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Science in HinduismLarge numbers and infinity
Posted By
Sarin
on
Oct 06, 2012
Latest Hinduism news
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In my previous article, we saw how Indian scholars laid out the foundation of modern mathematics. However, in my previous article, we saw about only the basic number system. In this article, we will go one step forward to see the concept of large numbers in ancient Indian literature including Vedic texts.
In all early civilizations, the first expression of mathematical understanding appears in the form of counting systems. Numbers in early civilization were typically done using symbols or series of lines separated by space. However, lots of difficulties were faced by the western concept of number system. Let us analyze their difficulties
Difficulties in Number system of western world
In the Western world, large numbers were not in use until quite recently with the advent of modern science in the nineteenth century.
In ancient times, Greeks number system were based on myriad (i.e 10,000) and their largest was a myriad myriad, i.e 100,000,000
Greek Number system
Later, Archimedes (287212 BC) devised a system of large numbers up to
, by using powers of a myriad myriad. However, Archimedes proposed his numbering system based on power of myriad and hence, would have faced notational difficulties in devising numbers system based on power of myriad. So, he stopped at this number because of not being able to derive any new ordinal numbers larger than 'myriad myriadth'.
Shortly, after Archimedes devised a system based on power of 10, Appollonious of perga devised a more practical number system which were not based on power of 10 but on naming power of a myriad, for example,
Would be a myriad squared.
Why there was a Need of a better number system
Using myriad, western scholars were not able to derive all possible large numbers. Hence, western scholars were facing difficulties in representing or deriving many mathematical concepts. Hence, there was a rising need of a better, More concise, more consistent and more precise number system. This is where the need of adopting the Indian number system came into place. Actually, when the westerners were thinking in ones or two, Indians had invented the whole place value system with numbers extending up to power of 621.
In fact, by the 7th century BCE, Indian scholars reintroduced the notion of infinity as the quantity whose denominator is zero.
There was no concept of infinity in western world and only by the end of 13
^{th}
century, Roman scholars introduced the concept of representing large numbers as millions i.e. 1,000,000 was expressed as
decies centena milia
, (ten hundred thousand); which was evidently called as million.
Let us now see how the Indians have dominated the number system
Number System in Ramayana
It may be surprising to know that Indian sages used large numbers up to power of 1062 and that too
millions of years ago with the first to use it were Sage Valmiki (Author of Ramayana)
Following verse from Ramayana said to be written at least 12 million years ago
in Tretâyuga
, presents a number system up to power of 1062, big enough to be represent infinity.
Verse:
�atam �atasahsrânam, kotim âhurmanisinah
Œatam k
otisahasrânam �ankurityabhidhiyate
Above verse can be precisely translated as
Œatam œatasahsram = One Koti
ie. Hundred hundred thousand = 100, 00,000 = 1 crore = 107
Œatam Kotisahsram = One Œanku
ie. Hundred thousand crore = 100, 000, 0000,000
= Œanku =
1012
1 Koti = 107 = 1 crore
1 Œanku = 1012 = 1 lakh crore
1 Mahaœanku = 1017
1 Vrndam = 1022
1 Mahavrndam = 1027
1 Padmam = 1032
1 Mahapadmam = 1037
1 Kharvam = 1042
1 Mahakharvam = 1047
1 Samudram = 1052
1 Ougham = 1057
1 Mahaugham = 1062
This number is actually the count of the monkey soldiers who built the historic Ram Sethu (Also known as Adam Bridge).
Monkey soldiers Building Ram sethu
While inhabitants of other continents were using stones and fingers to count, Vedic sages counted in trillions & trillions to measure the cosmic concepts of this universe.
Number system in Vedas:
Many Vedic texts points to the decimal number system.
Yajurveda describes the number system with place value up to 18 places, the highest called as parardha
^{.}
For Example, after preparing bricks for a Vedic ritual, Rishi(Sage)
Medhâtithi prays to the Lord of fire, Agni
Verse:
Imâ me Agna istakâ dhenava
Santvekâ ãa desa ãa satam ãa
Sahasram ćāyutam ãa niyutam ãa
Prayutam ćārbudam ãa nyarbudam ãa
Samudrasãa madhyam ćāntasãa
Parârdhasãaita me agna ishtakâ
Dhenavasant
vamutrâmushmimlloke .
