The number zero was invented in Ancient Pakistan

[roadrunner]

A lot of the Indians on here seem to be trying to leech Pakistani history by claiming references that say the number zero was invented in India, refer to modern day India.

This somehow justifies the leeching of Pakistani inventions like the number zero. So, to put this straight

So you can see that when no beads have been moved, you need a symbol to represent “0". This symbol is very important, in order to show that this is the number 15730 and not the much smaller number 1573. It was probably in using an abacus that the Hindus of the Indus valley in today’s Pakistan first invented zero.
From Zero to Hero - MSN Encarta

I would also like to point out some other Mathematical concepts our leechy friends try to steal by claiming that since their country today is called India, these inventions occurred within modern day India, when in fact they occurred in modern day Pakistan.

_________________________________________________________________________
Pingala's Binary numeral system - usage of Pascal's Triangle and Fibonnacci numbers - Discovered 300 BC in Ancient Pakistan.
_________________________________________________________________________
Panini's transformations and recursions - - Discovered 500 BC in the Indus Valley
_________________________________________________________________________
Negative numbers - used for the first time in Ancient Pakistan
_________________________________________________________________________

Many more things too. The name confusion caused at partition is nicely summarized:

"The first mathematics which we shall describe in this article developed in the Indus valley. The earliest known urban Indian culture was first identified in 1921 at Harappa in the Punjab and then, one year later, at Mohenjo-daro, near the Indus River in the Sindh. Both these sites are now in Pakistan but this is still covered by our term "Indian mathematics" which, in this article, refers to mathematics developed in the Indian subcontinent."
Indian mathematics

 

[Flintlock]

History of the Zero:

What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.

In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word "kha" for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.

We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.

We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.

Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-

The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.

Subtraction is a little harder:-

A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.

Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-

A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know who tried to extend arithmetic to negative numbers and zero.

In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta's book. He correctly states that:-

... a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.

However his attempts to improve on Brahmagupta's statements on dividing by zero seem to lead him into error. He writes:-

A number remains unchanged when divided by zero.

Since this is clearly incorrect my use of the words "seem to lead him into error" might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.

Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-

A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.

______________________________

Who was Brahmagupta?

Where was he from?

Ujjain, now in Uttar Pradesh, India.


Bhinmal, Rajasthan. OR Ujjain, Madhya Pradesh

Sources:

Plofker, Kim (2007). pp. 418–419. "The Paitamahasiddhanta also directly inspired another major siddhanta, written by a contemporary of Bhaskara: The Brahmasphutasiddhanta (Corrected Treatise of Brahma) completed by Brahmagupta in 628. This astronomer was born in 598 and apparently worked in Bhillamal (identified with modern Bhinmal in Rajasthan), during the reign (and possibly under the patronage) of King Vyaghramukha.

Brahmagupta biography

Brahmagupta the mathematician

Brahmagupta: Biography from Answers.com

Brahmagupta (print-only)

 

[acer]

^^^^^Sir I think "pai" is also defined by Aryabhata.. right?

 

[shrivatsa]

another genius Madhava

Although born in Cochin on the Keralese coast before the previous four scholars I have chosen to save my discussion of Madhava of Sangamagramma (c. 1340 - 1425) till last, as I consider him to be the greatest mathematician-astronomer of medieval India. Sadly all of his mathematical works are currently lost, although it is possible extant work may yet be 'unearthed'. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475, but this is only speculative. All we know of Madhava comes from works of later scholars, primarily Nilakantha and Jyesthadeva. G Joseph also mentions surviving astronomical texts, but there is no mention of them in any other text I have consulted.

His most significant contribution was in moving on from the finite procedures of ancient mathematics to 'treat their limit passage to infinity', which is considered to be the essence of modern classical analysis. Although there is not complete certainty it is thought Madhava was responsible for the discovery of all of the following results:

1) = tan - (tan3 )/3 + (tan5)/5 - ... , equivalent to Gregory series.

2) r= {r(rsin)/1(rcos)}-{r(rsin)3/3(rcos)3}+{r(rsin)5/5(rcos)5}- ...

3) sin = - 3/3! + 5/5! - ..., Madhava-Newton power series.

4) cos = 1 - 2/2! + 4/4! - ..., Madhava-Newton power series.
Remembering that Indian sin = rsin, and Indian cos = rcos. Both the above results are occasionally attributed to Maclaurin.

5) p/4 1 - 1/3 + 1/5 - ... 1/n (-fi(n+1)), i = 1,2,3, and where f1 = n/2, f2 = (n/2)/(n2 + 1) and f3 = ((n/2)2 + 1)/((n/2)(n2 + 4 + 1))2 (a power series for p, attributed to Leibniz)

6) p/4 = 1 - 1/3 + 1/5 - 1/7 + ... 1/n {-f(n+1)}, Euler's series.

