Mathematics & Computer Science

* Binary System of number representation:

A Mathematician named Pingala (c. 100BC) developed a system of binary enumeration convertible to decimal numerals [See 3]. He described the system in his book called Chandahshaastra. The system he described is quite similar to that of Leibnitz, who was born in the 17th century.


* Earliest and only known Modern Language:

Panini (c 400BC), in his Astadhyayi, gave formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 fundamental elements, like nouns, verbs, vowels and consonents, he put them into classes. The construction of sentences, compound nouns etc. was explained as ordered rules operating on underlying fundamental structures. This is exactly in congruence with the fundamental notion of using terminals, non-terminals and production rules of moderm day Computer Science. On the basis of just under 4,000 sutras (rules expressed as aphorisms), he built virtually the whole structure of the Sanskrit language. He used a notation precisely as powerful as the Backus normal form, an algabraic notation used in Computer Science to represent numerical and other patterns by letters.

It is my contention that because of the scientific nature of the method of pronunciation of the vowels and consonents in the Indian languages (specially those coming directly from Pali, Prakit and Sanskrit), every part of the mouth is exercised during speaking. This results into speakers of Indian languages being able to pronounce words from any language. This is unlike the case with say native English speakers, as their tongue becomes unused to being able to touch certain portions of the mouth during pronunciation, thus giving the speakers a hard time to speak certain words from a language not sharing a common ancestry with English. I am not aware of any theory in these lines, but I would like to know if there is one.

Invention of Zero

Although ancient Babylonians were known to have used what is often called "place holders" to distinguish between numbers like 809 and 89, they were nothing more than blank spaces or at times two wedge shapes like ". More importantly, they lacked the realization that zero has a place in the number system as well as it comes with a baggage of abstract interpretations. Hence, while they can be credited with intelligently solving a practical problem of avoiding misinterpreting two numbers like 809 and 89, they can hardly be credited with the invention of the complex notion of zero and the even more complex notion of the abstract idea of "nothingness".

The ancient Greeks were beginning their contributions to mathematics around the time zero as an empty place holder was being popularized by Babylonian mathematicians. The Greeks did not adopt what is called a positional number system, a system that gave a value to a number because of its relative position in the set of numerals. This is because the Greeks' achievements were based on geometry. This resulted into firstly, Greeks relating numbers with lengths of line segments, and secondly, decoupling numbers from any potential abstract interpretations. It is commonly thought that in Greek society numbers that required to be "named" were not used by mathematician- philosophers, but by merchants and hence no clever notation was needed. Thus even the eminent mathematician like Ptolemy used the then recent place holder "zero" more as a punctuation mark than any serious numeral. Although a few Greek astronomers began using the symbol "O", the symbol more familiar to us now, to denote place holders, zero was not thought of as a number by the Greeks.

The first notions of zero as a number and its uses have been found in ancient Mathematical treatise from India and thus India is correctly related to the immensely important mathematical discovery of the numeral zero. This concept, combined with the place-value system of enumeration, became the basis for a classical era renaissance in Indian mathematics. Indians began using zero both as a number in the place-value system of numerals as well as to denote an empty place (place holder). Obviously, the use as a number came later. Aryabhata devised a number system what has no zero yet a positional number system. There is however, evidence that z dot has been used in earlier Indian manuscripts to denote an empty position. Also contemporary Indian scriptures also tend to use zero in places where unknown values are registered, where we would use x. Later Indian mathematicians had names for zero, but no symbol for it. Aryabhata used the word "kha" for position and it was also used later as the name for zero.

The oldest known text to use zero is an Indian (Jaina) text entitled the Lokavibhaaga ("The Parts of the Universe"), which has been definitely dated to 25 August 458 BC [See 4] An inscription, created in 876AD, found in Gwalior, acts as the first use of zero as a number. Zero is not a "natural" candidate for being a number. It is a great leap from physical to abstract that one needs to bridge when dealing with zero. With zero also comes the notion of negative numbers and along with all these comes a series of related questions about arithmetic operations on natural numbers, both positive and negative and zero.

