Subject: [HM] "Binary Numbers in Indian Antiquity" by B. van Nooten
From: Dinesh Maheshwari (dsm@cypress.com)
Date: Tue Jan 25 2000 - 15:20:13 EST
Greetings,
Recently, I came across a summary of the article by B. van Nooten,
"Binary Numbers in Indian Antiquity", Journal of Indian Studies,
Volume 21, 1993, pp. 31-50
Does any one have an electronic copy of the article that could be emailed?
Thanks,
Dinesh Maheshwari
Advanced Design Methods
Cypress Semiconductor
SanJose, CA, USA
PS: The summary of the article is as follows :
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. 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).
Dinesh Maheshwari
Advanced Design Methods
Cypress Semiconductor
San Jose, CA, USA
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