Mathematics

Acharya Pingala

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17 September 2020                

Read time: 3 minutes

Introduction

Are you alarmed by the number of corona cases in the country?

More alarmed at the predictions of numbers in future?

How do they predict the number of infections? How did the lockdown help?

These numbers are predicted by a Mathematical principle of Binary expansion. Before we go into details of the theory, let me tell you a story. We Indians are clever storytellers teaching so much, so varied through a story. 

There is this old Indian legend about a King who was challenged to a game of chess by a visiting Sage. The King asked, "What is the prize if you win?”

The Sage said he would simply like some grains of rice: one on the first square, two on the second, four on the third and so on, doubling on each square. The King was surprised by this humble request. Well, the Sage won the game. So, how many grains of rice should he receive?

\(18,446,744,073,709,551,615\) grains (460 billion tonnes of rice).

Much more rice than what was produced in his entire kingdom or the world for the next 2000 years!

Ancient Picture

In the legend the Sage reveals himself to be Lord Krishna and tells the King that he doesn't have to pay the debt at once, but can pay him over time, just serve rice to pilgrims every day until the debt is paid off). This story reveals the magic of numbers – Binary exponential expansion along with an essential life skill, humility. The story doesn’t quite end here. The discovery of Binary numbers was also quite by accident.


Who is Pingala?

Binary numbers

Acharya Pingala discovered the immense possibilities of Binary numbers quite by accident. He was working on the meter or Chandah of Vedas. Our Vedas were composed in meters or Chandah. Most Indian Languages have the dheerga or long swar and the laghu or short swar. This combination of long and short sounds is the basis of Sanskrit prosody or meter or chandah. In \(3\)rd BCE, a work called Chandahsastra was authored by Pingala, the Mathematician. Chandaḥśāstra means the science of meters as used in poetry/ music in which it is recited. While studying chandah, he stumbled on the concept of the Binary number system.

  • The Chandahsastra or the study of composing poetry based on long and short syllables, i.e. Two syllables is the first known description of a binary numeral system.Pingala

  • Along with this, the discussion of the combinatorics of meter corresponds to the binomial theorem as well.
  • Though Pingala knew Binary numbers, he did not know the use of zero(\(0\)) Later mathematicians represented using 0 and 1 in, but Pingala used light (laghu) and heavy (guru) rather than \(0\) and \(1\) to describe syllables.
  • The Binary system of Pingala, starting at one (four short syllables—binary "\(0000\)"—is the first pattern) going to the nth pattern. It can be represented as \(\rm{n}−1\) (with increasing positional values).
  • Acharya Pingala, by working on the algorithm of different possibilities of laghu and guru swar, unwittingly discovered the different patterns of Binary numbers; a notation similar to Morse code.

Works of Pingala

A verse in classical Sanskrit literature has a verse or a pada. The meter of a verse or pada is determined by the arrangement of the long and short swaras. The last syllable of a foot of a meter is taken to be a dheergha or Guru swara. So, let us refer to the long/ dheergha swara as Guru.

E.g., Let us look at the second stanza of Bhavani Ashtakam composed by Adi Shankaracharya

Na Taato Na Maataa Na Bandhur-Na Daataa
Na Putro Na Putrii Na Bhrtyo Na Bhartaa |
Na Jaayaa Na Vidyaa Na Vrttir-Mama-Iva
Gatis-Tvam Gatis-Tvam Tvam-Ekaa Bhavaani ||1||

  • This quarter has \(12\) letters. The arrangement of these letters are in the pattern:

    Laghu+ Guru+ Guru or LGG. The pattern is LGG+ LGG+ LGG+ LGG….

  • This meter is called Bhujangaprayatam (like a snake advancing)

  • This stanza with \(12\) letters can be arranged in many different ways. How many different ways?

  • With just Laghu/ short and Guru/ long swar, and \(12\) letters we can calculate the number of arrangements as \(2^{12} = 4096.\) This means you can arrange the letters in \(4096\) ways to create a stanza.

So, Pingala, the Mathematician began to index, analyse and repair the chandahs in the Vedas. He developed a technique of Pratyay or an algorithm, Prastaar for all possible combinations of a syllable for a quarter with “n” letters. In other words, he generated difference combinations of sequences

Let us try and recreate how Pingala, the Mathematician might have gone about it:

For 1 syllable

1

G

2

L

For 2 syllable:

1

G

G

2

L

G

3

G

L

4

L

L

For 3 syllable

1

G

G

 G

2

L

G

G

3

G

L

L

4

G

G

L

5

L

L

G

6

L

G

L

7

G

L

G

8

L

L

L

For 4 syllable

1

G

G

 G

 G

2

L

G

G

G

3

G

L

G

G

4

L

L

G

G

5

G

G

L

G

6

L

G

L

G

7

G

L

L

G

8

L

L

L

G

9

G

G

G

L

10

L

G

G

L

11

G

L

G

L

12

L

L

G

L

13

G

G

L

L

14

L

L

G

L

15

G

L

L

L

16

L

L

L

L


If we replace the values as \(\rm{G}=0\) and \(\rm{L}=1.\) We would have:
 

1

0

0

 0

 0

2

1

0

0

0

3

0

1

0

0

4

1

1

0

0

5

0

0

1

0

6

1

0

1

0

7

0

1

1

0

8

1

1

1

0

9

0

0

0

1

10

1

0

0

1

11

0

1

0

1

12

1

1

0

1

13

0

0

1

1

14

1

1

0

1

15

0

1

1

1

16

1

1

1

1


Let us compare decimal numbers with the Pingala, the Mathematician’s Binary system and Modern Binary system.
 

Decimal

 Pingala’s Binary system

Binary system

0

0000

0000

1

1000

0001

2

0100

0010

3

1100

0011

4

0010

0100

5

1010

0101

6

0110

0110

7

1110

0111

8

0001

1000

9

1001

1001

10

0101

1010

11

1101

1011

12

0011

1100

13

1011

1101

14

0111

1110

15

1111

1111

*Observe that the Pingala’s Binary system is a mirror image of the Binary system

The Chandahsastra, written by Pingala, the Mathematician, has eight chapters, and these mathematical combinations and sequences are mentioned in the 8th chapter.

References to Pingala

  • ‘Vrittaratnakara‘ by Kedara in the 8th century has references to

  • In the 12th Century AD, Trivikrama referred to Chandahsastra in, ‘Tatparyatika.‘

  • 12th century AD and ‘Mritasanjivani‘ by Halayudha carries a commentary on Commentaries. It was Halayudha who used zero in the place of laghu. By this time, the use of zero was common in India and had travelled to many other parts of the world.

There is little historical reference available on Pingala; the Mathematician indicates that he was the younger brother of Pāṇini (4th century BCE), or of Patañjali, the author of the Mahabhashya (2nd century BCE).

It took many more centuries for this knowledge to reach Europe. In the town of Pisa in Italy, Leonardo, better known as Fibonacci, learnt the use of Binary numbers from Arabs. He was particular to mention that the Arabs had brought it from India. But his successors chose to call it as Arabic numerals.


Pingala Series

While exploring the number of possibilities of various combinations of the laghu and the guru, Pingala hit upon a series:

\(0,1,1,2,5,8,13,21,34,55…………\)

This was later called Fibonacci series.

This sequence is seen abundantly in nature: branching in trees, the arrangement of leaves on a stem etc.

The Fibonacci number was initially called mātrāmeru, by Pingala, the Mathematician. Now it is also known as the Gopala–Hemachandra number. Pingala is also credited with the binomial theorem for the index 2, i.e. for \((\rm{a} + \rm{b}) 2\), like his contemporary Greek Euclid.


Summary

The ancient Vedic system in India depended on experiential learning and oral transmission. The rules of language, i.e. Sanskrit, was such that poetry became an easy medium to express and propagate learning. So, even the most complicated learnings and observations were written as verse or sutras, making it easy to memorise and propagate. So, elaborate learning went into the organising of the verse. The Sanskrit Language was much more than aesthetics.

The combination of syllables had to be in a definite pattern or meter so that it became easy to recite and repeat. That is how Pingala, the Mathematician stumbled on the Binary numbers with its unlimited possibilities. So, when we wonder with awe at the huge leaps made by digital technology, let us pause for a moment and think of Pingala who imagined and recorded the unlimited possibilities of using combinations of just two swars - laghu and guru and from a grammarian became Pingala, the Mathematician!


Frequently Asked Questions (FAQs)

When was Pingala born?

He is said to have lived around 400-200 BC, maybe earlier.

When did Pingala die?

No clear documented information exists about the birth and death of Pingala, but it is believed that he lived around 400-200 BC.

Where was the Fibonacci sequence discovered?

While Fibonacci himself did not discover Fibonacci numbers (they were named after him), he did use them in Liber Abaci. The numbers originate back to ancient India and are credited to the Indian Mathematician Pingala.

Who is credited for the Binary System used by Computers?

While the modern binary number system goes back to Gottfried Leibniz, it can be noted that Chandahsastra presents the first known description of a binary numeral system which is credited to an Indian mathematician, Pingala.

What were the names that Pingala used for the binary numerals \(0\) and \(1\)?

Pingala used light (laghu) and heavy (guru) rather than 0 and 1 to describe syllables.


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