Saturday, February 19, 2011

Everything is Numerical trivia -Buddha, Vedic Maths & Pythagoras

I had a friend visiting me recently and every time as he sat in the car next to me as I was driving, he would keep pointing out a pattern of the number plates of the cars in front where the last two digits of the car registration number are the same as first two digits – 4949, 3434, 9191, 2323 and so on. It was amazing that he would invariably note such pattern even as we were talking or listening to music. Even though my friend is back in Pune, I keep looking for such pattern though not as vigorously (well, am in the driver’s seat!).
And I realize we are not the only ones inflicted with such weird addiction. I have been going through a very intresting book "Alex's Adventures in Numberland" by Alex Bellos and am enjoying it thoroughly for the magnificiant trivia and addiction simple life facts can be when sen differently in numbers.

source : Page 142 of Alex' book

Alex mentions about Maki Kaji who runs a Japanese magazine that specializes in number puzzles. And he is obsessed with photographing “arithmetically gratifying” car license plates and always carries a small camera and snaps every number plate for which the first pair multiplied by each other equals the second pair. For instance, 1101 can be thought of as 1*1=1 or 1202 as 1*2=2. Likewise 3412 can be 3*4=12. There are a total of 81 such possible combinations and he has so far collected more than 50. By the way, ever since I read this, I too have been looking for such a pattern and smile whenever such a pattern is visible. It could be a very useful tool for helicopter parents keen to ensure their kids pick up multiplication tables upto 9 quickly!
The idea that numbers can entertain is as old as maths itself. Mother Goose nursery rhyme of the early nineteenth century goes something like this:
As I was going to St Ives,
I met a man with 7 wives,
Every wife had 7 sacks,
Every sack had 7 cats,
Every cat had 7 kits,
Kits, cats, sacks, wives,
How many were going to St Ives?
(Answer is 7+7²+7³+7⁴ =2800)
Even by multiplying with 2, the numbers can swell quickly. Alex says that you keep a grain of wheat on the corner square of a chessboard, place 2 grains on the next and start filling up the rest of the board by doubling the grains of wheat per square. There are 64 boards and the final square would require 2ⁿ where n=63! The number will perhaps be more than the total annual grain production!!
Another example of recreational maths is Magic square (also called as lo shu in Chinese). The numbers 1-9 are arranged in such a manner that the sum of all rows, columns and diagonals add up to the same total =15. Chinese believe that it symbolizes the inner harmonies of the universe and used it for divination and worship. (table from page 216 of Alex's book)

The pattern, called as yubu, shows the movement of Taoist priests through a temple and also underlines some of the rules of feng shui.
In ancient India, comprehending very large numbers and coining words for them was a scientific and religious preoccupation. According to Lalitavistara Sutra when Buddha is challenged to express numbers greater than a hundred crore (koti), he replies:
One hundred koti is called an ayuta, a hundred ayuta make a niyuta, a hundred niyuta make a kakara, a hundred kankara make a vivara, a hundred vivara make a kshobhya, a hundred kshobhya make a vivaha (never knew Marriage is such a huge number!!), a hundred vivaha make a utsanga, a hundred utsanga make a bahula, a hundred bahula make a nagabala, a hundred nagabala make a titilambha, a hundred titilambha make a vyavasthanaprajnapati, a hundred vyavasthanaprajnapati make a hetuhila, a hundred hetuhila make a karahu, a hundred karahu make a hetvindriya, a hundred hetvindriya make a samaptalambha, a hundred samaptalambha make a gananagati, a hundred gananagati make a niravadya, a hundred nirvadya make a mudrabala, a hundred mudrabala make a visamjnagati, a hundred visamjnagati make a sarvajna, a hundred sarvajna make a vibhutangama and a hundred vibhutangama make a tallakshana.
In other words, Buddha could describe numbers upto 10ⁿ where n=53. But Buddha didn’t stop there.He went upto the dhvajagravati system and further upto the dhavjagranishamani system and the last number in the final system was 10ⁿ (n=421).
Vedic mathematics is based on 16 aphorisms or sutras:
1. By one more than the one before
2. All from 9 and the last from 10 (whenever one subtract a number from a power of ten for e.g. 1000-456 =544)
3. Vertically and Cross-wise
4. Transpose and apply
5. If the Samuccaya is the Same it is zero
6. If One is in ratio the Other is zero
7. By addition and by subtraction
8. By the Completion or Non-completion
9. Differential Calculus
10. By the Deficiency
11. Specific and General
12. The Remainders by the Last digits
13. The Ultimate and Twice the Penultimate
14. By One less than the One before
15. The Product of the Sum
16. All the Multipliers
There’s another method to calculate outcomes of multiplications. As an illustration, 892* 997 can be calculated as
Ist step – subtract both the number from 1000, 2nd step – Subtract diagonally, 3rd step – multiply the differential of these numbers among themselves and writing these two together is the answer
892 -108
997 -03
889 324
Answer thus is 889,324. However, I suppose the calculation gets difficult when the differentials are in double or triple digits.

Shankaracharya of Puri calls mathematics as the fountainhead of Indian philosophies. It was Pythagoras who coined the word “philosopher” and defines it as the finest type of man who gives himself up to discover the purpose and meaning of life itself. He seeks to uncover the secrets of nature.

Okay, back to some kid stuff again. “Excessive numbers” are the ones when sum of number’s divisors is greater than the number itself. For eg 12- divisors are 1,2,3,4 & 6 whose total is 16. “Defective numbers” are the ones when sum of number’s divisors is less than the number for e.g 10. “Perfect numbers” are the ones when the sum of divisors equals the number itself. For e.g. 6 with divisors 1,2 & 3. St Augustine observed thus that the number 6 is Perfect. The next Perfect number is 28, followed by 496, 8128, 33550336 and the next one is 8589869056.

Another interesting trivia is a concept of “friendly numbers” – pair of numbers such that each number is the sum of other’s divisors. For eg 220 and 284 are friendly numbers. Such numbers are said to be symbolic of friendship or a mathematical aphrodisiac. The other such pair is 17296 and 18416. The third pair is 9363584 and 9437056. Euler identified 62 such amicable pairs! A corollary to friendly numbers is “Social Numbers” – three or more number which form a loop in a manner that the divisor of first adds to the second number and so on. Another trivia is that 26 is the only number that lies between a square (25) and a cube (27).
Pythagoras realized that numbers are hidden in everything and proclaimed that “Everything is Number”. The pi was discovered based on the ratio of actual length of the river from source to mouth and their direct length as the crow flies. (which is also the ratio between the circumference of a circle and its diameter.

1 comment:

  1. Alex's Book is a Nice read and the chapter on
    Vedic Maths is awesome dedication to Zero and India! Nice book!