Translation
:
Oh Agni! Let these bricks be milk giving cows to me
Please give me one and ten and hundred and thousand
Ten thousand and lakh and ten lakh and
One crore and ten crore and hundred crore,
A thousand crore and one lakh crore in this world and other worlds too.
eka  1  one  10�
dasa  10  ten  101
satam  100  hundred  102
sahasram  1000  thousand  103
ayutam  10000  ten thousand  104
niyutam  100000  one lakh  105
prayutam 1000000  ten lakh  106  million
arbudam 10000000  one crore  107  ten million
nyarbudam 100000000  ten crore  108  hundred million
samudram 1000000000  hundred crore 109  billion
madhyam 1000000000  thousand crore 1010  ten billion
antam 100000000000  ten thousand crore1011  hundred billion
parardham 1000000000000 one lakh crore  1012  trillion
Even the concept of Fibonacci number can be found in Vedic verse translated as
The sun flower smiles at you with 34, 55 florets. (34, 55 are the numbers in the sequence of Fibonacci number.)
Sacred text of vedas
Concept of Infinity in vedas
Concept of infinity was used repeatedly in Vedic era. Latest being the vishwa roop darshan of lord Krishna where lord Krishna is shown as ananta, meaning “infinity” or “having no end”. Some of the other words used in vedic texts are
purnam, asamkhyata and aditi. For Ex:Word Asamkhyata
is used in Yajur Veda and Brihadaranyaka Upanishad to represent the number of mysteries of Indra as
ananta
. Following verse from yajurveda describes the mathematical concept of infinity. However, this verse (Shloka) is more metaphysical than matahemical
Verse:
pûrnamadah p ûrnamidam pûrnât pûrnamudacyate
p ûrnâsya pûrnamadaya p ûrnamevâvasishyate
Translation:
From infinity is born infinity.
When infinity is taken out of infinity, left over is only infinity.
Another Vedic verse with concept of infinity is
Om purnam adah purnam idam
Purnat purnam udacyate
Purnasya purnam adaya
Purnam
evâva�is
�
yate
Translation
OM
 the Complete Whole;
purnam
 perfectly complete; adahthat
; purnam
perfectly complete; idam

this phenomenal world; purnatthe allperfect; purnamcomplete unit; udacyate is produced; purnasya

Complete Whole; purnam

completely; adaya

having been taken away; purnamthe complete; eva even; avasisyate
i
s remaining.
Atharveda has some verses signifying the concept of one, arithmetic progression, and arithmetic series. Below are the numbers along with the term used in Vedas and other ancient Indian texts
Numerals greater then 100
Number
Sanskrit
200
Dvisata
300
Trisata
356
sat pancasat trisata
400
Catursata
500
Pancasata
1000
Sahasra
2000
Dvisahasra
3000
Trisahasra
4000
Catursahasra
10,000
dasasahasra, ayuta
20,000
Vimsatsahasra
30,000
Trimsatsahasra
100,000
satasahasra, laksha, lak
200,000
dvisatasahasra
300,000
trisatasahasra
1,000,000
prayuta, niyuta
10,000,000
koti, krore
100,000,000
arbuda, vrnda, nyarbuda
Numerals from Billion and above
Number
Sanskrit
1,000,000,000
abja, shatakoti, maharbuda, nikharva, nikarvaka, badva
10,000,000,000
Kharva
100,000,000,000
nikharva, akshita
1,000,000,000,000
mahaapadma, antya, antyam, nikharva
10,000,000,000,000
sha.nku
100,000,000,000,000
Jaladhi
1000,000,000,000,000
Antya
10,000,000,000,000,000
Madhya
100,000,000,000,000,000
Paraardha
WordNumeral Decimal System
Indian scholars expressed all large numbers using the decimal number system. The highest power of 10 named today is 1030
(Deca). But ancient Indian mathematicians had exact names for powers up to 1053.
