Halāyudha’s commentary includes a presentation of the Pascal’s triangle(called meruprastāra). Pingala’s work also contains the Fibonacci numbers, called mātrāmeru.
By Venkata Ramanan
Recently there was a News item that a Scientist has stated that Mythology is to be differntiated from Science and the attempt of the Government to include ‘Pseudo Science’ into Indian Education System.
He was speaking on ‘IIsc debunked Vimanas Theory.
He also observed that ‘the people who say that Hinduism/Vedas have said this before, why do they not say this before the facts are discovered by Science?What they say as facts from the Vedas can not be verified by experiment now”(the quote is not verbatim, i shall get it shortly).
I shall be posting a rebuttal to this shortly.
Be that as it may, let me reproduce something from the Vedic Period on Binomial System and Binary system, that is used for Modern Computing.
Ancient Indians used Mathematics extensively and relied on it so heavily that Indian Logic, Philosophy,Hindu Rituals and the Sanskrit Language have strong Mathematical base.
Meters, called Chandas are used in Prayers, literary works have a strict Mathematical base.
Pingala, younger brother of Panini, the Sanskrit grammarian, has devised Chanda Shastra that deals with these Meters.
He is dated to 2 BC, may be earlier.
Another Legend has it that he is the younger brother of Patanjali, who wrote the Yoga Sutra.
This assigns Pinagala to 4 BC.
Each number in the triangle is the sum of the two directly above it. (Photo credit: Wikipedia)
”
The Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables.The discussion of the combinatorics of meter corresponds to the binomial theorem. Halāyudha’s commentary includes a presentation of the Pascal’s triangle(called meruprastāra). Pingala’s work also contains the Fibonacci numbers, called mātrāmeru.
Use of zero is sometimes mistakenly ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, while Pingala used short and long syllables. As Pingala’s system ranks binary patterns starting at one (four short syllables—binary “0000”—is the first pattern), the nth pattern corresponds to the binary representation of n1, written backwards. Positional use of zero dates from later centuries and would have been known to Halāyudha but not to Pingala.
Formation of Binomial Triangle.Pingala Triangle.
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The Importance given to 2 by Pingala: Pingala in his rules to Sanskrit prosody has given undue importance to the number 2. Typically, he lays down that, Any power of two throughout divisible by two is equal to two raised to the power of two representing the number of twos the first power is divisible by two�, i.e, 2^{16} = _{2}2^{4}, 2^{32} = _{2}2^{5}, 2^{64} = _{2}2^{6 }and so on (VIII.407).
In grouping heavies and lights, Pingala adopts a unique method.
If we take Heavy = H and Light = L, for two syllables, we get the combination, as follows:
 1H
 1L
There are two combinations.
For 3 syllables, we get,
 3 H
 2H, 1L
 1H, 2L.
 3L.
There are four combinations.
For 4 syllables, we get,
 4H
 3H, 1L
 2H, 2L
 1H, 3L
 4L.
There are eight combinations.
For 5 syllables, we get,
 5H
 4H, 1L
 3H, 2L.
 2H, 3L
 1H, 4L
 5L
There are sixteen combinations.
Thus, this is the formation of Binomial Numbers, Triangle and Series. They are explained as follows:
(a + b)^{o} = 1
(a + b)^{1} = a + b
(a + b)^{2} = a^{2 }+ 2ab + b^{2}
(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}
(a + b)^{4} = a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + b^{4}
(a + b)^{5} = a^{5} + 5a^{4}b + 10a^{3}b^{2} + 10a^{2}b^{3} + 5ab^{4 }+ b^{5}
(a + b)^{6 }= a^{6} + 6a^{5}b + 15a^{4}b^{2} + 20a^{3}b^{3} + 15a^{2}b^{4 }+ 6ab^{5} + b^{6}
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1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 10 5 5 10 5 1
1 6 15 20 15 6 1

(a + b)^{n }= a^{n} + [n!/1!(n1)!] a^{(n1) }b + [n(n1)/2!(n2)!] a^{n(n1) }b^{2} + [n(n1)(n2)/3!(n3)!] a^{n(n1)(n2) }b^{3} + [n(n1)(n2)(n3)/4!(n4)!] a^{n(n1)(n2)(n3) }b^{4} +��+ b^{n} 
This has been explained in the context of prosody and similar exposition has been made in Vedic literature about the chanting of mantras with time scale. However, the mathematical significance has to be noted here. This Binomial triangle can rightly be called Pingala Triangle and the series Pingala series. Indian mathematicians have identified the series and arranged the numbers in the form of a pyramid, which they called asMeruprasthana and depicted as follows:
…
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 11 55 165 330 462 462 330 165 55 11 1
1 12 66 220 495 792 924 792 495 220 66 12 1

