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| Appendix | Bibliography | The Architects Plan | Cubit | Links | Zero
History pg. 1 A Brief History of Zero,
"in
the next two pages of my Zero Index", is needed as to
understand how an understanding of zero in relation to our our past will be
needed in order to better understand and affirm;
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 didn't 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 didn't recognize 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 a 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. Neither of these 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 a 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 un-baked 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 a 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 subtler that 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 didn't
need to name their numbers since they worked with numbers as lengths of
lines. Merchants, not mathematicians, used Numbers, which required to be
named for records, 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 favor 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 placeholder 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 it.
The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered 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 placeholder 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 that 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:- Positive or
negative numbers when divided by zero is a fraction the zero as denominator.
Zero divided by negative or positive numbers 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 to try 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 = 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 placeholder 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, Chu Shih-Chieh 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:-
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 21-century begin on 1 January 2001. Zero is still causing problems! Article by: J J O'Connor and E F Robertson BI666 New Genesis Links Below
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. At first sight we might be tempted to believe
that 
0 = 0.
