Binary Numbers In Ancient India (fwd)
M.G.G. Pillai
From: Norman <4...@globalnet.co.uk>
Newsgroups: soc.culture.indian, soc.culture.pakistan
Subject: BINARY NUMBERS IN ANCIENT INDIA
I found the following whilst surfing the net.
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Binary Numbers in Ancient India
Binary numbers form the basis for the operation of computers. Binary
numbers were discovered in the west by German mathematician Gottfried
Leibniz in 1695. However, new evidence proves that binary numbers were
used in India prior to 2nd century A.D., more than 1500 years before
their discovery in the west.
Ancient India had a tradition of scholarly learning. This tradition
continued till the beginning of current millennium. During the
millennium long foreign rule hostile to scholarly activities, a vast
body of scientific information was lost. Thankfully some of the ancient
literature has survived. Most of the scholarly work needed to preserve
the ancient learning was done in South India which remained free from
invasion for a significant time. Scholars are now rediscovering the
forgotten contributions of ancient India in the field of mathematics and
science. One of these discoveries is that of the use of Binary numbers
for the classification of meters.
The source of this discovery is a text of music by Pingala named
"Chhandahshastra" meaning science of meters. This text falls under the
category of "Sutra" or aphorismic statements. Detailed discussions of
these short but profound statements are found in later commentaries.
"Chhandahshastra" can be conservatively dated to 2nd century A.D. The
main commentaries on "Chhandahshastra" are "Vrittaratnakara" by Kedara
in probably 8th century, "Tatparyatika" by Trivikrama in 12th century
and "Mritasanjivani" by Halayudha in 13th century. The full significance
of Pingala's work can be understood by the explanations found in these
three commentaries.
Pingala (Chhandahshastra 8.23) describes the formation of a matrix in
order to give a unique value to each meter. An example of such a matrix
is as follows:
0 0 0 0 numerical value 1
1 0 0 0 numerical value 2
0 1 0 0 numerical value 3
1 1 0 0 numerical value 4
0 0 1 0 numerical value 5
1 0 1 0 numerical value 6
0 1 1 0 numerical value 7
1 1 1 0 numerical value 8
0 0 0 1 numerical value 9
1 0 0 1 numerical value 10
0 1 0 1 numerical value 11
1 1 0 1 numerical value 12
0 0 1 1 numerical value 13
1 0 1 1 numerical value 14
0 1 1 1 numerical value 15
1 1 1 1 numerical value 16
Following comments are in order:
1. Pingala's system of binary numbers starts with number one (and not
zero). The numerical value is obtained by adding one to the sum of
place values.
2. In Pingala's system the place value increases to the right, unlike
the modern notation in which it increases towards the left. This also
proves that these two systems developed independently.
Pingala (Chhandahshastra 8.24-25) also describes how to find the binary
equivalent of a decimal number. The procedure is as follows:
1. Divide the number by two. If divisible write 1, else write 0 on
ground.
2. If first division yielded 1, divide again by two. If divisible write
1, else write 0 to the right of first 1.
3. If first division yielded 0, add one to the remaining number and
divide by two. If divisible write 1, else write 0 to the right of first
0.
4. Continue this procedure till you get zero as the remaining number.
To illustrate this procedure let us find the binary equivalent of number
108.
Step 1: Divide by two. Divisible, so write 1. Remaining number is 54.
1
Step 2: Divide number 54 by 2. Divisible, so write 1 next to first 1.
Remaining number is 27.
1 1
Step 3: Divide 27 by 2. Indivisible, so write 0 to the right. Add 1 to
27 and divide by 2. Remaining number is 14.
1 1 0
Step 4: Divide 14 by 2. Divisible, so write 1 to the right. Remaining
number is 7.
1 1 0 1
Step 5: Divide 7 by 2. Indivisible, so write 0 to the right. Add 1 to 7
and divide by two. Remaining number is 4.
1 1 0 1 0
Step 6: Divide 4 by 2. Divisible, so write 1 to the right. Remaining
number is 2.
1 1 0 1 0 1
Step 7: Divide 2 by 2. Divisible, so write 1 to the right. Remaining
number is 0. So procedure terminates.
1 1 0 1 0 1 1
Now we can check that this number does represent 108 in Pingala's
system.
Now we can check that this number does represent 108 in Pingala's
system. Taking the sum of place values we get 107 (1*1 + 1*2 + 0*4 +
1*8 + 0*16 + 1*32 + 1*64). Adding 1 to this sum, we get 108, the number
we started with.
This subject has been discussed in detail in a scholarly article (B. van
Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies,
Volume 21, 1993, pp. 31-50). This article along with several other
articles pertaining to the contribution of ancient India to the field of
computer science has been published in a book titled "Computing Science
in Ancient India". More information about the book is as follows:
Computing Science in Ancient India
Edited by T. R. N. Rao and Subhash Kak
Published by Center for Advanced Computer Studies in 1998
University of Southwestern Louisiana 70504
Library of Congress Catalog Number: 98-86952
ISBN: 0-9666512-0-0
First edition of "Computing Science in Ancient India'' has been sold
out. It
is being republished by Munshiram Manoharlal and should be available in
a
couple of months.
Read the review of "Computing Science in Ancient India" by Dr. Raja Ram
Mohan Roy. You can contact Dr. Roy by sending an e-mail to
in...@goldenpub.com.