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 numeration systems.

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 mathematical system.

We have the following formal definition then of a mathematical system.

Mathematical System

A mathematical system is made up of the following three things:

1) a set of elements
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 commutativity.

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.


Ancient Egyptian Numeration

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 "simple grouping".





Hieroglyphic Number 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.


Example 1

Write the following numbers using the Egyptian hieroglyphic symbols:

(a) 234
(b) 2,104,120



(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. 

Example 2

Perform the indicated operation.






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 (405)



Traditional Chinese Numeration



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 all others.


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.



Example 3

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.  


Example 4

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.