3.1: The Evolution of Numeration Systems

Introduction

As we begin our journey through the history of mathematics, one question to be asked is “Where do we start?” Depending on how you view mathematics or numbers, you could choose any of a number of launching points from which to begin. Howard Eves suggests the following list of possibilities.[i]

Where to start the study of the history of mathematics…

• At the first logical geometric “proofs” traditionally credited to Thales of Miletus (600 BCE).
• With the formulation of methods of measurement made by the Egyptians and Mesopotamians/Babylonians.
• Where prehistoric peoples made efforts to organize the concepts of size, shape, and number.
• In pre-human times in the very simple number sense and pattern recognition that can be displayed by certain animals, birds, etc.
• Even before that in the amazing relationships of numbers and shapes found in plants.
• With the spiral nebulae, the natural course of planets, and other universe phenomena.

We can choose no starting point at all and instead agree that mathematics has always existed and has simply been waiting in the wings for humans to discover. Each of these positions can be defended to some degree and which one you adopt (if any) largely depends on your philosophical ideas about mathematics and numbers.

Nevertheless, we need a starting point. Without passing judgment on the validity of any of these particular possibilities, we will choose as our starting point the emergence of the idea of number and the process of counting as our launching pad. This is done primarily as a practical matter given the nature of this course. In the following chapter, we will try to focus on two main ideas. The first will be an examination of basic number and counting systems and the symbols that we use for numbers. We will look at our own modern (Western) number system as well those of a couple of selected civilizations to see the differences and diversity that is possible when humans start counting. The second idea we will look at will be base systems. By comparing our own base-ten (decimal) system with other bases, we will quickly become aware that the system that we are so used to, when slightly changed, will challenge our notions about numbers and what symbols for those numbers actually mean.

Recognition of More vs. Less

The idea of numbers and the process of counting goes back far beyond when history began to be recorded. There is some archeological evidence that suggests that humans were counting as far back as 50,000 years ago.[ii] However, we do not really know how this process started or developed over time. The best we can do is to make a good guess as to how things progressed. It is probably not hard to believe that even the earliest humans had some sense of more and less. Even some small animals have been shown to have such a sense. For example, one naturalist tells of how he would secretly remove one egg each day from a plover’s nest. The mother was diligent in laying an extra egg every day to make up for the missing egg. Some research has shown that hens can be trained to distinguish between even and odd numbers of pieces of food.[iii] With these sorts of findings in mind, it is not hard to conceive that early humans had (at least) a similar sense of more and less. However, our conjectures about how and when these ideas emerged among humans are simply that; educated guesses based on our own assumptions of what might or could have been.

The Need for Simple Counting

As societies and humankind evolved, simply having a sense of more or less, even or odd, etc., would prove to be insufficient to meet the needs of everyday living. As tribes and groups formed, it became important to be able to know how many members were in the group, and perhaps how many were in the enemy’s camp. Certainly it was important for them to know if the flock of sheep or other possessed animals were increasing or decreasing in size. “Just how many of them do we have, anyway?” is a question that we do not have a hard time imagining them asking themselves (or each other).

In order to count items such as animals, it is often conjectured that one of the earliest methods of doing so would be with “tally sticks.” These are objects used to track the numbers of items to be counted. With this method, each “stick” (or pebble, or whatever counting device being used) represents one animal or object.

In the picture to the right, you see each stick corresponding to one horse. By examining the collection of sticks in hand one knows how many animals should be present. You can imagine the usefulness of such a system, at least for smaller numbers of items to keep track of. If a herder wanted to “count off” his animals to make sure they were all present, he could mentally (or methodically) assign each stick to one animal and continue to do so until he was satisfied that all were accounted for.

Of course, in our modern system, we have replaced the sticks with more abstract objects. In particular, the top stick is replaced with our symbol “1,” the second stick gets replaced by a “2” and the third stick is represented by the symbol “3,” but we are getting ahead of ourselves here. These modern symbols took many centuries to emerge.

Another possible way of employing the “tally stick” counting method is by making marks or cutting notches into pieces of wood, or even tying knots in string (as we shall see later). In 1937, Karl Absolom discovered a wolf bone that goes back possibly 30,000 years. It is believed to be a counting device.[iv] Another example of this kind of tool is the Ishango Bone, discovered in 1960 at Ishango, and shown below.[v] It is reported to be between six and nine thousand years old and shows what appear to be markings used to do counting of some sort.

The markings on rows (a) and (b) each add up to 60. Row (b) contains the prime numbers between 10 and 20. Row (c) seems to illustrate for the method of doubling and multiplication used by the Egyptians. It is believed that this may also represent a lunar phase counter.

Spoken Words

As methods for counting developed, and as language progressed as well, it is natural to expect that spoken words for numbers would appear. Unfortunately, the developments of these words, especially those corresponding to the numbers from one through ten, are not easy to trace. Past ten, however, we do see some patterns:

Eleven comes from “ein lifon,” meaning “one left over.”

Twelve comes from “twe lif,” meaning “two left over.”

Thirteen comes from “Three and ten” as do fourteen through nineteen.

Twenty appears to come from “twe-tig” which means “two tens.”

Hundred probably comes from a term meaning “ten times.”

Written Numbers

When we speak of “written” numbers, we have to be careful because this could mean a variety of things. It is important to keep in mind that modern paper is only a little more than 100 years old, so “writing” in times past often took on forms that might look quite unfamiliar to us today.

As we saw earlier, some might consider wooden sticks with notches carved in them as writing as these are means of recording information on a medium that can be “read” by others. Of course, the symbols used (simple notches) certainly did not leave a lot of flexibility for communicating a wide variety of ideas or information.