2^{0}
2^{1}
2^{2}
2^{3}
2^{4}
2^{5}
2^{6}
2^{7}
2^{8}
2^{9}
2^{10}
2^{11}
2^{12}

1
2
4
8
16
32
64
128
256
512
1024
2048
4096

The basis of writing numbers can be easily explained:
1. Write one in the first square. 
1 

2. Draw two squares below, write 1 , 1 
1 
1 

3. Draw three squares, write 1, 1 in the first and last squares. 
1 
2 
1 









Add the adjacent numbers of the above row and write intermediate numbers i.e, 1+1=2.
4. 1, 1+2=3, 3+1=3, 1 
1 
3 
3 
1 

5. 1, 1+3=4, 3+3=6, 3+1+4, 1 
1 
4 
6 
4 
1 











Like, this, the squares can be continued with added numbers. The following Pingala Triangle is formed for 12 layers and it is mentioned as Meru Prasthana in the literature.

1



1

1



1

2

1



1

3

3

1



1

4

6

4

1



1

5

10

10 
Binary system explained.
“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 Other numbers have also been assigned zero and one combinations likewise. 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. In this system, the place value increases to the right, as against the modern notation in which it increases towards the left. The procedure of Pingala system is as follows: Divide the number by 2. If divisible write 1, otherwise write 0. If first division yields 1 as remainder, add 1 and divide again by 2. If fully divisible, write 1, otherwise write 0 to the right of first 1. If first division yields 0 as remainder that is, it is fully divisible, add 1 to the remaining number and divide by 2. If divisible, write 1, otherwise write 0 to the right of first 0. This procedure is continued until 0 as final remainder is obtained. Example to understand Pingala System of Binary Numbers : Find Binary equivalent of 122 in Pingala System : Divide 122 by 2. Divisible, so write 1 and remainder is 61. 1 Divide 61 by 2. Not Divisible and remainder is 30. So write 0 right to 1. 10 Add 1 to 61 and divide by 2 = 31. Divide 31 by 2. Not Divisible and remainder is 16. So write 0 to the right. 100 Divide 16 by 2. Divisible and remainder is 8. So write 1 to right. 1001 Divide 8 by 2. Divisible and remainder is 4. So write 1 to right. 10011 Divide 4 by 2. Divisible and remainder is 2. So write 1 to right. 100111 Divide 2 by 2. Divisible. So place 1 to right. 1001111 Now we have 122 equivalent to 1001111. Verify this by place value system : 1×1 + 0×2 + 0×4 + 1×8 + 1×16 + 1×32 + 1×64 = 64+32+16+8+1 = 121 By adding 1(which we added while dividing 61) to 121 = 122, which is our desired number. In Pingala system, 122 can be written as 1001111. Though this system is not exact equivalent of today’s binary system used, it is very much similar with its place value system having 20, 20, 21, 22, 22, 23, 24, 25, 26 etc used to multiple binary numbers sequence and obtain equivalent decimal number. Reference : Chandaḥśāstra (8.2425) describes above method of obtaining binary equivalent of any decimal number in detail. These were used 1600 years before westerners/arabs copied binary system from India through trade and invasion. We now use zero and one (0 and 1) in representing binary numbers, but it is not known if the concept of zero was known to Pingala— as a number without value and as a positional location.Pingala’s work also contains the Fibonacci number, called mātrāmeru, and now known as the Gopala–Hemachandra number. Pingala also knew the special case of the binomial theorem for the index 2, i.e. for (a + b) 2, as did his Greek contemporary Euclid..
This article is based on the research work of Dr.K.V.Ramakrishna Rao and material from the site’s Link provided second at the end of the Post.
http://www.allempires.com/forum/forum_posts.asp?TID=17915
http://dwarak82.blogspot.in/2015/01/fatherofbinarysystempingalagenius.html
https://ramanan50.wordpress.com/tag/computerlanguage/ 