Archive for October, 2012

Evolution of Indo/Arabic Numerals

October 24, 2012

Evolution of Indo/Arabic Numerals

L V Nagarajan

I am not attempting here to give an account of history about development of Number Systems. This is only to wonder how the present Indo/Arabic numerals came in to being. Arabs, the early residents of Arabian Peninsula, are always known to be the link between Europe and ancient India especially in carrying the ancient Indian thought and culture to the elite European community. This is not to say that Arabians themselves were devoid of higher thought and culture. It is the brighter Arabian minds which appreciated the importance of Indian and Oriental contributions and took them to the world along with their own achievements in similar fields. The Arabic numerals are one such great contribution to the world in general, and scientific community in particular. Later day research on ancient India resulted in naming these numerals as Indo/Arabic (or Hindu/Arabic) numerals.

Roman numerals are the earliest system of numerals known to the world. The Romans were active in trade and commerce, and from the time of learning to write they needed a way to indicate numbers. The system they developed lasted many centuries, and still sees some specialized uses today.

seven symbols/letters were used in Roman numerals to indicate following numbers

Roman Numeral Number
I One
V Five
X Ten
L Fifty
C Hundred
D Five Hundred
M Thousand

The list below illustrates how other numbers were constructed using the above 7 symbols/letters:

1 – 10              I,  II,    III,    IV,  V, VI,  VII,    VIII,    IX,   X

10 – 100        X, XX, XXX, XL, L, LX, LXX,  LXXX, XC, C

100 – 1000   C, CC,  CCC, CD, D, DC, DCC, DCCC, CM, M

A string of letters means that their values should be added together. For example, XXX = 10 + 10 + 10 = 30, and LXI = 50 + 10 + 1 = 61. If a smaller value is placed before a larger one, we subtract instead of adding. For instance, IV = 5 – 1 = 4. This is a major difference compared to the modern system. In addition there is no separate symbol for Zero.

The biggest Roman numeral is M, for 1000, so one easy way to write large numbers is to line up the M’s: MMMMMMM would be 7000, for instance. This system gets cumbersome quickly.

The system of numeration employed throughout the greater part of the world today was probably developed in India, but because it was the Arabs who transmitted this system to the West, these numerals have come to be called Arabic. After extending Islam throughout the Middle East, the Arabs began to assimilate the cultures of the peoples they had subdued. One of the great centers of learning was Baghdad, where Arab, Greek, Persian, Jewish, and other scholars pooled their cultural heritages and where in 771CE  an Indian scholar appeared, bringing with him a treatise on astronomy using the Indian numerical system.

Until that time the Egyptian, Greek, and other cultures used their own numerals in a manner similar to that of the Romans. Thus the number 321 was expressed like this:

Egyptian – I nn 999  (Right to left, the Arabic way)

Greek – HHH ÆÆ I

Roman – CCC XX I

The Egyptians actually wrote them from right to left. (Presently with the Indo/Arabic numerals they write numbers only from left to right, may be because of its non-Arabic origin)

The Indian contribution was to substitute a single sign for each cluster of similar signs. In this manner the Indians would render Roman CCC XX I as: 3 2 1. But however CCC I should mean 301, and not 31. Hence the  scholars perceived that a sign representing “nothing” or “naught” was required and Indians are credited with filling this need by inventing the symbol ‘zero’.

If the origin of this new method was Indian, it is not at all certain that the original shapes of the Arabic numerals also were Indian. In fact, it seems quite possible that the Arab scholars used their own numerals but manipulated them in the Indian way. The Indian way had the advantage of using much smaller clusters of symbols and greatly simplifying written computations. Their adoption in Europe began in the tenth century after an Arabic mathematical treatise was translated by a scholar in Spain and spread throughout the West.

Most of the ancient communities traditionally assigned numerical values to their letters and used them as numerals. This alphabetical system is still used by many, much as Roman numerals are used in the West for outlines and in enumerating kings, emperors, and popes. This part of evolution of numerals from letters is not apparently researched and discussed enough. Recently I read an essay about the way the letters of Tamil language were widely used to represent numbers, till as late as 19th century. This might have been the system that existed, since perhaps (not sure?) early Chola period,  9th Century.  This Tamil system of numerals is remarkable not only for using the letters for numerals but also for serving as a precursor for the evolution of modern numerals.

