This is a part of the series of lectures by Prof V Kannan from University of Hyderabad

1. Modern mathematics in old
Sanskrit books
Module 2
Large numbers

2. 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.

3. 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.

4. 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.

5. 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.

6. 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
“

7. 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

8. 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
• 100000010000100100010000
010001000001000100000100
010000000005
• 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

9. 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

10. From Valmiki
• 


H.
• 


/
• 


.
• 

/
• 


/
• 


शतं शतसहस्राणां कोटिमाहुममनीषिणः |शतं
कोटिसहस्राणां शङ्ख इत्यभिधीयत ||
Koti = 10 power 7
Shankha = 10 power 12
Mahashankha = 10 power 17
Brunda = 10 power 22
Mahabrunda = 10 power 27
Padma = 10 power 32
Mahapadma = 10 power 37
Kharva = 10 power 42
Mahakharva = 10 power 47
Samudram = 10 power 52
Ogha = 10 power 57
Mahaugha = 10 power 62.

11. From Valmiki Contd.
• 



.
• 


.
• 


.
• 
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.

12. Big numbers in a battlefield
• In Mahabharata war,
11+7 akshauhinis
participated.
• एकिैकरथा त्र्यश्वा पषतः
पञ्चपदाततका | पत्त्यङ््ैः
त्रि्ुणैः सववः क्रमादाख्या
यथोतरम् || सनामुखं
्ुल्म्णौ वाटहनी पृतना
चमः | अनीककनी
दशानीकक न्यक्षौटहणी ...
||
• 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.

13. 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.
• बाणास्थधमकतनां
मण्डलस्याष्िमान्तर |
• त्रिकोटि सप्तलक्ष त्रि
सहस्र द्षवस्तोपरर |
• एकषवंशतत संख्याका
वतमन्त धमकतवः ||
- Brihad vimana saastram,
Kriyasara tantram.

14. 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.

15. 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
ग्तयअङ्उशुछ्रुल्रु
• This gives the number
(according to his
coding), 57753336.

16. 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.

17. 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
• ित्समयामास तौ वीरौ कथान्त
शुकसारणौ |
• वकतुं अप्रस्तव स्तवं |
• 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.