|Writing down arithmetic
Kamal Lodaya, The Institute of Mathematical Sciences, Chennai
Many of us hate algebra. We remember the day when the X's and Y's made mathematics into impenetrable jargon.
Some others did not find algebra too bad. Manipulating equations this way and that made us feel that we had successfully entered the language of science. But then we remember encountering trigonometry! Then we shook our heads and said, no, maths is not for me.
Like all our school subjects, algebra and trigonometry took centuries to develop. But so did arithmetic---even our decimal number system. Counting, adding, subtracting, all are prehistoric. But what took thousands of years was how to write them down.
Right from the beginning, representing things with symbols has been associated with mathematics. We never learn in school why that has been so important. Let us take a look at numbers to begin with.
The picture shows how numbers were written in the Babylonian, Egyptian, Roman, Greek, Chinese and Mayan civilizations.
The Egyptians wrote on papyrus. Each bar | represented one. Five bars ||||| was 5. Like us, the Egyptians used a decimal system. Ten was represented by a different sign Ç, hundred by a sort of comma , and so on.
Over time, they had managed to devise signs for 1000, 10000, 100000 and 1000000. So they could write, let us say, the population of Chennai today (which is more than the total population of Egypt in those days). They could not write the population of India today, but if they needed to, they would have invented other signs for one crore, ten crores and so on.
The Romans of Italy used a similar system, using letters from their alphabet (which is the same as the capital letters in English) like X, C, M, to represent powers of ten. The Romans also used V, L, D for 5, 50, 500, to shorten the writing of a number.
The Greeks used their entire alphabet to write numbers. Alpha (a) was 1, beta (b) was 2 (the word “alphabet” comes from these two letters), to iota (i), 10. Next, kappa (i) was 20, and on to hundreds.
The Roman and Greek systems could not deal with numbers as large as India's population.
The Chinese invented the idea of reusing the same letters. Like the Greeks, they had letters for 1,2, ..., 9, and then for 10, 100, 1000, 10000, and so on. So a number like 12709 would be written roughly like we would read it out today: “one ten thousand, two thousands, seven hundreds, and nine”. If we represent this using our (Arabic) numerals and Roman letters, we would get something like 1T2M7C9. Since the Chinese were (and still are) masters at calculating with an abacus, one can see that they would actually be using the beads on the abacus in this way.
But the Sumerian civilization of Iraq (4000-2000 BC), before all these civilizations, had already made this invention. They went a step further: they realized that the letters (T, M, C in the example above, X is not used) were not necessary, the place where the numeral came could tell what was meant.
For instance, 9 has a different value in the number 90 than in the number 900. What we use today is the decimal system: in the first case, 9 represents 9 tens (=90); in the second case, 9 represents 9 hundreds. Since 10 is used as the base in the decimal system, each place is 10 times larger than the place to its right (900 is 10 times 90). So the right-most place is ones, the one to its left is tens, the next is hundreds and so on, in powers of ten.
The Sumerians used the number 60 as a base (powers of 60) instead of 10, used by the Chinese. Thus 12709 was partitioned as 10800 + 1860 + 49, that is, 3 X 3600 + 31 X 60 + 49, and the number written (in our notation) as '3 31 49'. This is what is written in the picture with Sumerian signs. The base 60 notation of the Sumerians was very useful for fractions, and we still use it today, when we divide a degree or an hour into 60 minutes, and a minute into 60 seconds.
The same invention was also made by the Maya civilization of Mexico, which was not in contact with any of the Old World civilizations, by the 4th century AD. The Mayans used base 20, and 12709 would become 8000 + 4400 + 300 + 9, or 1 X 8000 + 11 X 400 + 15 X 20 + 9, and written as '1 11 15 9'.
One of the great advantages of such a place value system is that the arithmetic operations can be taught as rules. (If you think that is easy, try adding TMDLXVIII and DCCXXIII!) After the Sumerians, the Babylonians ruled Iraq. Babylonian children were taught addition and subtraction, much as they are taught today in primary school.
One interesting thing is that the Babylonians used tables for multiplication, division, fractions, square roots and so on. So arithmetic was very easy, just a matter of looking up tables. Here is a sample school problem, written in our notation: “I multiplied the length and the width to find the area. The answer was '3 2'. Then I added the length and width together. The answer was 27. Find the length, width and area”
The Babylonian school child would recognize this as an “area” exercise, and proceed to use the standard method taught. (Notice that '3 2' is 3 X 60 + 2, or 182.)
To solve this problem, the child would look up the square root table. The nearest squares to 182 are 13 X 13 = 169 (too small) and 14 X 14 = 196 (too large). So maybe the field was rectangular.
Looking at the multiplication table shows that 13 X 14 = 182. Length plus width equals 27, so this is the correct answer.
The tables were prepared by Babylonian mathematicians. Maybe they thought the multiplication rules were too complicated to teach.
Egyptian scribes were taught multiplication, by doubling and adding. Here is a calculation of 226 X 13:
1+4+8 = 13
But there is one difficulty that a place value system like this introduces. How would the Babylonians handle a number like 10802? This is 3 X 3600 + 2, but it cannot be written '3 2', since that would be 182! The Babylonians used to leave a blank position for the missing power of 60. We can symbolize their notation as '3 2'. Sometimes they wrote '3 × 2'.
The Mayans found a more visible letter. They used a head (either or a profile) to fill the blank position. This can be thought of as the 0 numeral.
