Modern mathematics in old Sanskrit books

Module 2Large numbers

Beginning Remarks

• Handling large numbers is a sign of advanced civilization.

• Nowadays we use such large numbers as Avogadro number (having 24 digits) in Chemistry and Physics, the number of bits in a computer hard disc (having 13 or 14 digits), the number of atoms in the universe (having 85 digits), the speed of light, the distance between stars, etc.

Need for large numbers

• Large numbers are needed in such subjects as Astronomy, Cryptography, cosmology, computer science, etc.

• Therefore one expects: Thousand years ago, there was no need for handling large numbers. But the fact is that in India, Astronomy, cosmology, were advanced subjects thousands of years ago, and so very large numbers were profusely used.

Quote

• "The Hindus cultivate numerous other branches of science and literature, and have a nearly boundless literature. I however could not comprehend it with my knowledge”. –

Al Beruni, "India", p.74. • “I have composed a treatise showing how far,

possibly, the Hindus are ahead of us in this subject (arithmetic)” p.84.

About Al Beruni • one of the greatest scholars of

medieval Islamic era • Expert in history, chronology,

linguistics, physics, mathematics, astronomy, and natural sciences

• In 1017 he traveled to the Indian subcontinent and became the most important interpreter of Indian science to the Islamic world. He is given the titles the "founder of Indology"

• Full name: Abū al-Rayhān Muhammad ibn Ahmad al-Bīrūnī[

• 973-1048

• He was conversant in Khwarezmian, Persian, Arabic, Sanskrit, and also knew Greek, Hebrew and Syriac.

Al Biruni writes

• “I have studied the names of the numbers in various languages with all kinds of people whom I have been in contact and I have found that no nation goes beyond thousand. The Arabs too stop there …Those however who go beyond ,are the Hindus. They extend the names until the eighteenth order “

Ten Indian books mentioning very large numbers

• Valmiki Ramayanam• Maha Bharatam• Srivishnu puranam• Bhagavatam• Vimana shastram

• Jambudvipa Prajnapti• Lalitha Vistharam• Ganita sara sangraham• Aryabhateeyam• Siddhanta Deepika

Which Numbers? Just a sample

The book • Valmiki Ramayanam

• Maha Bharatam• Srivishnu puranam• Bhagavatam• Vimana shastram• Jambudvipa Prajnapti• Lalitha Vistharam• Ganita sara sangraham• Aryabhateeyam• Siddhanta Deepika

A number mentioned there• 1000000100001001000100000100

01000001000100000100010000000005

• 3 padmas + 10000.• 30,67,20,000.• 9,51,00,000.• 3,07,03,221• 3,16,227• 10 raised to the power 53.• 11000011000011• 57,75,336 • 16,43,524

Two Quotes to contrast

• “The time and number sense of ancient Indians was extraordinary. They had a long series of number names" – Pt.J.Nehru.

• We have it indeed on the authority of African explorers that many Hittentot tribes do not have in their vocabulary the names for numbers larger than three. … If the number is more than three, he will answer many.” -One, two, three, infinity, facts and speculations, by G.Gamow

From Valmiki• S:t:ö S:t:s:h+aN:aö

kaðeXm:ahØm:ün:ie\:N:H. • S:t:ö kaðeXs:h+aN:aö S:£ Ety:eB:D:iy:t:ð/ • S:t:ö S:£s:h+aN:aö m:haS:£ Eet: sm:àt:m:Î. • m:haS:£s:h+aN:aö S:t:ö b:ànd Eet:

sm:àt:m:Î/ • S:t:ö b:ànds:h+aN:aö m:hab:àndem:et:

sm:àt:m:Î/ • m:hab:ànds:h+aN:aö S:t:ö p:¼em:haðcy:t:ð. • S:t:ö p:¼s:h+aN:aö m:hap:¼em:et:

sm:àt:m:Î/ • m:hap:¼s:h+aN:aö S:t:ö K:v:üem:haðcy:t:ð. • S:t:ö K:v:üs:h+aN:aö m:haK:v:üem:et:

sm:àt:m:Î/ • m:haK:v:üs:h+aN:aö s:m:ØdÓm:eB:D:iy:t:ð. • S:t:ö s:m:ØdÓs:ah+m:aðG: Ety:eB:D:iy:t:ð. • S:t:m:aðG:s:h+aN:aö m:haòG: Eet:

ev:Â:Øt:H/

शतं� शतंसहस्रा�णां�� को�टि मा�हुमा�नी�षि�णां� |शतं� को�टि सहस्रा�णां�� शङ्� ख इत्यभि�धी�यतं� ||

Koti = 10 power 7Shankha = 10 power 12Mahashankha = 10 power 17Brunda = 10 power 22Mahabrunda = 10 power 27Padma = 10 power 32Mahapadma = 10 power 37Kharva = 10 power 42Mahakharva = 10 power 47Samudram = 10 power 52Ogha = 10 power 57Mahaugha = 10 power 62.

From Valmiki Contd.

