The greatest contribution of India to mathematics is Zero, is what you often hear. But that is plain wrong. Zero was a natural by-product of an even more greater mathematical invention by ancient Indians, the Place Value (Positional Notation) System. Let me explain.
Have you tried doing mathematics using Roman Numerals? Each number is represented using a symbol like I for one, II for two, V for five, L for 50 etc. But if we don’t devise a scientific method to do simple arithmetic, then we will end up with a million different symbols for million different numbers. L, C, D, M , then X bar, L bar, C bar etc. This is the case with Roman numeral system. No useful mathematics could be done by this. Nor would have any better science been possible if one were to follow this system.
What Indians did was to invent Place Value system (Positional Notation) where we take a limited set of symbols 1,2,3,4,5,6,7,8,9. Then use a combination of these symbols to represent bigger numerical values like 32767, 2718 etc. This was done by giving a value for each position in incremental of 10. Aryabhata clearly mentions in his book Aryabhatiyam(1), sthaanam sthaanam dasha gunam meaning “from place to place increase 10 times”.
For ex in 11, 1 in the unit’s place has a value of one, 1 in the ten’s place has a value of 10, and hence the value of 11 is ten+one=eleven. This was a ingenious method and not only any number could be represented using just 9 symbols, one could even do complicated math using these numbers, starting with multiplication, division and proceeding even further to everything we have achieved in modern mathematics.
Try doing multiplication in roman numeral system and you will understand the genius of the India Place Values. Place value system was an amazing invention. As Albert Einstein once said
We should be thankful to Indians who taught us how to count, without which no worthwhile scientific discovery would have been possible
Note that Aryabhatta did not invent the place value system, he was merely restating an already existing knowledge. Nor did he use the Brahmi numeral symbols 1 to 9, he used the traditional Sanskrit method of using letters to denote numbers. Place value system is an even ancient invention of India, and is in built into the very language of Sanskrit. For instance, the name for Eleven is eka dasha in Sanskrit where eka means one and dasha means ten. Note that this Sanskrit place value system is from the left, while modern place value system is from the right. So in Sanskrit when we say eka dasha, the left most value is 1 (eka), and the next value is ten (dash). While in modern notation 11, the right most value is 1, and the one to its left is 10.
The invention of place value system (Positional Notation) created a new problem. There were places with no values at all, and there was no way to represent them via numeric symbols.  For example, to represent hundred and one, the rightmost 1 has a value of one, the left most 1 has a value of 100. But the middle place has no value, its a null or empty. It had to be written something like 1 1 (meaning 1space1) where the space in between represented a positional notation which had no value.
To overcome this difficulty, Indians invented 0, just to represent an empty place. It was not a number in itself, but just a notation implying “no value here”.
In Sanskrit, Shunya means nothing or emptiness, and zero was given the name Shunya. The Arab merchants who came to India for trade called it Sifr or Safira which means “empty” in Arabic. These Arab merchants took the Indian arithmetic to Europe when they went to trade the Indian goods there. Sifr in Arabic became Zero in the European languages. Since this numeral system came to Europe from India (Hindus) through Arab merchants, the Europeans called it Hindu-Arabic Numerals.
In fact, initially when this Indian number system was introduced in Europe, many European regions initially banned it. Florence in Italy banned its usage in 1299 CE. Most Europeans in those days were illiterate and also superstitious and were under the impression that there was some “black magic” in the Indian number system, and hence they banned it and continued using their awkward and difficult Roman Numeral System. Those were the times when scientists like Galileo were prosecuted in Europe, Giordano Bruno was burnt alive by the church for  supporting the Copernican theory that Sun was the center of the Universe.
In fact the strange Zero was considered to be a creation of Satan, devil. For them the Arabic word for zero, sifr, sounded like some secret code. Hence the Europeans started using the term “cipher” for secret codes, and decipher became solving secret codes. It was only much later that the Indian numeral system became acceptable in Europe. Today, the entire science and technology of the world, including the Computers run on this ingenious system invented in ancient India.
To summarize, Indians invented Zero as a natural by-product of their greatest mathematical invention – the place value system or the positional notation system. The oldest reference of using Sunya to represent zero appears in the works of the ancient Indian scholar Pingala (around 500 BCE) in his binary numeral system using long and short syllables.
French mathematician Georges Ifrah writes in his book Universal History of Numbers
Thus it would seem highly probable under the circumstances that the discovery of zero and the place-value system were inventions unique to the Indian civilization. As the Brahmi notation of the first nine whole numbers (incontestably the graphical origin of our present-day numerals and of all the decimal numeral systems in use in India, Southeast and Central Asia and the Near East) was autochthonous and free of any outside influence, there can be no doubt that our decimal place-value system was born in India and was the product of Indian civilization alone
Georges Ifrah also explains that the use of zero is implicit in Aryabhatta’s works where it is used in representing powers of ten with null coefficients.
Notes
1. Aryabhatiyam was a later name given to the available works of Aryabhata. We do not know the original name of his works. His disciple Bhaskara I calls it Ashmaka Tantra.