WordNumeral
Decimal Equivalent
Ekam
10
^{0}
Dashkam
10
^{1}
1 Shatam
10
^{2}
1 Shahashram
10
^{3}
10 Dash Shahashram
10
^{4}
Laksha
10
^{5}
Dash Laksha
10
^{6}
Kotihi
10
^{7}
Ayutam
10
^{9}
Niyutam
10
^{11}
Kankaram
10
^{13}
pakoti
10
^{14}
Vivaram
10
^{16}
Pararadahaa
10
^{17}
kshobhya
10
^{18}
Nivahata or
vivaha
10
^{19}
Utsangaha or
kotippakoti
10
^{21}
Bahulam
10
^{23}
Naagbaalaha
10
^{25}
Titlambam
10
^{27}
nahuta
10
^{28}
Vyavasthaanapragnaptihi or
titlambha
10
^{29}
Hetuhellam or
vyavasthanapajnapati
10
^{31}
Karahuhu
10
^{33}
Hetvindreeyam or
ninnahuta
10
^{35}
Sampaata Lambhaha or
hetvindriya
10
^{37}
Gananaagatihi or
samaptalambha
10
^{39}
Niravadyam or
gananagati
10
^{41}
akkhobini
10
^{42}
Mudraabalam or
niravadya
10
^{43}
Saraabalam
10
^{45}
Vishamagnagatihi
10
^{47}
Sarvagnaha or
bindu
10
^{49}
Vibhutangaama
10
^{51}
Tallakshanaam
10
^{53}
abbuda
10
^{56}
nirabbuda
10
^{63}
ahaha
10
^{70}
ababa
10
^{77}
atata
10
^{84}
soganghika
10
^{91}
uppala
10
^{98}
kumuda
10
^{105}
pundarika
10
^{112}
paduma
10
^{119}
kathana
10
^{126}
mahakathana
10
^{133}
asamkhyeya
10
^{140}
dhvajagranishamani
10
^{421}
Numbers above infinity
Below are some of the numbers representing Infinity
�
bodhisattva
(
बोधिसत्व
or
बोधिसत्त
) 10
^{37218383881977644441306597687849648128}
�
lalitavistarautra
(
ललितातुलनातारासूत्र
) 10
^{200}
infinities
�
matsya
(
मत्स्य
) 10
^{600}
infinities
kurma
(
कुरमा
) 10
^{2000}
infinities
varaha
(
वरहा
) 10
^{3600}
infinities
narasimha
(
नरसिम्हा
) 10
^{4800}
infinities
vamana
(
वामन
) 10
^{5800}
infinities
parashurama
(
परशुराम
) 10
^{6000}
infinities
rama
(
राम
) 10
^{6800}
infinities
khrishnaraja
(
कृष्णराज
) 10infinities
kaiki
(
काईकी
or
काइकी
) 10
^{8000}
infinities
balarama
(
बलराम
) 10
^{9800}
infinities
dasavatara
(
दशावतारा
) 10
^{10000}
infinities
bhagavatapurana
(
भागवतपुराण
) 10
^{18000}
infinities
avatamsakasutra
(
अवताम्सकासुत्रा
) 10
^{30000}
infinities
mahadeva
(
महादेव
) 10
^{50000}
infinities
prajapati
(
प्रजापति
) 10
^{60000}
infinities
jyotiba
(
ज्योतिबा
) 10
^{80000}
infinities
Large number in other texts
Many ancient Indian texts talks about Lord Brahma being the creator of this universe and have often mention Lord Brahma to be trillions of years old. Below is the image extracted from such texts?
Hindu units of time in a logarithmic scale
When the world was thinking of one and two, Indian sages were dealing in trillions which is very evident in their religious thought. For example, in Vedas which is atleast ten thousand years old (as per carbon dating), we find Sanskrit words and verses corresponding to powers of 10 up to a trillion and even 10
^{62}
. (Sanskrit words crores and lakhs, referring to 10,000,000 and 1,00,000 respectively, are in common use even today).
Large numbers in Surya Prajnapti
Another mathematical text Surya Prajnapti (wriiten around 400 BC) separates all numbers in three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders as:
•
Infinite: nearly infinite, truly infinite, infinitely infinite
•
Innumerable: nearly innumerable, truly innumerable and innumerably innumerable
Enumerable: lowest, intermediate and highest
Concept of Transfinite number
The concept of infinity is used in mathematical theory of limits. It is defined a number greater than any finite number. However, in the late 19th century, mathematicians started studying transfinite number. Transfinite number is a number which is not only greater than any finite number but also greater than infinity. Incidentally, it was the Indian mathematician ‘Jaina’ who first considered the concept of transfinite numbers in 400 BCE.