• Ov:ö kaðeXs:h+ðN: S:£an:aö c: S:t:ðn: c:.

• m:haS:£s:h+ðN: t:T:a b:àndS:t<n: c:.

• m:hab:ànds:h+ðN: t:T:a p:¼S:t:ðn: c:.

• m:hap:¼s:h+ðN: t:T:a K:v:üS:t:ðn: c:.

• s:m:ØdÓðN: c: t:ðn:òv: m:haòG:ðn: t:T:òv: c:.

• O\: kaðeX m:haòG:ðn:/

The number here is 10^10+10^14+10^20+

10^24+10^30+ 10^34+10^40+ 10^44+10^52+ 10^57+10^62+ 5.

Big numbers in a battlefield

• In Mahabharata war, 11+7 akshauhinis participated.

• एको� �!कोरथा� त्र्यश्वा� पभि'� पञ्चपदा�षितंको� | पत्त्यङ्गैः!� षि-गु/णां!� सर्वैः1� क्रमा�दा�ख्य� यथा�'रमा� || स�नी�मा/ख� गु/ल्मागुणां5 र्वैः�षिहनी� प6तंनी� चमा7� | अनी�षिकोनी� दाश�नी�षिको न्यक्षौ5षिहणां� ... ||

• an Akshauhini, by calculation, contains 21,870 elephants, 21,870 chariots, 65,610 Horses, and 109,350 foot soldiers.

• The ratio is 1 chariot : 1 elephant : 3 cavalry : 5 infantry soldiers.

In Aero science

• The number of meteors in the eighth level of the celestial sphere is 30703221.

• Taken from the book “Science in Sanskrit” published by Samskrita Bharati, Delhi, 2007.

• बा�णां�स्थधी7माको� तं7नी�� माण्डलस्य�ष्टमा�न्तंर� |

• षि-को�टि सप्तंलक्षौ षि- सहस्रा षिCस्तं�परिर |

• एकोविंर्वैःFशषितं स�ख्य�को� र्वैःतं�न्तं� धी7माको� तंर्वैः� ||

- Brihad vimana saastram, Kriyasara tantram.

In Siddhanta Deepika

• This is a book written by Parameswara (1370-1460).

• Here the dates of many eclipses are listed.

• The number of days in Kaliyuga is given, to describe the exact date.

• 1643524• This is the first in a list

of fourteen such 7-digit numbers.

• 9th November 1398 is a day of solar eclipse. This is actually the 1643524th day since the beginning of Kaliyuga.

In Aryabhateeyam

• In a full yuga consisting of 43,20,000 solar years, how many revolutions does the moon make?

• Aryabhata gives the answer in his own coded language as चय षिगुयियअङ्� उश/छ्रुJल्रु/

• This gives the number (according to his coding), 57753336.

A discussion on the Ramayana-number

An objection• Obviously so many monkeys

cannot be there. This number is so large that the entire earth is insufficient to accommodate them.

• This proves that Valmiki has made false statements.

A reply• No Valmiki does not make

this assertion. Instead, Valmiki writes that shuka and Sarana, two funny characters in his book make this statement.

• They are portrayed as exaggerators, whimsical, but talented.

Discussion continues

English translation• At the end of the

conversation, Ravana threatened both Shuka and Sarana.

• You have the audacity to praise the enemy camp unduly.

Sanskrit sloka• �त्स�य�मा�स तं5 र्वैः�र5 कोथा�न्तं�

श/कोस�रणां5 |

• र्वैःक्तुं/� अप्रस्तंर्वैः� स्तंर्वैः� |• They did so because, they

were (i)terribly afraid of the monkey-army (ii) thankful to them for releasing him without killing and (iii) ready to irritate their master Ravana.

Discussion continues

• Still, in spite of the fact that this is an exaggerated false account, our point is made: Very large numbers were employed (many milleniums ago) fancifully and freely.

Two other instances

Jains• The Jaina religious works

date from 500 B.C. • According to a certain

measurement of time, one Purvi = 75600000000000 years. Note that there are 11 zeros here.

Buddhists• Lalita Vistara is a Buddhist

work. It was written as early as first century B.C. In it Buddha (Bodhi satva) is talking to a mathematician by name Arjuna. He reveals his talent by enumerating large numbers, namely by stating the names of powers of ten, upto 10 power 53.

Link with Module 1

• In the first module we saw that two large numbers made their appearance as solutions to the equation 61x^2+1=y^2.

• These are: y=1766319049, x=226153980• In this module we continue it by asserting that

Like Bhaskara, many of his predecessors unhesitatingly handled very large numbers.

In Ganitasara sangraha

• Mahaveeracharya of 9th century mentions the number 11000011000011 and points out two properties: (i) It is same when read from left to right also. (ii) Its factorisation is 333333666667 x 33.

Conclusion

• There are many other old Sanskrit books that employ many different large numbers.

• They were employed in Astronomy, arithmetics, geography, history, war-science, and the like.

• They demonstrate that Indian civilisation was highly advanced since several thousands of years.