The development of the notion of zero began, in my opinion, when mathematicians tried to answer these questions. Three Hindu mathematicians, Brahmagupta, Mahavira and Bhaskara tried to answer these in their treatise. In the 7th century Brahmagupta attempted to provide rules for addition and subtraction involving zero. The sum of zero and a negative number is negative, the sume of a positive number and zero if positive, the sum of zero and zero is zero. A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is nagative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.

Brahmagupta also says that any number multiplied by zero is zero. But problems arise when he tries to explain division. While he is unsure about what division of a number by zero means, he wrongly gives zero divided by zero to be zero. Brahmagupta's is the first attempt from any mathematician to explain the arithmetic operations on natural numbers and zero.

In the 9th century, Mahavira updated Brahmagupta's attempts at defining operations using zero. Although he correctly finds out that a number multiplied by zero is zero, but wrongly says that a number remains unchanged when divided by zero.

The next valiant attempt came from Bhaskara in the 11th century. Division of zero still remained an illusive mystery. 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.

This, in its face value seems correct, by suggesting that any number when divided by zero is infinity, Bhaskara suggeted that zero multiplied by infinity is any number, and hence all numbers are equal, which is not correct. But Bhaskara did correctly find out that the square of zero is zero, as is the square root. The Indian numeral system and its place value, decimal system of enumeration came to the attention of the Arabs in the seventh or eighth century, and served as the basis for the well known advancement in Arab mathematics, represented by figures such as al-Khwarizmi. Al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning that described the Indian place-value system of numerals based on numerals 1 through 9 and 0. Scholars like ibn Ezra and al-Samawal used the notion of zero from al-Khwarizmi's work. In the 12th century al- Samawal extended arithmetic operations using zero as follows.

If we subtract a positive number from zero the same negative number remains, ... if we subtract a negative number from zero the same positive number remains.

Zero also reached eastwards from India to China, where Chinese scholars Chin Chiu-Shao and Chu Shih-Chieh made use of the same symbol O for a places-based system in the 12th and 13th centuries respectively. From the time of Han (206 to 220 BC), Chinese scholars used a place-value system called the suan zi ("calculation using rods") that was a regular system that used horizontal and vertical lines that used to denote the nine numerals. Ifrah says that "Thus one could be forgiven for assuming that following the links established between India and China at the beginning of the beginning of the first millennium BC, Indian scholars were influenced by Chinese mathematicians to create their own system in an imitation of the Chinese counting method." [See 4] He goes on to argue that in suan zi, the zero appeared at a much later date. Thus the notion of zero helps one to recognize the originality of the Indian mathematicians vis-a-vis their Chinese counterparts. Ifra also establishes that the Chinese scholars overcame the difficulties the absence of zeros caused in trying to represent numbers like 1,270,000 often either using characters of their ordinary counting system (a non-positional system that did not require the use of a zero) or simply by empty spaces. After providing a sequence of clues, [in 4], Ifrah continues "It was only after the eighth century BC, and doubtless due to the influence of the Indian Buddhist missionaries, that Chinese mathematicians introduced the use of zero in the form of a little circle or dot (signs that originated in India),...".

Zero reached Europe in the twelfth century when Adelard of Bath translated al-Khwarizmi's works into Latin [See 1]. Fibonacci was one of the main mathematicians who accepted the concepts of zero in Europe. He was an important link between the Hindu-Arabic number system. In his treatise Liber Abaci ("a tract about the abacus"), published in 1202, he described the nine Indian symbols together with the symbol O for zero, but it was not widely accepted until much later.

Significantly, Fibonacci spoke of numbers 1 through 9, but a "sign" O. Although he brought the notion of zero to Europe, it is clear that he was not able to reach the sophistication of Indians like Brahamagupta, Mahavira and Bhaskara, nor of the Arabic mathematicians like al-Samawal. The Europeans were at first resistant to this system, being attached to the far less logical Roman numeral system (notably the Romans never propounded the idea of zero), but their eventual adoption of this system arguably led to the scientific revolution that began to sweep Europe beginning by the middle of the second millennium. However, it was not until the 17th century that zero found widespread acceptance through a lot of resistance.

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