A particular case of the above series when t =1/3 gives the expression:
7) p = 12 (1 - {1/(3 3)} + {1/(5 32)} - {1/(7 33)} + ...}

In generalisation of the expressions for f2 and f3 as continued fractions, the scholar D Whiteside has shown that the correcting function f(n) which makes 'Euler's' series (of course it is not in fact Euler's series) exact can be represented as an infinite continued fraction. There was no European parallel of this until W Brouncker's celebrated reworking in 1645 of J Wallis's related continued product.

A further expression involving p:
8) pd 2d + 4d/(22 - 1) - 4d/(42 - 1) + ... 4d/(n2 + 1) etc, this resulted in improved approximations of p, a further term was added to the above expression, allowing Madhava to calculate p to 13 decimal places. The value p = 3.14159265359 is unique to Kerala and is not found in any other mathematical literature. A value correct to 17 decimal places (3.14155265358979324) is found in the work Sadratnamala. R Gupta attributes calculation of this value to Madhava, (so perhaps he wrote this work, although this is pure conjecture).

Of great interest is the following result:
9) tan -1x = x - x3/3 + x5/5 - ..., Madhava-Gregory series, power series for inverse tangent, still frequently attributed to Gregory and Leibniz.

It is also expressed in the following way:
10) rarctan(y/x) = ry/x - ry3/3x3 + ry5/5x5 - ..., where y/x 1

The following results are also attributed to Madhava of Sangamagramma:
11) sin(x + h) sin x + (h/r)cos x - (h2/2r2)sin x

12) cos(x + h) cos x - (h/r)sin x - (h2/2r2)cos x

Both the approximations for sine and cosine functions to the second order of small quantities, (see over page) are special cases of Taylor series, (which are attributed to B Taylor).

Finally, of significant interest is a further 'Taylor' series approximation of sine:
13) sin(x + h) sin x + (h/r)cos x - (h2/2r2)sin x + (h3/6r3)cos x.
Third order series approximation of the sine function usually attributed to Gregory.

With regards to this development R Gupta comments:

...It is interesting that a four-term approximation formula for the sine function so close to the Taylor series approximation was known in India more than two centuries before the Taylor series expansion was discovered by Gregory about 1668. [RG5, P 289]


Although these results all appear in later works, including the Tantrasangraha of Nilakantha and the Yukti-bhasa of Jyesthadeva it is generally accepted that all the above results originated from the work of Madhava. Several of the results are expressly attributed to him, for example Nilakantha quotes an alternate version of the sine series expansion as the work of Madhava. Further to these incredible contributions to mathematics, Madhava also extended some results found in earlier works, including those of Bhaskaracarya.

The work of Madhava is truly remarkable and hopefully in time full credit will be rewarded to his work, as C Rajagopal and M Rangachari note:

...Even if he be credited with only the discoveries of the series (sine and cosine expansions, see above, 3) and 4)) at so unexpectedly early a date, assuredly merits a permanent place among the great mathematicians of the world. [CR /MR1, P 101]

Similarly G Joseph states:

...We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition. [GJ, P 293]

With regards to Keralese contributions as a whole, M Baron writes (in D Almeida, J John and A Zadorozhnyy):

...Some of the results achieved in connection with numerical integration by means of infinite series anticipate developments in Western Europe by several centuries. [DA/JJ/AZ1, P 79]

There remains a final Kerala work worthy of a brief mention, Sadrhana-Mala an astronomical treatise written by Sankara Varman serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19th century and the author stands as the last notable name in Keralese mathematics.

In recent histories of mathematics there is acknowledgement that some of Madhava's remarkable results were indeed first discovered in India. This is clearly a positive step in redressing the imbalance but it seems unlikely that full 'credit' will be given for some time, as that will possibly require the re-naming of various series, which seems unlikely to happen!

Still in many quarters Keralese contributions go unnoticed, D Almeida, J John and A Zadorozhnyy note that a well known historian of mathematics makes:

...No acknowledgement of the work of the Keralese school. [DA/JJ/AZ1, P 78]
(Despite several Western publications of Keralese work.)
9 III. Madhava of Sangamagramma

 

[Vinod2070]

Zero is attributed to Aryabhatta. Aryabhatta was born in Pataliputra, Magadha (modern day Bihar). He probably never even saw the North Western parts of India.

How is zero in any way related to "ancient Pakistan"!

 

[third eye]

Wow !

Thats a lot of inputs on zero's & negative numbers !!

 

[slugger]

A lot of the Indians on here seem to be trying to leech Pakistani history by claiming references that say the number zero was invented in India, refer to modern day India.

This somehow justifies the leeching of Pakistani inventions like the number zero.

