Different Numeral

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Different Numeral Systems In History
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• I choose the topic Different Numeral Systems In
  History because I thought it would be
  interesting to try and find a bit about how we
  got our numeral system and find out about a few
  others too. There are many different numeral
  systems but I chose the 5 systems in this project
  because they looked interesting because of the
  interesting information I found and strange way
  their symbols were wrote.
• On the wall I have displayed 2 systems, the
  Mayan System that is wrote from 0-10 and the
  Arabic System that is wrote from 0-9.
• The Numeral System I chose for this project are
• - Babylonian
• - Mayan
• - Egyptian
• Indian
• - Arabic

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Babylonian Numerals
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The Babylonian civilization in Mesopotamia
replaced the Sumerian civilization and the
Akkadian civilization. In terms of their number
system the Babylonians got the ideas from the
Sumerians and from the Akkadians. From the number
systems of these two earlier civilizations came
the bases of the 60 numeral, that is the
sexagesimal system. Yet neither the Sumerian or
the Akkadian system was a positional system this
advancement by the Babylonians, was undoubtedly
their greatest achievement in terms of developing
the number system.

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Some would say that it was their biggest
achievement in mathematics. The Babylonian number
system went up to 60 and when people first heard
this their reaction was what a lot of special
number symbols they must have had to learn. This
comment was based on knowledge of our own decimal
system which is a system with nine special
symbols and a zero symbol. Instead of having to
learn 10 symbols like we do to use our numbers,
the Babylonians only had to learn two symbols to
produce their 60 number system. Even though
the Babylonian system was a 60 number system, it
had some signs of a 10 number system within it.
This is because the 59 numbers, which go into one
of the places of the system, were built from a
'unit' symbol and a 'ten' symbol.
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Here are the 59 symbols built from two symbols
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Why did the Babylonians have a number system
with only 60 numbers? The easiest answer to
this is that they inherited the 60 number system
from the Sumerians but that is not a good answer.
It only leads to the question why the Sumerians
used 60 number system . First, we dont have to
go back further the the Sumerians because it is
fairly certain that the sexagesimal system was
originated by the Sumerians. The second point is
that modern mathematicians were not the first to
ask these questions. Theon of Alexandria tried to
answer this question in the fourth century AD and
many historians of mathematics such as Neugebauer
and Moritz Cantor have offered an opinion since
then without any coming up with a really
convincing answer. There were many different
theorys created based on many different types of
mathematics
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Egyptian Numerals
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The Egyptians had a bases 10 system of
hieroglyphs for numerals. This means that they
had separate symbols for one unit, ten, one
hundred, one thousand, ten thousand, hundred
thousand, and one million.
Numeral Hieroglyphs
To make up the number 276, for example, fifteen
symbols are required two "hundred" symbols,
seven "ten" symbols, and six "unit" symbols. The
numbers appears as this
276 in hieroglyphs.
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I would like to point out that the hieroglyphs
did not remain the same throughout the two
thousand or so years of the ancient Egyptian
civilization. This civilisation is often broken
down into three distinct periods Old Kingdom -
around 2700 BC to 2200 BCMiddle Kingdom - around
2100 BC to 1700 BCNew Kingdom - around 1600 BC
to 1000 BC Numeral hieroglyphs were somewhat
different in these different periods, yet
retained a broadly similar style.
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Another number system, which the Egyptians used
after the invention of writing on papyrus, was
composed of hieratic numerals. These numerals
allowed numbers to be written in a far more
compact form. Yet using the system required many
more symbols to be learned. There were separate