Large number in TaittiriyaSamhita
TaittiriyaSamhita written before 1000 B.C uses terminology for numbers up to order 10 ** 19 as
1, 10, 10**1, 10**2, 10**3, 10**4, 10**5, 10**6, 10**7, 10**8, 10**9, 10**10,
10**11, 10**12, 10**13, 10**14, 10**15, 10**16, 10**17, 10**18, 10**19.
However the terminology used in the medieval age varied slightly than those used in TaittiriyaSamhita. Also, as discussed above, the Valmiki Ramayana has specified the terminology for numbers up to the order of 10 **60. However, Point to be noted here is that term ‘samudra’ here denotes 10**9 whereas in the TaittiriyaSamhita, ‘Samudra’ represents 10**50 and many other terms for large numbers used by valmiki differs from those used in TattriyaSamhita. Also, except this context, all other numerical terms used in valmiki Ramayana are same as those used in TaittiriyaSamhita e.g. Niyuta and Nyarbuda, Arbuda, Madhya, Antya, Samudra and Parardha. Probably, because of limited use of sequential large numbers, some of the numerical terms disappeared in the later literature and those which were found do not represent the same as used in ancient literature. For ex:
Samaranganasutradhara (11th century A.D.) uses padma for 10**13, , Sanku for 10 ** 12, Kharva for 10**10 and Vrnda for 10**9.
In the Taittiriya Upanishad, there is a section which signifies the quasimathematical relationship between the bliss of a Brahman (Enlightened individual) and the bliss of a young man who have all materialistic pleasure of this world. It says that a young handsome man who is fit, strong, healthy, and is the king of the world, is one unit of human bliss. Further, many human attributes are used to provide a series of multiplication to measure human bliss in 10010 units equal to spiritual enlightenment or salvation (Moksha).
Verses of TaittiriyaSamhita
Large numbers in Jainism
Jain scholars had a big fascination of large numbers. In fact, they were the first, scientific thinker to theorize that “all infinities are not the same or equal”. This idea was established in the modern world only in the late nineteenth century when Cantor initiated his theory of sets.
Beside concept of infinities, Jain scholars were also aware of indices related theory, though they did not used any convenient notation as we use in mathematics today. Instead of using notation, Jain used simple statements to specify theory of indices. Instead of calling square roots and squares, they used the term “first square root” or “second square” etc. For ex: consider the following statement from their texts: The first square root multiplied by the second square root is the cube of the second square root. According to what we learned about indices, this statement can be denoted
a
^{1/2}
x a
^{1/4}
= (a
^{1/4 }
)
^{3 }
Large numbers in Buddhism
Buddhist literature also describes finite, infinite, determinate and indeterminate numbers. Buddhist mathematicians classified mathematics as Garna (Simple Mathematics) or Sankhyan (Higher Mathematics). Numbers were of three types: Asankheya (uncountable), Sankheya (countable), and Anant (infinite).
According to Lalitavistara Sutra(a Buddhist work), there was a contest involving writing, arithmetic and archery, where buddha was defeated by great mathematician Arjuna who showed off his mathematical skills by citing the words representing numbers in power of ten up to 1 'tallakshana', equals to 1053. It also states that these words were just one of the series of counting system which geometrically expands up to 10421, that is, a 1 followed by 421 zeros.
The Decimal System in Harappa
No one knows the exact period of invention of number system and the modern world asks for proof of Vedas being millions of years old. So, let us go on the basis of evidences. As per the evidences,
a decimal system existed during the preHarappa period. Archaeological evidences with weights corresponding to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500 have been found in Harappa and MohenjoDaro. However, the most notable characteristic of these weights and measures are their remarkable accuracy.
For Ex: A bronze rod marked with 0.367 inches exactly measures to .367 inches. Scales with unit of .367 inches were used in those days for proper planning and construction of towns, roads, drains, palaces, towers as per the guidelines laid out by the architect/King. This discovery of an accurate weight based system points to the existence of trade and commerce in the Harappa society.