But surely, everybody knows that it is the Indians who invented zero? The Indians invented zero the number; not the numeral. Although the Maya and the Babylonians used the 0 numeral, their counting began with 1. They never thought of a number to represent nothing. This came from the Indians.
Like the Babylonians, the Indians used a dot '×' for the zero numeral. The actual numeral sign we use today, 0, was developed by the Arabians, like the rest of the signs 1, 2, ... 9. A new “mnemonic” system was designed by the Indian astronomer and mathematician, Aryabhatta, using the Sanskrit alphabet, much like the Greeks did. He made use of the difference between vowels and consonants in the Indian alphabet. The picture shows his system, from his book Aryabhatiya, written in 499 AD, when he was 23 years old.
The numbers from 1 to 25 are represented by the letters from k (क) to m (म). The letters y (य), etc., have special values. In the table below, those who know Sanskrit or Hindi can read k, kh, g, etc., as क, ख, ग, etc.
k 1 kh 2 g 3 gh 4 ng 5
ch 6 chh 7 j 8 jh 9 nj 10
tt 11 tth 12 dd 13 ddh 14 dn 15
t 16 th 17 d 18 dh 19 n 20
p 21 f 22 b 23 bh 24 m 25
y 30 r 40 l 50 v 60 sh 70
sy 80 s 90 h 100
The vowels were used to represent even powers of 10 (read a=अ, i=इ, etc):
a i u ru lu e o ai au
1 100 10000 10 lakh ...
The vowels go up in powers of a hundred. (The long and short forms of the vowels, like a=अ and aa=आ, mean the same thing.)
Now one can use a pronounceable sequence of letters to represent numbers. For instance, the number of years in a yuga, 4320000, can be written as (2 + 30) X 10000 + 4 X 10 lakhs, that is, '(kh+y) X u + gh X ru', or khyughru. It could also be written ghrukhyu, or even ghrutthanu.
Indian mathematics followed an oral, not a written, tradition. Knowledge was memorized and handed down the generations in verses. So the ability to pronounce a number was important. Here is a verse from Aryabhatta:
makhi bhakhi fakhi dhakhi dnakhi njakhi
ngakhi hasgna skaki kisyga shghaki kighva
ghlaki kigra hakya dhaki kicha
sga shgna ngva kla pta fa chha kalãrdhajyã.
What does this mean? It says that the kalãrdhajyã, that is, the differences of the half-jyã, are
225 224 222 219 215 210
205 199 191 183 174 164
154 143 131 119 106
93 79 65 51 37 22 7.
What could this extraordinary sequence of 24 numbers mean? A bowstring is what jyã means, so a half jyã could stand for the line BC in the picture. This is a distance called the sine (also from the Latin for “bowstring”) in trigonometry.
Suppose we let the circumference of the circle in the picture be 360 degrees, or 360 X 60 = 21600 minutes. As we move the point B from D upwards along the circle, the distance BC increases. (At first it is zero.) We are moving 90 degrees, 5400 minutes, along the circumference. Divide this into 24 parts. So each time we move 225 minutes. How many minutes would the length BC be as we keep moving 225 minutes at a time?
225, 449, 671, 890, 1105, 1315,
1520, 1719, 1910, 2093, 2267, 2431,
2585, 2728, 2859, 2978, 3084,
3177, 3256, 3321, 3372, 3409, 3431, 3438.
And now, if you calculate the differences from one number to the next (assuming zero to begin with), you will get the table above! With this sort of calculation, it is no surprise to learn that Aryabhatta knew the value of the ratio of the circumference to the diameter of a circle, today famous as p, correct to four decimal places, p=3.1416. It was a thousand years before this value was bettered.
Aryabhatta was the first of a line of Indian astronomer mathematicians. He seems to have come from Maharashtra (some say, from Kerala), although he worked at Kusumapura (Bihar). One of his contemporaries was Varahamihira, who compiled Indian astrology, astronomy and mathematics into the Pancha Siddhanta (505 AD). This is the first work where the number zero appears.
They were followed by the brilliant Brahmagupta (born 598 in Sind, Pakistan), who freely used negative numbers, giving rules like “a negative number times a negative number is a positive number”.
Aryabhatta also introduced the Indian method of solving equations, kuttaka (pounding, or pulverizing), which is not easy. (The Greek mathematician Euclid had also written methods to solve such problems.)
Here is one of Aryabhatta's problems:
Find a number which, divided by 137, leaves a remainder 10; but divided by 60, leaves no remainder.
Here is how you pound out the answer:
Divide 137 by 60 to get 2, remainder 17.
Divide 60 by 17 to get 3, remainder 9.
Divide 17 by 9 to get 1, remainder 8.
Divide 9 by 8 to get 1, remainder 1.
Now you have to work your way backwards:
18 is a number which multiplying by the last remainder 1 and
subtracting 10 (from the original problem), gives a number divisible by 8.
18 X 1 + 1 = 19.
19 X 1 + 18 = 37.
37 X 3 + 19 = 130.
130 X 2 + 37 = 297.
297 X 60 = 17820.
17820 is one answer, as you can check against the problem. More answers can be found.
John McLeish, Number, Flamingo Books, Harper Collins, 1991.
D.M. Bose, S.N. Sen and B.V. Subbarayappa (eds), A concise history of science in India, Indian National Science Academy, 1989.