Introduction

The mathematical achievements of ancient India have largely remained hidden, but thanks to the internet age are coming to the surface in recent times. Many books have been written about the advanced mathematics, including trigonometry and calculus discovered in India which reached Europe in the Middle Ages through the Arabs. Here, I want to go into an even more basic mathematical idea which turns out to be one of the biggest leaps of imagination also. This is the ancient Indian numeric system.

Other Civilizations

The most basic indicator of the mathematical abilities and imagination of any civilization is the numeric system used by them and the largest number which they have arrived at. The following are some of the numbers used by ancient civilizations.

The largest number having any representation in ancient China was 10,000. That’s it.

Chinese wrote Ten Thousand as:

Similarly, the largest named number for the ancient Greeks was a Myriad, which was 10,000. To be fair, Archimedes did write a paper calculating the number of sand grains in the universe and therefore did imagine numbers as large as 10⁶³, but that knowledge remained unused and quickly forgotten.

Even the Romans, Persians and Egyptians never went beyond One Million.
Roman one million:

Persian one million – میلیون
Egyptians wrote one million this way:

Of course, Arabs received all knowledge from India, so they should not even be counted. Compared to the above numbers from other civilizations, ancient India went far ahead.

Some of the largest numbers are referred to and defined in theValmiki Ramayan. Valmiki defines these numbers while describing the size of Sri Ram’s army in the Yuddha Kanda, (६-२८-३३)

शतम् शतसहस्राणाम् कोटिमाहुर्मनीषिणः || शतम् कोटिसहस्राणाम् शङ्कुरित्यभिधीयते |

“Wise men call a Shata Shata Sahastra as a Koti. A Shata Sahastra Koti is reckoned as a Shanku.”

Shata is a hundred and Sahastra is a thousand. So a Koti is 10,000,000 (10 Million) and Hundred Thousand Koti is a Trillion. These days, what we call as “one lakh crore” for lack of proper terminology, actually had a name in ancient times – a “Shanku”!

So One Shanku (10¹²) is basically One Trillion.

Valmiki goes further to define even larger numbers:

शतम् शङ्कुसहस्राणाम् महाशङ्कुरिति स्मृतः || महाशङ्क्य्सहस्राणाम् शतम् वृन्दमिहोच्यते |
शतम् नृन्दसहस्राणाम् महावृन्दमिति स्मृतम् || महावृन्दसहस्राणाम् शतम् पद्ममिहोच्यते |
शतम् पद्मसहस्राणाम् महापद्ममिति स्मृतम् || महापद्मसहस्राणाम् शतम् खर्वमिहोच्यते |
शतम् खर्वसहस्राणाम् महाखर्वमिति स्मृतम् || महाखर्वसहस्राणाम् समुद्रमभिधीयते |
शतम् समुद्रसाहस्रमोघ इत्यभिधीयते || शतमोघसहस्राणाम् महौघ इति विश्रुतः |

Using Shatam Sahastram (same as Laksha) as Hundred Thousand for easy understanding:

Hundred Thousand Shanku = Maha Shanku =  10¹⁷

Hundred Thousand Maha Shanku = Vrinda =  10²²
Hundred Thousand Vrinda = Maha Vrinda =  10²⁷
Hundred Thousand Mahavrinda = Padma =  10³²
Hundred Thousand Padma = Mahapadma =  10³⁷
Hundred Thousand MahaPadma = Kharva =  10⁴²
Hundred Thousand Kharva = Maha Kharva =  10⁴⁷
Hundred Thousand MahaKharva = Samudra =  10⁵²
Hundred Thousand Samudra = Augha =  10⁵⁷
Hundred Thousand Aughas = Maha Augha =  10⁶²

How big is Maha Augha?

Does anyone know what 10⁶² is called in the modern metric system? Probably not, because no one uses numbers as large as these, except scientists working at the astronomical scale (counting stars and galaxies) or atomic scale while measuring the number of atoms in the universe.

Yes, that’s how big this number is.

So, considering that there are 10²³ stars and estimated 10⁸⁰ atoms in the known universe, that’s the scale we are talking about. This is far far ahead of any civilization in all of earth’s history, that came up with a numeric system.

Even Further

Later on, the Buddhist monks went even further. The Lalitavistara Sutra (a Mahayana Buddhist work) recounts a contest which included writing, arithmetic, wrestling and archery. In it the Buddha was pitted against the great mathematician Arjuna and showed off his numerical skills by citing the names of the powers of ten, up to 1 ‘tallakshana‘, which equals 10⁵³, but then going on to explain that this is just one of a series of counting systems that can be expanded geometrically. The last number at which he arrived at after going through nine successive counting systems was 10⁴²¹, that is, a 1 followed by 421 zeros! But he did not stop there, he actually named all the numbers up to that, with 10⁴²¹ being called as “dhvajagranishamani (ध्वजाग्रनिशमनी)”. That is astounding!
 

Very small numbers

I can bet many people have no idea that ancient Indians were interested in very small numbers also, especially when defining the concept of Time. 

How small? 

Well, त्रुति (Truti) was equal to “0.031 µs (micro second)” = 3.1 x 10^-8

रेणु (Renu) = 60 Truti = 0.86 µs = 8.6 x 10^-7

लव (Lava) = 60 Renu = 0.11 ms = 1.1 x 10^-4
लीक्षक (Leekshak) = 60 Lava = 6.696 ms = 6.696 x 10^-3

And so on. I am sure it must come as a surprise to many people that in ancient India we had names for numbers that went to negative powers of ten also!

Infinity

After naming bewilderingly large numbers, it would have been surprising if ancient Indians did not come up with a shloka or two to define Infinity.

Om poornam-adah poornam-idah poorna-aat poornam-udachyate,
Poorna-asya poornam-aadaaya poornam-evaa vashishyate
 

That is whole, This is whole, From the whole comes the whole. Even if the whole is taken away from whole, still the wholeness remains.

Yes. That is the exact definition of Infinity, that many of us use in our daily prayers!

Where are we now?

So, forget about using any of these numbers in our daily lives, the mind boggles just at the thought of the mathematical imagination of our forefathers. Remember, Ramayan and Lalita-Vistara were religious scriptures, so it is an ode to the mathematical genius of ancient Indians, that they found it very normal to define huge mathematical numbers even in our religious texts.

Maybe, in those days being good in mathematics was just commonplace.

Compared to this, how does it sound now, when we hear people using simple terms like “lakh crore” and “crore core” being used in modern India when our own ancestors were way smarter than us in using larger numbers?

References / Footnotes

• http://www.valmikiramayan.net/utf8/yuddha/sarga28/yuddha_28_frame.htm

• https://en.wikipedia.org/wiki/Indian_numbering_system

• https://en.wikipedia.org/wiki/The_Sand_Reckoner

• https://en.wikipedia.org/wiki/History_of_large_numbers

• https://en.wikipedia.org/wiki/Hindu_units_of_time

• https://archive.org/details/TheFlowerOrnamentScriptureATranslationOfTheAvatamsakaSutraByThomasClearypdfdtyxxytd

• http://www.storyofmathematics.com/

• http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Chinese_numerals.html

• http://www.universetoday.com/36302/atoms-in-the-universe/