history of Zero
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Ancient Indian Mathematics index History Topics Index
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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 = . 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.
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.
You can see a separate article about Mayan mathematics.
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!
References (14 books/articles)
Article by: J J O'Connor and E F Robertson
----- Original Message -----
From: "He-Man" <[EMAIL PROTECTED]>
To: <wanita-muslimah@yahoogroups.com>
Sent: Sunday, November 20, 2005 8:26 AM
Subject: Re: [wanita-muslimah] Re: SI yang mana?
>
> Angka nol baru dikenal bangsa Arab setelah Muhammad bin Qasim menaklukkan
> Sind dan Punjab (india).
>
> ----- Original Message -----
> From: "asrh" <[EMAIL PROTECTED]>
> To: <wanita-muslimah@yahoogroups.com>
> Sent: Sunday, November 20, 2005 12:00 PM
> Subject: Re: [wanita-muslimah] Re: SI yang mana?
>
>
>> wah pak He-Man yang nemu angka nol justru orang Arab, Muhammad bin Ahmad,
>> he-he..
>> jangan anti dengan Arab deh, wong Qur'an itu bahasa Arab.
>> saya juga setuju dengan pak Satriyo, bukan Arabnya, bukan Islamnya, tapi
>> ummatnya, ya kite-kite ini :-)
>>
>
>
>
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