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 0
2 = 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 12
th
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 12
th 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!