Care to draw out the boundaries of Pakistan dynasty and existence of Pakistan pre-August 14 1947

Also do ennumerate how the present-day Pakistan and demography, whose existence was made possible under the rationale of providing Muslims with a land of their own, contributed to a civilization that has nothing to do with Islam

The fact of the matter is that the origin of the Indus Valley civilization was based on on Hinduism and Jainism and not Islam for which Pakistan stands

The origins of Hinduism and Jainism emerge out of present day India and the spread of Hinduism and Jainism of which the Indus Valley Civilization was a part is simply an indication of the progress and geographical reach of Civilizations and Dynasties that conducted its governance and state policies using the frameworks of religion that had nothing to do with Islam

Also when an Indian talks about India's contribution and progress by giving references to ancient Indian dynasties and works of people like Aryabhatta and others, he is doing so as counter to claim made by Pakistanis who try are hell-bent on proving that Arabic and Persian mathmaticians accomplished things that clearly finds mentions and practice in the works under Dynasties that did not practice Islam

So instead of blaming Indians, who are simply defending accomplishment of the Indian sub-continent you should try talking some sense in to Pakistanis who derieve pleasure in taking away the acievements made in the geographical area of the Indian sub-continent and associating them with Arabs and Persians, who contributed nothing to the spread of the empirs and civilizations with roots in prsent day India


The fact is Islam is the history, present and future of Pakistan

Any Pakistani accomplishment must be claimed by refering to the following timeline

An analysis of the timeline of Pakistan

History of Pakistan -> 14th August 1947 - 7th February 2009

Present Day Pakistan -> 8th February 2009

Future of Pakistan -> 9th February 2009 - infinity

 

[Flintlock]

^^^^^Sir I think "pai" is also defined by Aryabhata.. right?

No, definitons of pi have existed for a long time before in other civilizations, but Aryabhata gave a very precise measurement to 3 decimal places (more precise than those before him).

Yajnavalkya gave an approximation upto 2 decimal places in 9th century BC

Later, Nilakantha Somayaji from Kerala discovered that pi is "incalculable" , i.e. an irrational number.

 

[Vinod2070]

Mathematics

[edit] Place Value system and zero

The number place-value system, first seen in the 3rd century Bakhshali Manuscript was clearly in place in his work.[4] ; he certainly did not use the symbol, but the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients[5]

However, Aryabhata did not use the brahmi numerals; continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities (such as the table of sines) in a mnemonic form[6].

[edit] Pi as Irrational

Aryabhata worked on the approximation for Pi (&#960, and may have realized that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ.
"Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached."

Aryabhata used the word āsanna (approaching), appearing just before the last word, as saying that not only that is this an approximation, but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 by Lambert)[7].

After Aryabhatiya was translated into Arabic (ca. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra[1].

Aryabhata - Wikipedia, the free encyclopedia

 

[Flintlock]

So you can see that when no beads have been moved, you need a symbol to represent “0". This symbol is very important, in order to show that this is the number 15730 and not the much smaller number 1573. It was probably in using an abacus that the Hindus of the Indus valley in today’s Pakistan first invented zero.
From Zero to Hero - MSN Encarta

The modern concept of the zero was given by Brahmagupta in the 5th century AD.

Before him Aryabhata had already given an incomplete definition of zero, which was later improved by Brahmagupta and others.

This article is regarding the concept of zero as it could have developed by those who used an abacus (it might not, we do not know)

The the author does not provide any sources for his claim. Unless you provide a source, the article does not prove anything. Infact, its his personal opinion that the zero "was probably invented by those who used an abacus" There are no ancient texts to prove his claim. Its his own theory.

 

[roadrunner]

Let's not get carried away now.

I'll answer some of your fictitious points now.

The modern concept of the zero was given by Brahmagupta in the 5th century AD.

Brahmagupta invented the rules governing the usage of the number zero.

Brahmagupta was born in Multan, and he is part of Ancient Pakistai history.

• Né en 598 au nord-ouest de l’Inde, à Multan, aujourd’hui au Pakistan, Brahmagupta passera une grande partie de sa vie dans la ville de Bhîlmal sous la protection du souverain Gurjara.
  Brahmagupta
  
  Rough translation: Brahamgupta was born in 598 in Northwest India, Multan, in today's Pakistan, Brahmagupta spent a great deal of his life in the town of Bhilmal under the protection of the Gurjara
  
  
  
• The great mathematician of India Brahmagupta (Born in Multan and lived during 598-660 A.D) wrote on nature of Zero in his book “Bramhagupta Siddhanth” History Of Zero

Zero is attributed to Aryabhatta. Aryabhatta was born in Pataliputra, Magadha (modern day Bihar). He probably never even saw the North Western parts of India.

False.

Aryabhata only used the number zero in the 4th century AD. The number zero was invented long before this, in around 400 BC or even before.

