Fwd: A history of zero

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hinducivilization , " S.Kalyanaraman " <kalyan97 wrote:

 

A history of Zero

 

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One of the commonest questions which the readers of this archive ask

is: Who discovered zero? Why then have we not written an article on

zero as one of the first in the archive? The reason is basically

because of the difficulty of answering the question in a

satisfactory form. If someone had come up with the concept of zero

which everyone then saw as a brilliant innovation to enter

mathematics from that time on, the question would have a

satisfactory answer even if we did not know which genius invented

it. The historical record, however, shows quite a different path

towards the concept. Zero makes shadowy appearances only to vanish

again almost as if mathematicians were searching for it yet did not

recognise its fundamental significance even when they saw it.

 

The first thing to say about zero is that there are two uses of zero

which are both extremely important but are somewhat different. One

use is as an empty place indicator in our place-value number system.

Hence in a number like 2106 the zero is used so that the positions

of the 2 and 1 are correct. Clearly 216 means something quite

different. The second use of zero is as a number itself in the form

we use it as 0. There are also different aspects of zero within

these two uses, namely the concept, the notation, and the name. (Our

name " zero " derives ultimately from the Arabic sifr which also gives

us the word " cipher " .)

 

Neither of the above uses has an easily described history. It just

did not happen that someone invented the ideas, and then everyone

started to use them. Also it is fair to say that the number zero is

far from an intuitive concept. Mathematical problems started

as 'real' problems rather than abstract problems. Numbers in early

historical times were thought of much more concretely than the

abstract concepts which are our numbers today. There are giant

mental leaps from 5 horses to 5 " things " and then to the abstract

idea of " five " . If ancient peoples solved a problem about how many

horses a farmer needed then the problem was not going to have 0 or -

23 as an answer.

 

One might think that once a place-value number system came into

existence then the 0 as an empty place indicator is a necessary

idea, yet the Babylonians had a place-value number system without

this feature for over 1000 years. Moreover there is absolutely no

evidence that the Babylonians felt that there was any problem with

the ambiguity which existed. Remarkably, original texts survive from

the era of Babylonian mathematics. The Babylonians wrote on tablets

of unbaked clay, using cuneiform writing. The symbols were pressed

into soft clay tablets with the slanted edge of a stylus and so had

a wedge-shaped appearance (and hence the name cuneiform). Many

tablets from around 1700 BC survive and we can read the original

texts. Of course their notation for numbers was quite different from

ours (and not based on 10 but on 60) but to translate into our

notation they would not distinguish between 2106 and 216 (the

context would have to show which was intended). It was not until

around 400 BC that the Babylonians put two wedge symbols into the

place where we would put zero to indicate which was meant, 216 or

21 '' 6.

 

The two wedges were not the only notation used, however, and on a

tablet found at Kish, an ancient Mesopotamian city located east of

Babylon in what is today south-central Iraq, a different notation is

used. This tablet, thought to date from around 700 BC, uses three

hooks to denote an empty place in the positional notation. Other

tablets dated from around the same time use a single hook for an

empty place. There is one common feature to this use of different

marks to denote an empty position. This is the fact that it never

occured at the end of the digits but always between two digits. So

although we find 21 '' 6 we never find 216 ''. One has to assume

that the older feeling that the context was sufficient to indicate

which was intended still applied in these cases.

 

If this reference to context appears silly then it is worth noting

that we still use context to interpret numbers today. If I take a

bus to a nearby town and ask what the fare is then I know that the

answer " It's three fifty " means three pounds fifty pence. Yet if the

same answer is given to the question about the cost of a flight from

Edinburgh to New York then I know that three hundred and fifty

pounds is what is intended.

 

We can see from this that the early use of zero to denote an empty

place is not really the use of zero as a number at all, merely the

use of some type of punctuation mark so that the numbers had the

correct interpretation.