symbols for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20,
30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400,
500, 600, 700, 800, 900, 1000, 2000, 3000, 4000,
5000, 6000, 7000, 8000, 9000
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Hieratic Numerals
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With this new system numbers could be formed of
a few symbols. The number 9999 had just 4
hieratic symbols instead of 36 hieroglyphs. One
major difference between the hieratic numerals
and our own number system was the hieratic
numerals did not form a positional system so the
particular numerals could be written in any
order.
Here is one way the Egyptians wrote 2765 in
hieratic numerals
Here is a second way of writing 2765 in hieratic
numerals with the order reversed
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Like the hieroglyphs, the hieratic symbols
changed over time but they underwent more changes
with six distinct periods. Initially the symbols
that were used were quite close to the
corresponding hieroglyph but their form diverged
over time. The versions I have given of the
hieratic numerals date from around 1800 BC. The
two systems ran in parallel for around 2000 years
with the hieratic symbols being used in writing
on papyrus, as for example in the Rhind papyrus
and the Moscow papyrus, while the hieroglyphs
continued to be used when carved on stone.
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Mayan Numerals
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A common culture, calendar, and mythology held
the civilisation together and astronomy played an
important part in the religion which underlay the
whole life of the people. Of course astronomy and
calendar calculations require mathematics and
indeed the Maya constructed a very sophisticated
number system. We do not know the date of these
mathematical achievements but it seems certain
that when the system was devised it contained
features which were more advanced than any other
in the world at the time.
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Mayan Numerals
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People are almost certain that the reason for a
20 numeral system arose from ancient people who
counted on both their fingers and their toes.
Although it was a base 20 system, called a
vigesimal system, one can see how five plays a
major role, again clearly relating to five
fingers and toes. In fact it is worth noting that
although the system is base 20 it only has three
number symbols (perhaps the unit symbol arose
from a pebble and the line symbol from a stick
used in counting). Often people say how
impossible it would be to have a number system to
a large base since it would involve remembering
so many special symbols. This shows how people
are conditioned by the system they use and can
only see variants of the number system in close
analogy with the one with which they are
familiar. Surprising and advanced features of the
Mayan number system are the zero. The Mayan
Numeral System is probably the most easiest the
understand that I have researched.
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Indian Numerals
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We certainly know that today's symbols took on
forms close to what they use to have in Europe in
the 15th century. It was the advent of printing
which motivated the standardisation of the
symbols. However we must not forget that many
countries use symbols today which are quite
different from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and
unless one learns these symbols they are totally
unrecognisable as for example the Greek alphabet
is to someone unfamiliar with it. One of the
important sources of information which we have
about Indian numerals comes from al-Biruni.
During the 1020s al-Biruni made several visits to
India. Before he went there al-Biruni already
knew of Indian astronomy and mathematics from
Arabic translations of some Sanskrit texts. In
India he made a detailed study of Hindu
philosophy and he also studied several branches
of Indian science and mathematics.
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Historians trace the various symbols for
numerals which al-Biruni saw back to the Brahmi
numerals which came into being around the middle
of the third century BC. Now these Brahmi
numerals were not just symbols for the numbers
between 1 and 9. The situation is much more
complicated for it was not a place-value system
so there were symbols for many more numbers. Also
there were no special symbols for 2 and 3, both
numbers being constructed from the symbol for 1.