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i need help i do not under stand this
Vedic Mathematics
Voltaire, the famous French writer and philosopher) stated that “Pythagoras
went to the Ganges to learn geometry.” Abraham Seidenberg, author of the
authoritative “History of Mathematics,” cr edits the Sulba Sutras as inspiring all mathematics of the ancient world fromBabylonia to Egypt to Greece.
As Voltaire & Seidenberg have stated, many highly significant mathematical concepts have come from the Vedic culture, such as:
The theorem bearing the name of the Greek mathematician Pythagorus is found in
the Shatapatha Brahmana as well as the Sulba Sutra,
the Indian mathematical treatise, written centuries before Pythagorus was born.
But, the fact that it was first discovered/invented in India, what importance does it have? it would have been discovered/invented eventually. It doesn’t convey any special attribute to the Indian culture or people, as it would have been discovered/invented anyway, by an Indian, or by someone else.
Celebrating as a contribution to humanity is good. Thinking that being first conveys any special attribute is incorrect and meaningless.
There is no such thing as a monopoly of intelligence, morals, thought, IQ, etc by any race or people in the world. Anywhere on earth you find examples of outliers, irrespective of where it is (country), race, culture, etc. There are, of course, impediments for that intelligence, morals, etc to bear fruits depending on the culture, government, institutions, beliefs, etc that are localized to geographical location, countries, religions, and forms of government. Understanding and pointing those out is important, as allows people to work towards changing them for better systems.
But, the deep premise of the article (it was first invented here, it means we are special) is flawed and irrelevant, when thinking broadly about the human race.
Your post is flawed, as when it comes to Pythagoras theorem, you would say… He did it first, regardless of the possibility of someone discovering this in the future.
Likewise, Galileo and his discoveries.
It seems there are a lot of Indians who will go put of their way to negate anything India has achieved, perhaps it is a part of the inferiority complex a lot of Indians have embedded in their psyche.
Theere may be questions about Zero if the Phoenicians originated or the Indians — that is thingness out of nothingness — but the decimal system is entirely Indian — a leap of mathematical imagination that transcends even higher than zero.
Please help me to settle the argument: who first had the idea of the Place Value System? Indian or Chinese? Some of my friends said that it was Aryabhata (476–550 CE, http://en.wikipedia.org/wiki/Aryabhata), some said that Chinese Rod-Numeral System (200 BCE, http://en.wikipedia.org/wiki/Counting_rods) already had the concepts of negative number, decimal system, the place of zero, and the Place Value System.
The pro-Chinese group mentioned the following Wikipedia statement about Indian mathematics (http://en.wikipedia.org/wiki/Indian_mathematics)
It has been hypothesized that the Indian decimal place value system was based
on the symbols used on Chinese counting boards from as early as the middle of
the first millennium BCE. According to Plofker 2009,
These counting boards, like the Indian counting pits, …, had a decimal place
value structure … Indians may well have learned of these decimal place value
“rod numerals” from Chinese Buddhist pilgrims or other travelers, or they may
have developed the concept independently from their earlier non-place-value
system; no documentary evidence survives to confirm either conclusion.”
Its actually the other way round – Chinese learnt the place value based math from the Buddhist monks of India just like the way they learnt the martial art techniques which has its roots in the Kalarippayattu martial art from Kerala, India. As a matter of proof, the Shaolin Temple which was the center of martial arts in China was founded by Bodhidharma, a Buddhist monk from Southern India who visited China.
Coming back to the place value based system, the oldest available proof of this system is in the Sanskrit language itself, including in the vedic texts which are at least 3000 years older than Buddhism itself. In Sanskrit, there is no separate words for numerical values greater than 10 except for large multiples of 10 like 100, 1000 etc. So in Sanskrit there is no separate word like “eleven, twelve” etc instead 11 is called “eka dasha” which translates to “one ten” ie eka is the name for one and dasha is the name for ten in Sanskrit. Similary 12 has no separate word twelve, but dva dasha where dva is also the name of “two”.
Also note that unlike modern written numerical system where we have units, tens, thousands moving from right to left, in Sanskrit language it is from left to right.
Also note that modern numerals are called Hindu-Arabic in the western world because it spread there through the Arabic merchants who learnt it from the Hindus of India which has been documented in their texts as well. But in India you never find ANY such documentation about having learnt it from Chinese or anywhere else.
So, the place value based system is a part and parcel of the Sanskrit language itself which is as ancient to the times of Buddha, as is the times of Buddha for us today.
Note that even the Japanese Zen philosophy is India’s dhyana philosophy which first went to China and became Chan there and from there went to korea and finally to Japan as Zen.
did you not know Buddha was Chinese too ??… you are either lazy or ignorant, why don’t you do your own research to see how the ‘zero’ was invented.. by the way can you tell me when the Arabs became civilized?? Have you heard of an ancient Arabic University, but I know in India Thakshila and Nalanda the 2 oldest universities in the world. For your info ‘Lin
Yutang, Chinese scholar and author, wrote that: “India was China’s teacher
in trigonometry, quadratic equations, grammar, phonetics… ” and so
forth.’ . Read some books instead of playing ‘nintando’ and learn then you can tell your Chinese Nintando friends all about your findings.