 

Now the ancient Greeks began their contributions to mathematics

around the time that zero as an empty place indicator was coming

into use in Babylonian mathematics. The Greeks however did not adopt

a positional number system. It is worth thinking just how

significant this fact is. How could the brilliant mathematical

advances of the Greeks not see them adopt a number system with all

the advantages that the Babylonian place-value system possessed? The

real answer to this question is more subtle than the simple answer

that we are about to give, but basically the Greek mathematical

achievements were based on geometry. Although Euclid's Elements

contains a book on number theory, it is based on geometry. In other

words Greek mathematicians did not need to name their numbers since

they worked with numbers as lengths of lines. Numbers which required

to be named for records were used by merchants, not mathematicians,

and hence no clever notation was needed.

 

Now there were exceptions to what we have just stated. The

exceptions were the mathematicians who were involved in recording

astronomical data. Here we find the first use of the symbol which we

recognise today as the notation for zero, for Greek astronomers

began to use the symbol O. There are many theories why this

particular notation was used. Some historians favour the explanation

that it is omicron, the first letter of the Greek word for nothing

namely " ouden " . Neugebauer, however, dismisses this explanation

since the Greeks already used omicron as a number - it represented

70 (the Greek number system was based on their alphabet). Other

explanations offered include the fact that it stands for " obol " , a

coin of almost no value, and that it arises when counters were used

for counting on a sand board. The suggestion here is that when a

counter was removed to leave an empty column it left a depression in

the sand which looked like O.

 

Ptolemy in the Almagest written around 130 AD uses the Babylonian

sexagesimal system together with the empty place holder O. By this

time Ptolemy is using the symbol both between digits and at the end

of a number and one might be tempted to believe that at least zero

as an empty place holder had firmly arrived. This, however, is far

from what happened. Only a few exceptional astronomers used the

notation and it would fall out of use several more times before

finally establishing itself. The idea of the zero place (certainly

not thought of as a number by Ptolemy who still considered it as a

sort of punctuation mark) makes its next appearance in Indian

mathematics.

 

The scene now moves to India where it is fair to say the numerals

and number system was born which have evolved into the highly

sophisticated ones we use today. Of course that is not to say that

the Indian system did not owe something to earlier systems and many

historians of mathematics believe that the Indian use of zero

evolved from its use by Greek astronomers. As well as some

historians who seem to want to play down the contribution of the

Indians in a most unreasonable way, there are also those who make

claims about the Indian invention of zero which seem to go far too

far. For example Mukherjee in [6] claims:-

 

.... the mathematical conception of zero ... was also present in the

spiritual form from 17 000 years back in India.

 

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 = & #8734;. 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 & #8734; 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 & #8730;0 = 0.

 

Perhaps we should note at this point that there was another

civilisation which developed a place-value number system with a

zero. This was the Maya people who lived in central America,

occupying the area which today is southern Mexico, Guatemala, and

northern Belize. This was an old civilisation but flourished

particularly between 250 and 900. We know that by 665 they used a

place-value number system to base 20 with a symbol for zero. However

their use of zero goes back further than this and was in use before

they introduced the place-valued number system. This is a remarkable

achievement but sadly did not influence other peoples.

 

The brilliant work of the Indian mathematicians was transmitted to

the Islamic and Arabic mathematicians further west. It came at an

early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of

Reckoning which describes the Indian place-value system of numerals

based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first

in what is now Iraq to use zero as a place holder in positional base

notation. Ibn Ezra, in the 12th century, wrote three treatises on

numbers which helped to bring the Indian symbols and ideas of

decimal fractions to the attention of some of the learned people in

Europe. The Book of the Number describes the decimal system for

integers with place values from left to right. In this work ibn Ezra

uses zero which he calls galgal (meaning wheel or circle). Slightly

later in the 12th century al-Samawal was writing:-

 

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.

