The Journey of ZERO

                                                     The Journey of Zero

“THE importance of the creation of the zero mark can never be exaggerated .This which gives us airy nothing not merely a local habitation, a name, a picture, a symbol but helped power is characteristics of the Hindu race from which it sprang .It is like coining the NIRVANA into the dynamos .No single mathematical creation has been ever patent for the general on go of intelligence and power.

                                                                                                --G B Halsted

We all know 1+1=2, 2+1=3, 3+1 = 4, 4+1= 5, and 5+1 =6 and so on … but think if we reverse the process  then, 6-1 =5 ,5-1 =4, 4-1 =3 ,3-1 =2 ,2-1 =1 and what about 1-1 ? If we talk now then certainly it is ZERO. But think for a while was it so simple to answer in the early time?

If you are said to differentiate between 2403 and 243 you can clearly answer it because the discovery of place value system has made it possible. Before the introduction of place value system writing big-big numbers were a problem. It does not mean that the early civilization did not have any place value system. The Babylonian though had a positional system but the concept of zero was not fully developed. In the sexagesimal system used by the Babylonian 1, 25 could mean 1 x 60 +25 = 85 and 1,0,25 could mean 1 x60² + 25 = 3625, but it was quite confusing and there was a greater chance of misunderstanding because they used to leave a gap to indicate zero. Around 200BC the Babylonian had developed a symbol for zero to denote the absence of a figure but this was not used in calculation.

According to Boyer, "The Babylonians seem at first to have had no clear way in which to indicate an "empty" position--that is, they did not have a zero symbol, although they sometimes left a space where a zero was intended. ... By about the time of the conquest by Alexander the Great, however, a special sign, consisting of two small wedges placed obliquely, was invented to serve as a placeholder where a numeral was missing. ... The Babylonian zero symbol apparently did not end all ambiguity, for the sign seems to have been used for intermediate empty positions only. There are no extant tablets in which the zero sign appears in a terminal position. This means that the Babylonians in antiquity never achieved an absolute positional system.

                                                                     The Babylonian zero

Ptolemy in his Almagest written around 130AD used the Babylonian sexagesimal system and also 0 as the empty place holder. The Greek texts of 500 AD had the symbol O for zero taken from the Greek word ouden meaning nothing. The Mayans were the first to fully utilize the principle of place value system. The Maya civilization as early as 1^st century BC used   for zero (which looks like a half-closed eye) but it did not spread beyond Mesoamerica.

As far as the modern zero concept is concerned it is purely Indian. As far back as 200BC we get the earliest known reference to zero in Acharya Pingla’s book Chanda-Shastra where he writes “In Gayatri chanda one pada has six letters when the number is made half it becomes three. Remove one from it and make it half to get one .Remove one from it thus gets the zero.”

         In mathematical words

                                                1/2{1/2(6)-1}-1=0

Aryabhatta (476AD), a great Indian mathematician and astronomer, had devised a number system without zero though it was a positional system. He used the word kha for the positional system which in the later phase become a name for zero. Another renowned mathematician of India Varahmihir (505AD-587AD) in his famous book Pancasiddhantika written in 575AD had many a times used the place value system containing zero which is believed to be fairly proven fact for the use of zero for the first time.

It was the Indian mathematician Brahmagupta who in his book Brahma-Sputa-Siddhanta defined zero, which he wrote at the age of 30 i.e. in 628 AD where he writes

a-a = 0 (in modern symbol).

In the Khmer inscription of 683AD found at Trapeang Bay in Cambodia a heavy point appears for zero where as in an inscription found at Sumatra in 683AD the small circle appears for zero.

The first record of Indian use of zero can be found on the inscription on a stone tablet at the town of Gwalior of King Jayavardhana II written around 876AD, where the number 270 and 50 are written as we write today; the only difference is that the 0 is smaller and slightly raised. The description is as follows:-

Rama Rock showing 0 in 270

They have planted a garden 187 by270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple.

 In Bakshali Manuscript (300AD) some numbers are found with zeros with heavy dots.    

                                                            The Bakshali Manuscript

Laplace truly said-

‘The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.’