This system of numerals were in use till 19th century in most of the Tamil documents and  records. Tamils used 12 symbols or letters to denote whole numbers 1-9, 10, 100 and 1000. These symbols or letters are given below:.

Numbers Present Digits Tamil Symbols Unicode
One        1        ௧ &#3047
Two        2        ௨ &#3048
Three        3        ௩ &#3049
Four        4        ௪ &#3050
Five        5        ௫ &#3051
Six        6        ௬ &#3052
Seven        7         ௭ &#3053
Eight        8         ௮ &#3054
Nine        9         ௯ &#3055
Ten       10         ௰ &#3056
Hundred      100         ௱ &#3057
Thousand      1000         ௲ &#3058

As these symbols themselves are not important in the present context, I am dealing with Tamil’s number system with known symbols as below

Numerals 1 to 9, X for 10,  C for 100 and M for 1000.

With the above symbols let us see how Tamils wrote other numbers

27 was written as 2X7  –  (More precisely as ௨௰௭)

327 was written as 3C2X7

5327 was written as 5M3C2X7

5307 was written as 5M3C7 – (Symbol for Zero is not used)

234 021 was written as 2C3X4M2X1

1 Million will be  1MM or MM – (Just 2000 in Roman system)

3, 234, 521 will be  3M2C3X4M5C2X1 or (3M 2C3X4)M 5C2X1

1 Billion was written as 1MMM or MMM

and so on.

First let us look at the numbers without a zero in between. If you remove the symbols X,C and M from the sequence it exactly coincides with the present system. This may exactly be the reason why a symbol for zero was invented by Indians because they needed it the most. This symbol zero (0) helped them to totally remove X, C and M from their numerals (but still using their implied presence) . This is perhaps the way the place system of numerals was evolved and gifted to rest of the world. Tamils were definitely a part of the larger Indian Science and Culture and hence this could have been the number system that existed all over ancient India. The ancient Tamil inscriptions just give the evidence of the same. The reason this system was used till 19th century could be the same why English people still call their Queen as Elizabeth II. They find the older systems as authentic in recording history.

It may also be interesting to know that Tamils had letters even for representing about 15 fractions. There are also combination of symbols to represent fractions as low as 1/1,838,400. There are separate names for each of these fractions. After the advent of decimal systems in coinage, weights and measures since 1960s, most of these fractions have gone out of use.

In a lighter vein, there is a poem attributed to Avvayar(?) which mocks at perhaps another inferior poet thus:

௭ட்டேகால் லக்ஷணமே, எமனேறும் வாகனமே

Ettekaal Lakshaname, Emanerum Vahaname

Ettekaal = 8 and 1/4. In Tamil, the symbols are அ and வ

Avalakshaname – Means ‘the ugly one’

Emanerum Vahaname – Lord Yama’s mount, the buffalo

Such pun on number/symbols are aplenty in Tamil Language literature

Ref:

1. http://www.novaroma.org/via_romana/numbers.html

2. http://mathforum.org/dr.math/faq/faq.roman.html

3. http://www.islamicity.com/education/ihame/default.asp?Destination=/education/ihame/22.asp

4. http://ramanchennai.wordpress.com/category/archaeology-history/

5. http://tamilelibrary.org/teli/numeral.html

LVN / 24 Oct 2012

 

Thirukkural – 178

October 19, 2012

Thirukkural – 178

அஃகாமை செல்வத்திற்கு யாதெனின் வெஃகாமை

வேண்டும் பிறன்கைப் பொருள்

Ahkkamai selvatthiRku yadhenin vehkkamai

VeNdum piran kai PoruL

 

Ahkkamai – conservation, retention

Selvam – Ones prosperity, wealth

Vehkkamai – not to crave for, not to grab or not to seize

Piran Kai PoruL – belongings in other’s hands.

 

What is the best way to retain one’s own prosperity? One should not yearn for, or grab, or seize the belongings in other’s hands.

You are considered rich only when there is no need for  you to earn more. Desire to earn by unfair means renders you among the poor.

 

Wealth in your hand should never decay – then

Grabbing from others should never be the way

– by L V Nagarajan