 

The Indian ideas spread east to China as well as west to the Islamic

countries. In 1247 the Chinese mathematician Ch'in Chiu-Shao wrote

Mathematical treatise in nine sections which uses the symbol O for

zero. A little later, in 1303, Zhu Shijie wrote Jade mirror of the

four elements which again uses the symbol O for zero.

 

Fibonacci was one of the main people to bring these new ideas about

the number system to Europe. As the authors of [12] write:-

 

An important link between the Hindu-Arabic number system and the

European mathematics is the Italian mathematician Fibonacci.

 

In Liber Abaci he described the nine Indian symbols together with

the sign 0 for Europeans in around 1200 but it was not widely used

for a long time after that. It is significant that Fibonacci is not

bold enough to treat 0 in the same way as the other numbers 1, 2, 3,

4, 5, 6, 7, 8, 9 since he speaks of the " sign " zero while the other

symbols he speaks of as numbers. Although clearly bringing the

Indian numerals to Europe was of major importance we can see that in

his treatment of zero he did not reach the sophistication of the

Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and

Islamic mathematicians such as al-Samawal.

 

One might have thought that the progress of the number systems in

general, and zero in particular, would have been steady from this

time on. However, this was far from the case. Cardan solved cubic

and quartic equations without using zero. He would have found his

work in the 1500's so much easier if he had had a zero but it was

not part of his mathematics. By the 1600's zero began to come into

widespread use but still only after encountering a lot of

resistance.

 

Of course there are still signs of the problems caused by zero.

Recently many people throughout the world celebrated the new

millennium on 1 January 2000. Of course they celebrated the passing

of only 1999 years since when the calendar was set up no year zero

was specified. Although one might forgive the original error, it is

a little surprising that most people seemed unable to understand why

the third millennium and the 21st century begin on 1 January 2001.

Zero is still causing problems!

 

Books

 

R Calinger, A conceptual history of mathematics (Upper Straddle

River, N. J., 1999).

G Ifrah, From one to zero : A universal history of numbers (New

York, 1987).

G Ifrah, A universal history of numbers : From prehistory to the

invention of the computer (London, 1998).

G G Joseph, The crest of the peacock (London, 1991).

R Kaplan, The nothing that is : a natural history of zero (London,

1999).

R Mukherjee, Discovery of zero and its impact on Indian mathematics

(Calcutta, 1991).

Articles:

 

 

S Giuntini, A discussion concerning the nature of zero and the

relation between imaginary and real numbers (Italian), Boll. Storia

Sci. Mat. 4 (1) (1984), 25-63.

R C Gupta, Who invented the zero?, Ganita-Bharati 17 (1-4) (1995),

45-61.

P Mäder, " Wie die Puppe ein Adler sein wollte, der Esel ein Löwe,

die Äffin eine Königin - so wollte die Null eine Ziffer sein! " Ein

Überblick zur Geschichte der Zahl Null, in Jahrbuch Überblicke

Mathematik, 1995 (Braunschweig, 1995), 39-64.

R N Mukherjee, Background to the discovery of the symbol for zero,

in Proceedings of the Symposium on the 1500th Birth Anniversary of

Aryabhata I, New Delhi, 1976, Indian J. Hist. Sci. 12 (2) (1977),

225-231.

K Muroi, The expressions of zero and of squaring in the Babylonian

mathematical text VAT 7537, Historia Sci. (2) 1 (1) (1991), 59-62.

L Pogliani, M Randic and N Trinajstic, Much ado about nothing - an

introductive inquiry about zero, Internat. J. Math. Ed. Sci. Tech.

29 (5) (1998),729--744.

S Ursini Legovich, The origin of the zero in Central American

civilization. Comparative analysis with the Hindu case (Spanish),

Mat. Enseñanza No. 13 (1980), 7-20.

M Ja Vygodskii, L'origine du signe de zéro dans la numération

babylonienne (Russian), Istor.-Mat. Issled. 12 (1959), 393-420.

 

Article by: J J O'Connor and E F Robertson

 

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html

 

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