics of the Babylonians were facilitated by the use of the Sexagesimal base; for 60 has many divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. So, it is also remarkable that the smallest integer divisible by all integers from 1 to 6. We are still using this ancient Babylonian system in our modern-day usage of 60 seconds in a minute, 60 minutes in an hour, 360 degrees (60x6) in a circle. In a somewhat similar reason 12 has been used for a long period and still now; for, 12 has factors 1, 2, 3, 4 and 6. For example 12 months in a year, 12 inches in a feet, 2x12 hours a day, twelve zodiac signs, 12 eggs a dozen etc.

The Babylonians also had another revolutionary mathematical concept, a circle character for zero, although its symbol was really still more of place holder than a number in its own right.

Babylonians used a true place-value system in their numbers. The digits were   written towards left represented larger values, much like modern decimal system (of base 10) but using base 60.

The Babylonian number system began about 5000 years ago with tally marks just as most ancient math system did, From the idea of our present-day decimal system in which ten different numerals or symbols, namely, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, are used to represent different numbers, the most likely question arises in our mind that did the Babylonians then has to use 60 different symbols for their sexagesimal or 60 base number system? The answer is an emphatic ‘NO.’ They had to learn only two symbols, a wedge shaped symbol (    ) for the ‘unit’ and a wedge pointing to the left (    ) sign for 10. Therefore, although the Babylonian number system was a positional 60 base system, it had some vestiges of a base 10 system within it. From adjoining table No.2, it is amply clear how the Babylonians used these two symbols to write their first 59 non-zero digits [Table No.-2 :Babylonian numerals]

                                                      Table No.-2 :Babylonian numerals

So, any number less than 10 had a wedge that pointed down, say 5 will be written as                  ,   10 as      ,    20 as           ,    46 as                    ,   64 as   

when they wrote 60, they would put a single wedge mark in the second place of the numeral, like                

and for 120 ,

Now, if we adopt a notation where we separate the numerals by commas so for example 22, 11, 23, represents the Sexagesimal number.

22x60² +11x60+23 which, in decimal notation is 79883 and will be represented in Babylonian numerals as 

But there are some potential problems with this system of numerals. Since the number two is represented by two symbols representing one unit (    ), like (          ) and the Babylonian Sexagesimal number 61 or (1, 1) is represented by the one character or symbol for a unit in the first place and second identical symbol in the second place, as     (      ), so the number 2 and (1, 1) or 61 have essentially the same representation. Although this was no problem to the Babylonians. They removed this ambiguity by writing the two symbols of unit touching together (     ) to represent 2. For writing the number 61 there was a space between these two symbols of unit character. So, spacing of the symbols allowed one to tell the difference.

A much more serious problem was that, there was no symbol for zero to put into an empty position. If we look at the representation of the Sexigesimal numbers 1 and (1, 0) or 60 in decimals, we will find that these two numbers have identical representation and there was no way that spacing could help. But in reality the later Babylonian civilizations invented a symbol to indicate an empty place which may be considered as an idea of zero.

An empty place in the middle of a number likewise was also a problem. Perhaps they solved this problem by allowing more space between the numerals for an empty place, which in our decimal representation is zero.

The Babylonian neither technically have a digit for, nor a concept of, the number zero. Although they understood the idea of nothingness, it was not seen as a number. What the Babylonians had instead was a space to mark the nonexistence of a digit in a certain place value.

The main contribution of the Sumerians and Babylonians was the development of writing with their unique cuneiform script which has already been mentioned earlier. They preserved their knowledge of counting and numerals, including mathematical ideas and inventions on clay tablets. It helped to pass down these treasures through generations. The clay tablets discovered and translated by archeologists revealed information about the daily life of these ancient people.

From these tablets modern historians delved deep into the past and explored the sophisticated mathematical techniques* of these people. It paved the way of the very foundation of the explosion in mathematics of the later Greeks.

[*These tablets lead us to go back to as early as 300 BC and we find that the Sumerians had developed a complex system of metrology, the science of measurement. We have evidence that from 2600 BC onward they constructed multiplication and division tables, tables of squares, square roots, cube roots. Later Babylonian tablets dating from 1800 to 1600 BC cover topics, such as idea of fractions, algebra, methods for solving linear, quadratic and even cubic equations. In one such tablet the value of Ö2 is inscribed to an astonishing five decimal places. There are also tablets which give the list of squares of numbers up to 59, the cubes of numbers up to 32 as well as tables of compound interests. From such a tablet, the value of π, (the ratio of the circumference of a circle to its diameter is found to be 3  or 3.125 which a reasonable approximation of the modern value of 3.14159.

The Babylonians had also interest in geometry. They used geometric shapes in their buildings and designs. The leisure games based on geometry, such as backgammon, one of the oldest board game with dice, was very popular in their society. They also applied geometry for practical purposes of finding the areas of rectangles, triangles, trapezoids, as well as the volumes of simple shapes such as bricks and cylinders, although  not pyramids, as they had no idea of it.

The famous and controversial Plimpton 322 clay tablets dated around 1800 BC, suggests that Babylonians knew the secret of right-angled triangles, namely the square on the hypotenuse equals the sum of the squares on the other two sides many centuries before the famous Greek geometer Pythagoras.]

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 Reference : Internet.
                   Table No 2:Babylonian numerals.
                    Attribution: Creative Cosmos (https://commons.wikimedia.org/wiki/File:Babylonian_numerals.svg)