Knowing the evolution of Sanskrit in Ancient Pakistan, and the term given to zero in early Sanskrit, one can conclude that the number zero (Shunya) evolved in Ancient Pakistan on the Indus. This is confirmed by the MSNEncarta reference - a neutral reference.

 

[slugger]

A few more posts here and this thread should be eligibile for a change in title to

Achievements of ancient Indian Empires and Civilizations

 

[Flintlock]

Earliest Documented evidence for the use of Zero in the place value system:

The oldest known text to use a decimal place-value system, including a zero, is the Jain text from India entitled the Lokavibhâga, dated 458 AD. This text uses Sanskrit numeral words for the digits, with words such as the Sanskrit word for void for zero .[19] The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuja Temple at Gwalior in India, dated 876 CE.[20][21] There are many documents on copper plates, with the same small o in them, dated back as far as the sixth century AD, but their authenticity may be doubted.[6]

Sources:

Ifrah, Georges (2000), p. 416.
Ifrah, Georges (2000), p. 400.
Feature Column from the AMS

 

[Vinod2070]

Brahmagupta invented the rules governing the usage of the number zero.

Brahmagupta was born in Multan, and he is part of Ancient Pakistai history.

Brahmagupta was a Hindu born and worked in Ujjain. Absolutely nothing to do with North West India of the time. Flintoff has given the reference already:

Brahmagupta, whose father was Jisnugupta, wrote important works on mathematics and astronomy. In particular he wrote Brahmasphutasiddhanta (The Opening of the Universe), in 628. The work was written in 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala which today is the city of Bhinmal. This was the capital of the lands ruled by the Gurjara dynasty.

Brahmagupta became the head of the astronomical observatory at Ujjain which was the foremost mathematical centre of ancient India at this time. Outstanding mathematicians such as Varahamihira had worked there and built up a strong school of mathematical astronomy.

Brahmagupta biography

When did Ujjain in Central India become a part of "ancient Pakistan"!

Aryabhata only used the number zero in the 4th century AD. The number zero was invented long before this, in around 400 BC or even before.

Knowing the evolution of Sanskrit in Ancient Pakistan, and the term given to zero in early Sanskrit, one can conclude that the number zero (Shunya) evolved in Ancient Pakistan on the Indus. This is confirmed by the MSNEncarta reference - a neutral reference.

The reference is indeed neutral but it says this:

It was probably in using an abacus that the Hindus of the Indus valley in today’s Pakistan first invented zero.

Too inconclusive! Do we have any evidence that abacus was in use in Indus valley? The only part that is correct is that Hindu mathematicians discovered zero.

 

[AgNoStiC MuSliM]

Care to draw out the boundaries of Pakistan dynasty and existence of Pakistan pre-August 14 1947

Also do ennumerate how the present-day Pakistan and demography, whose existence was made possible under the rationale of providing Muslims with a land of their own, contributed to a civilization that has nothing to do with Islam

The fact of the matter is that the origin of the Indus Valley civilization was based on on Hinduism and Jainism and not Islam for which Pakistan stands

The origins of Hinduism and Jainism emerge out of present day India and the spread of Hinduism and Jainism of which the Indus Valley Civilization was a part is simply an indication of the progress and geographical reach of Civilizations and Dynasties that conducted its governance and state policies using the frameworks of religion that had nothing to do with Islam


The fact is Islam is the history, present and future of Pakistan

Any Pakistani accomplishment must be claimed by refering to the following timeline

An analysis of the timeline of Pakistan

History of Pakistan -> 14th August 1947 - 7th February 2009

Present Day Pakistan -> 8th February 2009

Future of Pakistan -> 9th February 2009 - infinity

This particular debate has been going on on another thread for a while now, and many of these arguments have been addressed there.

The history of a people does not change with the faith they adopt, if that were the case, the Greeks, primarily orthodox Christians now, woudl have no claim to their marvelous history intertwined with Greek mythology.

Pakistan's current demography and existence as a Muslim/Islamic state similarly has no effect on the non-Islamic history of our people - it is still Pakistani history.

The IVC did not practice Hinduism, again, these arguments have been gone over multiple times, but even if it did. that does not make that history any less Pakistani.

As to your time line, its absurd. The people inhabiting the lands of Pakistan did not shoot out of nowhere in 1947. They have millenium of history behind them, and that history is theirs, and therefore that of the State of Pakistan.

I would also like to point out that a similar time line can also be made for India

History of India -> 15th August 1947 - 7th February 2009

Present Day India -> 8th February 2009

Future of India -> 9th February 2009 - infinity

Lets stay away from the general arguments such as those made in your post on this thread - these arguments have been made multiple times and responded to and refuted on at least two other threads in the history section. Reading through those long threads would be a good idea to avoid repeating the same arguments time and time again.

Limit this thread to the arguments over the number Zero.