Other mediums on which “writing” may have taken place include carvings in stone or clay tablets, rag paper made by hand (12^th century in Europe, but earlier in China), papyrus (invented by the Egyptians and used up until the Greeks), and parchments from animal skins. And these are just a few of the many possibilities.

These are just a few examples of early methods of counting and simple symbols for representing numbers. Extensive books, articles and research have been done on this topic and could provide enough information to fill this entire course if we allowed it to. The range and diversity of creative thought that has been used in the past to describe numbers and to count objects and people is staggering. Unfortunately, we don’t have time to examine them all, but it is fun and interesting to look at some systems in more detail.

Egyptian Hieroglyphs

The system of ancient Egyptian numerals was used in Ancient Egypt from around 3000 BCE until the early first millennium CE. It was a system of numeration based on multiples of ten written in hieroglyphs. The Egyptians had no concept of a place-valued system such as the decimal system. The hieratic form of numerals stressed an exact finite series notation, ciphered one-to-one onto the Egyptian alphabet.

These symbols represent the powers of ten. From left to right: 1 (stroke), 10 (heel bone), 100 (scroll), 1000 (lotus flower), 10,000 (finger), 100,000 (tadpole), 1,000,000 (astonished person).

They used a simple grouping system by combining multiple symbols of the same value to add them up.

Example 1

Convert to our decimal system:

Solution

Since there are 3 scrolls, 4 heel bones, and 7 strokes, this represents the number 347.

The Egyptians also had a method for multiplication. This method is based on the fact that every positive integer can be written as the sum of different powers of 2.

Powers of 2

Powers of 2	Value
\(2^0\)	1
\(2^1\)	2
\(2^2\)	4
\(2^3\)	8
\(2^4\)	16
\(2^5\)	32
\(2^6\)	64

Example 2

Write the numbers 35 and 79 as a sum of powers of 2.

Solution

\(35=2^5+2^1+2^0=32+2+1\)

\(79=2^6+2^3+2^2+2^1+2^0=64+8+4+2+1\)

Example 3

Multiply \(35 \times 79\)

Solution

\(35 \times 79\ = 35 \times (64+8+4+2+1)\)

\(35 \times 79\ = 35 \times 64 + 35 \times 8 + 35 \times 4 + 35 \times 2 + 35 \times 1\)

Now make a table of Powers of 2 times 35 just by doubling 35 many times.

Powers of 2 Times 35

Powers of 2	Times 35
1	35
2	70
4	140
8	280
16	560
32	1120
64	2240

Thus \(35 \times 79\ =2240+280+140+70+35=2765\).

Roman Numerals

Even though Roman Numerals are rare in today’s society, they are still used and expected to be understood. They are taught in grades three through five, depending on the district. They can be seen in clocks, the Super Bowl, Film Credits for the copyright date like MCMLXII, preface of textbooks and others like Star Wars Episode VI and WWII.

Basic Table of Roman Numerals

Value	Symbol
1	I
5	V
10	X
50	L
100	C
500	D
1000	M

An line over a numeral means to multiply by 1000. You'll see in this next table.

The Complete Table of Roman Numerals

	1	2	3	4	5	6	7	8	9
ONES	I	II	III	IV	V	VI	VII	VIII	IX
TENS	X	XX	XXX	XL	L	LX	LXX	LXXX	XC
HUNDREDS	C	CC	CCC	CD	D	DC	DCC	DCCC	CM
THOUSANDS	M	MM	MMM	\(\overline{\mathrm{IV}}\)	\(\overline{\mathrm{V}}\)	\(\overline{\mathrm{VI}}\)	\(\overline{\mathrm{VII}}\)	\(\overline{\mathrm{VIII}}\)	\(\overline{\mathrm{IX}}\)
TEN THOUSANDS	\(\bar{X}\)	\(\overline{X X}\)	\(\overline{X X X}\)	\(\overline{X L}\)	\(\bar{L}\)	\(\overline{L X}\)	\(\overline{L X X}\)	\(\overline{L X X X}\)	\(\overline{X C}\)
HUNDRED THOUSANDS	\(\bar{C}\)	\(\overline{C C}\)	\(\overline{C C C}\)	\(\overline{C D}\)	\(\bar{D}\)	\(\overline{D C}\)	\(\overline{D C C}\)	\(\overline{D C C C}\)	\(\overline{C M}\)

Example 4

For all numbers except 4 and 9, we ADD the Roman Numerals together, in order from left to right, greatest value to lowest value.

\[\begin{aligned}
11 &=10+1=X I \\
8 &=5+1+1+1=V I I I \\
123 &=100+10+10+1+1+1=C X X I I I \\
3816 &=1000+1000+1000+500+100+100+100+10+5+1=M M M D C C C X V I \\
7002 &=5000+1000+1000+1+1=\overline{V I I }I I
\end{aligned} \nonumber \]

Example 5

For any number that includes a 4 or a 9, we subtract. When we are looking at a Roman Number expression and we see a Roman character OUT OF ORDER, which is the clue to SUBTRACT!

\[\begin{aligned}
4 &=5-1=I V \\ 9 &=10-1=I X \\
40 &=50-10=XL \\ 99 &=(100-10)+(10-1)=X C I X \\
400 &=500-100=C D
\end{aligned} \nonumber \]

Notice \(99 \neq 100-1\) because you can only subtract numerals from the next two higher numerals.

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[i] Eves, Howard; An Introduction to the History of Mathematics, p. 9.

[ii] Eves, p. 9.

[iii] McLeish, John; The Story of Numbers - How Mathematics Has Shaped Civilization, p. 7.

[iv] Bunt, Lucas; Jones, Phillip; Bedient, Jack; The Historical Roots of Elementary Mathematics, p. 2.

[v] http://www.math.buffalo.edu/mad/Ancient-Africa/mad_zaire-uganda.html