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The Brahmi numerals have been found in
inscriptions in caves and on coins in regions
near Poona, Bombay, and Uttar Pradesh. Dating
these numerals tells us that they were in use
over quite a long time span up to the 4th century
AD. Of course different inscriptions differ
somewhat in the style of the symbols.
Here is one style of the Brahmi numerals
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We will look at a number of different theories
concerning the origin of the Brahmi numerals.
There is no problem in understanding the
symbols for 1, 2, and 3. However the symbols for
4, ... , 9 appear to us to have no obvious link
to the numbers they represent. There have been
quite a number of theories put forward by
historians over many years about the origin of
these numerals. In Ifrah lists is a number of the
hypotheses which have been put forward.

• The Brahmi numerals came from the Indus valley
  culture of around 2000 BC.
• The Brahmi numerals came from Aramaean numerals.
• The Brahmi numerals came from the Karoshthi
  alphabet.
• The Brahmi numerals came from the Brahmi
  alphabet.
• The Brahmi numerals came from an earlier
  alphabetic numeral system, possibly due to
  Panini.
• The Brahmi numerals came from Egypt.

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Ifrah examines each of the six hypotheses in
turn and rejects them, although one would have to
say that in some cases it is more due to lack of
positive evidence rather than to negative
evidence. So, Ifrah proposed his own theory
... the first nine Brahmi numerals constituted
the vestiges of an old indigenous numerical
notation, where the nine numerals were
represented by the corresponding number of
vertical lines ... To enable the numerals to be
written rapidly, in order to save time, these
groups of lines evolved in much the same manner
as those of old Egyptian Pharonic numerals.
Taking into account the kind of material that was
written on in India over the centuries (tree bark
or palm leaves) and the limitations of the tools
used for writing (calamus or brush), the shape of
the numerals became more and more complicated
with the numerous ligatures, until the numerals
no longer bore any resemblance to the original
prototypes.
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This was a nice theory, and it could be true,
but there seems to be absolutely no positive
evidence in its favour. The idea is that they
evolved from
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The Brahmi numerals developed into the Gupta
symbols between the early 4th century AD to the
late 6th century AD.
The Gupta numerals evolved into the Nagari
numerals, sometimes called the Devanagari
numerals. This form developed from the Gupta
numerals beginning around the 7th century AD and
continued to develop from the 11th century
onward.
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Arabic Numerals
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Indian numerals form the basis of the European
number systems which are now widely used. The
numbers didnt go directly from India to Europe
but rather came first to the Arabic/Islamic
peoples and from them to Europe. The story of
this transmission is not, however, a simple one.
The eastern and western parts of the Arabic world
both saw separate developments of Indian numerals
with relatively little interaction between the
two. By the western part of the Arabic world we
mean the regions comprising mainly North Africa
and Spain. Transmission to Europe came through
this western Arabic route, coming into Europe
first through Spain.
The first sign that the Indian numerals were
moving west was in 662 AD
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At first the Indian methods were used by the
Arabs with a dust board. In fact in the western
part of the Arabic world the Indian numerals came
to be known as Guba (or Gubar or Ghubar) numerals
from the Arabic word meaning "dust". A dust board
was used because the arithmetical methods
required the moving of numbers around in the
calculation and rubbing some out some of them as
the calculation proceeded. The dust board allowed
this in the same sort of way that one can use a
blackboard, chalk and a blackboard eraser. Any
student who has attended lectures where the
lecturer continually changes and replaces parts
of the mathematics as the demonstration
progresses will understand the disadvantage of
the dust board!
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We have seen in the article Indian numerals that
the form of the numerals themselves varied in
different regions and changed over time. Exactly
the same happened in the Arabic world. Here
is an example of an early form of Indian numerals
being used in the eastern part of the Arabic
empire. It comes from a work of al-Siizi, not an
original work by him but rather the work of
another mathematician which al-Siizi copied at
Shiraz and dated his copy 969.
The numerals from al-Sizji's treatise of 969
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The numerals had changed their form about 100
years later when a copy of one of al-Birunis
astronomical texts was made. Here are the
numerals as they appear in a 1082 copy.
The numerals from al-Biruni's treatise copied
in 1082
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A closer look will show that between 969 and
1082 the biggest change in the numerals was the
fact that the 2 and the 3 have been rotated
through 90 . There is a reason for this change
which came about because of the way that scribes
wrote, for they wrote on a scroll which they
wound from right to left across their bodies as
they sat cross-legged. The scribes therefore,
instead of writing from right to left (the
standard way that Arabic was written) wrote in
lines from top to bottom. The script was rotated
when the scroll was read and the characters when
then in the correct orientation.
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Perhaps because scribes did not have much
experience at writing Indian numerals, they wrote
2 and 3 the correct way round instead of writing
them rotated by 90 so that they would appear
correctly when the scroll was rotated to be read.

Here is an example of what the scribe should
write
and here is what the scribe actually wrote
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The form of the numerals in the west of the
Arabic empire look more familiar to those using
European numerals today which is not surprising
because it was this form of the Indian number
system that reached Europe.
al-Banna al-Marrakushi's form of the numerals