§4.1 Historical Numeration Systems
The earliest episodes from history make it
clear that the need for counting was the basis for the development of
In the first MGF course (1106) you were
introduced to the concept of sets - basically a collection of elements. A set
by itself may have no particular structure to it. But when we add ways of
combining the elements (aka operations) and ways of comparing the
elements (aka relations) we obtain a
We have the following formal definition then of a mathematical system.
system is made up of the following three things:
1) a set of
2) one of more operations for combining the elements
3) one or more relations for comparing the elements
For example let's consider the set of whole
numbers, i.e., 0, 1, 2, 3, . . .
Now let's apply a basic operation on this
set, say addition. We know many simple things involving this operations, like
1 + 3 = 4; 9 + 13 = 22; and so forth.
Now it is also true that 3 + 1 = 4, and
that 13 + 9 = 22. This is a relation this system has - we call it
So you can now have a general idea of
what we mean when we talk about a mathematical system and the three things it
is made up of.
If I were permitted the luxury of traveling
back in time, a sabbatical certainly to be celebrated by some of my colleagues
and students, indulging myself on a visit to any era in the history of man,
undoubtedly I would choose to see the spectacular Egyptian Empire of some
3500+ years ago.
This great civilization which emerged in the catch basin of
the Nile River brought order to chaos by replacing the hunter-gatherer way of
living with a more stable and sedentary lifestyle centered around permanent
villages supported by planned farming. This in turn created something never
before experienced by man – leisure time.
Although the majority of the population toiled in the fields
all day, the kings, priests, merchants, and scribes found time at the end of
the day to think about the mysteries of nature and science. They devised new
methods of communication, of managing their affairs, and a building prowess
unparalleled in its day as evident by the impressive monuments still standing
in the north desert regions of Africa.
All of this could not have developed without a method of
writing and calculating. Such a method is found in both the ancient Egyptian
hieroglyphics (which translates as sacred signs), and in the cursive
hand of the accounting scribes. In the tombs of the ancient pharaohs and on
the scrolls of papyrus these symbols have been found.
The hieroglyphic system of writing is a picture script in
which each character represents something: a pharaoh, a place, or a number
value. In this section we will look at how the Egyptians used hieroglyphics to
form their system of numbering. I might point out that the Egyptian
hieroglyphs will provide us with an example of what is call an additive
numbering system - what your authors call
Symbols of the Egyptians
The Egyptians symbols for numbers were basically just powers
of ten. This would in effect make their system, although additive, a decimal
system. The prefix deci is derived from the Latin decem
meaning "ten". The fact that ten is often found by ancient cultures to be
their choice of a numbering base is no doubt attributed to the fact that man
has ten fingers to count upon, which I'm sure they did then, since some of us
may still do it now.
The number "one" is represented, as it is in many other
civilizations numbering systems, by a vertical stroke, or perhaps it is a
picture of a staff. A picture resembling a horseshoe is used to represent a
collection of ten vertical strokes. Other pictures were used to represent
each new power of ten up to 10,000,000.
Write the following numbers using the
Egyptian hieroglyphic symbols:
(a) Choosing the appropriate symbols from
above we find that 234 is written
(b) Again choosing the
appropriate symbols we find that 2,104,120
Notice that the number of
Egyptian symbols used matches the sum of the digits in each number. That is
the three number symbols we use for 234, add up to 9 (2 + 3 + 4 = 9) and the
Egyptian version of this same number uses 9 symbols.
So, how many Egyptian
symbols would be needed to write the number 999?
The basic arithmetic operations of addition and
subtraction caused little problem for the Egyptians. In adding, it was
necessary only to collect symbols and exchange ten like symbols for the next
higher symbol. The next example illustrates an Egyptian addition problem.
Perform the indicated
You can try as your author's
suggest to add by grouping - but you can also convert each number into it's
Hindu Arabic equivalent (aka our own numbering system) then add the two
together and then rewrite using the Egyptian hieroglyphs.
Either way you should get
Ancient Oriental civilizations either existed simultaneously
with, or are somewhat younger than those of Mesopotamia and Egypt. Scholars
are not sure of the flow of influence with regards to these ancient people,
however, many believe that their development of mathematics is independent of
The traditional Chinese (brush form) system of numeration
shares some of the best features of both Egyptian hieroglyphic and Greek
alphabetic numerals. It is an example of a vertically written multiplicative
grouping system based on powers of ten. Symbols for this system are given
below. Note that this system also was used
extensively, even today, in many parts of the orient.
Show the traditional Chinese method of
writing the number 7829.
The Chinese are renowned for their intellectual activities,
and so it was through a natural evolution of mathematical maturity that they
saw they could omit the powers of ten in their representations of numbers. In
this way, exactly as we do today, they were letting each character assume two
values: (1) its known value, and (2) a value based on its position (place)
within the written number.
Using traditional Chinese brush stroke
numerals write the number 7829 omitting the powers of ten.
Professorial note: A
Disagreement About Zero (0)
Your authors and I do not
agree on the zero symbol they have included. Fact is the ancient Chinese were
quick to follow the Hindus and Arabs in using a zero symbol - in fact the very
one we use today.
So, when you see the symbol
they use to indicate zero just accept it. I cannot go back and rewrite their
textbook - but I have sent them and their editor a note.