History of Zero 
Zero is
known to have been discovered in three separate cultures at approximately the
same time – around 400 to 300 BC. The
causes were distinctly different.
One came
from a Hindu monk who meditated in the Himalayas’ in India and suddenly thought
as he looked at the stars – “what if there were none?” When he reported this to his fellow monks it
sparked a great philosophical consideration perhaps culminating with Jean Paul
Sartre’s “Being and Nothingness”.
At
almost this same time Alexander the Great was conquering all lands between
Greece and India. He was descended from Heracles and was tutored by Aristotle. He had the vision to impose taxation
(tribute) upon the conquered peoples (rather than just annihilate). This resulted in a very basic need for the
ability of subjects to say “I’ve paid your taxes – I owe you nothing.” Some believe this was also the first time
subtraction became popular as taxes were sometimes paid in increments.
The third (and
probably first to use) were the Mayans who developed several calendars. There was
a 365 day civil year, a 260 day religious year and, key to their invention of
zero, the complicated Long Count calendar which measured time from the start of
the Mayan civilization (August 12, 3113 B.C.) and completes a full cycle on
December 21, 2012. (http://www.mediatinker.com/blog/archives/008821.html 8.15.2011)
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 occurred 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 recognize 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.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
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:-
Subtraction
is a little harder:-
Brahmagupta then says that any number when multiplied by zero is zero
but struggles when it comes to division:-
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.
Bhaskara wrote over 500 years after Brahmagupta.
Despite the passage of time he is still struggling to explain division by zero.
He writes:-
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 12thcentury al-Samawal was
writing:-
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.