Systems within systems within systems.
any reason for particularly choosing base 10 system???
Easyness in calculations..
A theory states that the base 10 evolves from the fact that humans have 10 fingers that can aid in counting
What will be representation of 47363632 in your quoted “Sanskrit Notation”??
chaturkoti trisaptaati laksha trishashti sahasra shatshata dvaatrimshatha
meaning 4 crore 73 lakh 63 thousand 6 hundred thirty two
I have been reading your articles on Vedas, education system
If we look at the history, Why didn’t we advance with so much knowledge? British destroyed this.ok. But Our ancestors must have thought about protecting and continuing this knowledge system. The entire human race could have been benefitted from this knowledge and it would have been originated from this land, India.
You still have access to that ancestral knowledge, only if you studied Sanskrit and in a Gurukul where all that is still being taught.
Is there a Gurukul in India where I can send my daughter to study these. Is it free education like olden gurukuls?
Gurudev one basic query… as you write dasha means 10, how did one knew as how to represent dasha i.e 1 and 0 as 10 to call it dasha ? do you see the catch here, i mean empty, zero is shunya, they didn’t say eka shunya for representing 10… so probly shunya seems to be older than place value since word dashaavtar is also pretty old, also ravana had 10 heads dashanan…. wat do you think ??
Sainath, you are bit confused. Let me clear it. In the original Sanskrit there was no need for shunya while representing place value based numbers, for the simple reason that each numeral in the number was pronounced along with its place value.
So for instance to represent 10, they just said dasha which meant 10. Each power of 10 has a name in Sanskrit, so 100 is Shata. Now to represent 101, it was simply ekaadhikashata which is eka (1) + adhika (more than) + shata (100). In other words 101.
So as you can see, since each numeric symbol was spelled along with its place value, there was no need to use zero or shunya in the Sanskrit language. Remember that the original Sanskrit was only spoken, not written. Knowledge was passed down through generations orally, which is why the vedic texts are available even today. So the language itself was evolved to be self contained without the need of any written documentation. And in the original Sanskrit, shunya was a word which represented not the number zero, but concepts like void, empty, etc.
The need for zero came when numbers started being represented in written form, using symbols. Since in this case individual symbols were used, and their position denoted the place value, there was a need for zero to indicate empty positions.
So Dasha, Shata, were just names for powers of 10. Which is why scientists like Carl Sagan were so amazed to see such an ancient language having individual names for numbers as big as a hundred quadrillion (paraardha).
Hope I have answered your question.
Gurudev yes i may be confused, i understand what you have written, but let me try again, you said “So for instance to represent 10, they just said dasha which meant 10” and my question is for this “to represent 10….meant 10”, though if we take that the language was only spoken someone has to think of 9+1 i.e 10 items i.e to associate word with the number as eka to 1, dasha to 100 and so…am i still confused :)
In my previous reply, I have explained why you dont require a zero while speaking numeric values in Sanskrit, like say 101, etc. Hope you have understood it.
Now regarding dasha, shata etc as I said in my comment earlier, in Sanskrit for each power of 10 there is a name. Just like you have eka for 1, pancha for 5 etc, similarly for 10 it is dasha, shata for 100, sahasra for 1000 etc.
So to summarize,
1) There is no zero in Sanskrit naming of numeric values, instead you have a name for every power of 10
2) Similar to numeric symbols in base 10 system, you have a name for each of them like
1=eka
2-dvi
3=tri
4=chatur
5=pancha
6=shash
7=sapta
8=ashta
9=nava
He He :)… same reply in a different manner…np leave it we will discuss this on chat sometime, i understand the enumeration of numerals with a specific name…may be i am unable to frame properly… :)
anyways thanks for the reply gurudev
If you are asking, how to understand dasha means 10, well that is something that is “defined as 10”. its a proper name, just like here “sainath” means you. In any language or science or maths or any subject you first start with a set of basic definitions, symbols and meanings, and then create combinations which would then automatically imply what they mean without the need for any definition. So while dasha means 10, eka means 1, you don’t need a definition for what ekadhika dasha means. Hope you got it.
As chanakya says a country is destroyed when its knowledge is destroyed.!! Im really happy to se such place..
If i could do something to improve this.
Hi Sainath, like Gurudev said for each power of 10 there is a name, and i think you are confused about 0, let me clear you out, 0 is numeric representation, 10, 100, 1000, you repeat zero which increases value and 1 is fixed right, but in sanskrit there are different names, dasha, shata, are you getting, so there is no same word repeating and increasing the value, what you might be thinking. So dasha or dashanan doesn’t proves that logic of 0 was there.
Please correct me if I got you wrong.
In this age where Western theory and culture has spell bounded the minds of people and society. And the Indian government has failed to bring about Renaissance. This kind of article brings hope in us and I can say that it’s not “The End” yet.
Keep posting….Thanks