The name for zero:

The Hindu name for zero used in inscription and manuscript to indicate the blank position was Sunya, meaning void or empty. The Arabic text used the word Sifr meaning vacant.Abraham ben Meir ibn Ezra (1092-1167) used galgal for zero in a description on a decimal system of numeration. In the Latin translation of Arabic text it was called zephirum. Dionysius Exiguus used the word nulla to indicate zero in his Easter tables. In Fibonacci’s Liber Abaci zephirum for zero was used where as Italian used zeuro. Cipher is also found in English as early as 1399. The first printed treatise on zero De arithmetrica opusculum was written in 1491 by Philippus Calandrus.  Maximus Planudes called it tziphra and this was used by Fine (1530) in 16^th century.

 In the further journey zero was given the name such as zepiro, cifra which finally led the foundation of the word    zero. The modern symbol of zero by 0 might have been taken from the Greek name omicron used by Buteo in 1559.

ACCORDING TO AN AMERICAN MATHEMATICAL MAGAZINE:

“IT is humorously said   The Hindu contributed nothing to mathematics .In the whole history of mathematics there has been no more a revolutionary step than the one which the HINDU took when they invented zero.”

It is crystal clear that there is no trace found who exactly discovered zero but every mathematician in the world agrees on the fact that it is a Hindu symbol and it had its origin in India and the rest of the world adopted it in due course. This symbol has not only made writing the big numbers easy but one can clearly state the positional value of a number.

In 80205 there are two zeros but we can differentiate between them as the first zero represents the tens position and the second zero represents the thousand position and the whole number can be read as eighty thousand two hundred and five.

So whether we call zero as sunyam, cipher, or give any name to it, as long as mathematics exists there is no harm in accepting the fact that this Hindu symbol has brought the revolution. The computer understand only binary system (0, 1), even in the circuit problem, or in Boolean algebra zero has created the history.

Operation of zero 

The Indian mathematician Brahmagupta, born in Ujjain in 598 AD in his book Brahmasphutasiddhanta, tells about the following operation:-

1.      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.

2.      A negative minus zero is a negative. A positive minus zero is a positive .Zero minus zero is a zero. A negative subtracted from zero is a positive. A positive subtracted from zero is a negative.

3.      The product of zero multiplied by a negative or positive is zero. The product of zero multiplied by zero is zero.

4.      Zero divided by zero is zero

 

In modern mathematical notation the above laws can be written as

 i) 0+ (-a) = -a, a+0 =a, 0+0 =0

 ii) (-a)-0= -a, a-0 =a, 0-0 =0, 0-(-a) = a, 0- (a) =-a

iii) 0 x (±a) = 0, 0 x 0 = 0

iv) 0/0 =0

Though the fourth law is wrong but it was indeed a brilliant attempt on part of Brahmagupta to give the rule of mathematical operation of zero to the world.  

In the Sanskrit text written below Mahivira an Indian mathematician born around 800AD writes about the operation of zero in his Ganita Sara Samgraha.

rfM+r% [ksu jkf’k% [ka  lks·fodkjh ârks ;qr%A

ghuks·fi [ko/kkfn% [ka ;ksxs [ka ;ksT; :ide~ AA

 Tarita khen Rashi khä sobikkari hato yutä

Hinopi khabdhadhí khä yoge khä yojya rupkam

When any number is multiplied by any zero the result is zero and when any number is divided by zero the result is zero or when zero is added to or zero is subtracted from any number the result is always the same.

a+0=a                          a-0=a                           a x 0=0                        a/0= 0

He is wrong here in saying that a/0 =0. 

In his book Trisatika Sridhar writes “When zero is added to any number or zero is subtracted from any number the result does not change but when any number is multiplied with zero the result is always zero.

a  x 0=0, a+0 =a ,a-0 =a

Sridhar didn’t speak any thing about a number having denominator zero.

These brilliant efforts of Indian mathematicians were translated in Arabic text by al- Khwarizmi where he describes about the Indian place value system based on ten numerals and probably this was the first work where zero is used as the place value system.

Rajesh Kumar Thakur
rkthakur1974@gmail.com