### Origins of Tamils?[Where are Tamil people from?/PART :28

Compiled by: Kandiah Thillaivinayagalingam

When we write ‘three hundred’, we use two symbols: a ‘three’ and a couple of ‘zeros’. But what if, like the Sumerians/Babylonians, we had no symbol for ‘zero’?  When they wrote ‘sixty’ it was  , which is exactly the same as ‘one’! Their symbol    can mean ‘ten’  or ten x 60  or ten x 60 x 60  or . . . So why wasn’t this a big problem for them? It seems that the context helped them.By ‘context’ I mean the sentences in which they found the numbers.So we give an example below to explain this.

"A poor fisherman had only small boat and with luck he would catch << fish in a day. In a month he’d catch about    fish. During this time he needed  fish to feed his family. The rest he sold."Now we give below possible explanation:

Boats:  can mean 1, or 1 x 60, or 1 x 60 x 60 or..... But he was poor, so he couldn’t have had 60 boats.He must have had only one boat.Fish caught in a day:<< can mean 20, or 20 x 60, or 20 x 60 x 60 or .... He had only 1 small boat; he couldn’t have stored 1200 fish on it.So the fisherman caught 20 fish per day.
Fish caught in a month: A month has about 30 days, so he caught about 600 fish.    can mean 9 x 60 + 50 = 590 which is about right,  or  9 x 60 x 60 + 50 x 60 =  which is way too much. He caught about 590 fish a month.

Fish to feed his family: can mean 5 (way too few),  or 5 x 60 = 300 (looks right),  or 5 x 60 x 60 = 18,000 (way too large).

Fish sols in a month:The fisherman had 1 boat. He caught about 20 fish a day, 590 per month. He used 300 to feed his family and sold the rest.So in a month he sold 590-300=290 fish, which is 9 or 10 per day.

We use positional notation to represent not only very big numbers but also very small ones: that is, we have the decimal representation of fractions. For example, we can write 1/4 as 0.25, with 2 in the tenths place and 5 in the hundredths place. The Mesopotamian people took the analogous step: they introduced a 1/60 place, a 1/3600 place and so on.Moreover, due to 60's many factors, Mesopotamian people sexagesimal representations were typically much simpler than our decimals. For example, since 1/4 = 15/60 this fraction could be written by the Mesopotamian people with just 15 in the 1/60 place. An even more telling example is 1/3, which didn't bother the  Mesopotamian people at all, while we have to resort to an infinite decimal.

The above table of reciprocals illustrates the benefits of 60 as a base, and also that 60 was not a cure-all. The reciprocal 1/7 could only be expressed as an infinite repeating sexagesimal, just as we would write it as an infinite repeating decimal. Indeed, tables of reciprocals often left out such problematic examples, though they did have methods for obtaining approximations to these fractions.Hence  the table had gaps in it since 1/7, 1/11, 1/13, etc. are not finite base 60 fractions.

Sumerian treated division as "multiplication by the reciprocal." Instead of computing 19 divided by 12, they would compute 19 times the reciprocal of 12.That is, in order to divide they made use of fact that a/b = a  (1/b).To this end,they also constructed tables of inverse,like the one given above.They also had tables of squares,cubes,square roots ,cube roots and tablets for geometrical exercises,division problems from around 2600 BC onwards, However Their geometry was sometimes incorrect..Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC, give squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives 82= 1,4 which stands for = (1 x 60) + 4 = 64 & and so on up to 59² = 58, 1 = ( 58 x 60) +1 = 3481.Also These mathematicians used the formula, ab =  [(a + b)2 - a2 - b2]/2  to make multiplication easier as they already knows squares of the numbers. Later Babylonian tablets dating from about 1800 to 1600 BC cover topics as varied as fractions, algebra, methods for solving linear, quadratic and even some cubic equations with the help of tables,studied circular measurement.,One Babylonian tablet[YBC 7289] gives an approximation to √2 accurate to an astonishing five decimal places.  Yet another gives an estimate for π of 3 1⁄8 [3.125], a reasonable approximation of the real value of 3.1416.

In the history of mathamatics tamils had great achievements. The below verses/poems indicates that they knew how to calculate area of circles even before pythagorouss  & others. This was stated in a text called Kakkai Paadiniam  of pre christian era by female poet Kakkai Paadiniar.In this text she explained the area of a circle in a poem

"விட்டமோர் எழு செய்து திகைவர நான்கு சேர்த்து சட்டென இரட்டிச் செய்யின் திகைப்பன சுற்றுத் தானே"- Kakkai Paadiniam.This poem is said to be written by Kakkai Paadiniar as per professor Kodumudi S Shanmugam.

Here  it says:Circumference of circle = 2*(7+4)*D/7.

Here  π = 22 / 7 (This is an approximate value for π

Also Kanakkathikaram Poem 50 explain how to calculate circumference of a circle.

“ விட்ட மதனை விரைவா யிரட்டித்து
மட்டு நான்மா வதினில் மாறியே – எட்டதனில்
ஏற்றியே செப்பியடி லேறும் வட்டத்தளவும்
தோற்றுமெப் பூங்கொடி நீ சொல் “

Explanation

விட்டம்தனை விரைவா யிரட்டித்து =Twice of diameter
= 2r + 2r = 4r (Diameter= 2r )
மட்டு நான்மா வதினில் மாறியே       =by multiplying by 4
எட்டதனில் ஏற்றியே                           =by multiplying by 8
செப்பியடி = divide by 20

Circumference of circle                             = ( 4r x 4 x 8 )  / 20
=  32 / 5 r
=  2 ( 16/5) r
= 2 π r

Here  π = 16 / 5 = 3.2 (This is an approximate value for π
PART :29 WILL FOLLOW