1.2.2 - Denominational Systems

1.2.3

Denominational Systems

Introduction

In tracing the earliest history of numbers, one inevitably is lead to also investigate, in turn, the early development of mathematics, writings, and even civilization itself. The earliest development of Number is lost to time, as the development occurred in human minds prior to any form of writing. We can only make educated guesses of how man arrived from a number-sense to a solid notion of one-to-one correspondence along with a object free tally system. We can't even say precisely when man made his first tally mark, although the evidence tells us it happened no later than around 35,000 B.C. The next natural question seems to be : when did man first begin to develop numbers beyond a simple tally system?

In searching for the very first post-tally Numeration System we are brought to the very beginning of recorded history. The Sumerians, the earliest known civilization, who are purported to have lived in Mesopotamia as far back as 4500 B.C., are often credited as the first civilization to develop a true form of writing. The earliest written records actually only go as far back as 2900 B.C. but we have enough to actually discern several phases of development from simple pictograms all the way to a syllabic based writing system. In parallel to this we also know something about the development of mathematics, and specifically number notation in Mesopotamia. This suggests that this really was the cradle of civilization, and probably the first place to develop numbers beyond simple tallies.

However I think we must be careful here. Sumerian civilization was fairly sophisticated for a proto-civilization, with a government, divisions of labor including weaving, leather work, metal work, masonry and pottery, a military class, a priest class, and even those specializing in mathematics. We have tablets that appear to be assignments in mathematics, so there might have even been school systems as well. If civilization developed gradually from agrarian villages, then even the city of Sumer may have developed from even earlier civilizations now lost to time.

Consider that the oldest Japanese potteries date from 10,000 B.C. and a site known as Gobekli Tepe is the oldest known religious structure said to have been built some time around 9500 B.C. So what we see in Sumer would seem to be only a culmination and conglomeration of previous develops over the last few thousand years since the last Ice Age which ended about 12,000 B.C.

Keep in mind that we haven't found every ancient civilization there is to find. New sites are occasionally found. In fact, the information historians have gathered about the Sumerians is itself relatively recent. Prior to this historians had credited the Egyptians with many firsts, including the development of mathematics. With that in mind it seems to me we can't be absolutely certain that the Sumerians were the first civilization to possess a written language. It could well be then that some earlier Number Systems existed in the distant past, but we simply don't know.

Sumer is already incredibly ancient. In fact historians only learned about it from ancient Babylonian accounts of what even at that time was ancient history! Perhaps the Sumerians in turn knew of even earlier civilizations that we now know nothing about!

However we might as well put speculation aside and begin with the Sumerian Number System. There system is quite simple and yet quite elegant and sophisticated, especially for an exceptionally early notation. In fact I would go so far as to say the Sumerian Number System was more sophisticated than even the later Egyptian Number System.

The notations that I am about to describe are considered part of the development of "Babylonian Mathematics". This is because the major city of Babylonian inherited many of the mathematical developments of earlier civilizations in Mesopotamia. "Babylonian Mathematics" is really an umbrella term that covers a long period of time and several civilizations. The earliest to have settled the area are usually referred to as the Sumerians, whose city was Sumer. They were later conquered and absorbed by the Akkadians,

The Development of the Sumerian Number System

The Sumerian Number System developed over a long period of time, going through many distinct phases. This gradual development, as you will see, probably gives a very accurate road map for how man went from the concrete notions of discrete objects to the abstract notion of numeric symbols.

The earliest development of the Number System was closely tied to the monetary system. Originally the monetary system in Mesopotamia would have been quite literal. Items of intrinsic value were traded for their use value, such as grains. Food, especially perishable food, however makes for a poor form of currency. It can only retain value for as long as it remains edible. Consequently, at some point physical grains were replaced by grain tokens for trade. Different tokens would be used to represent different kinds of goods. This represents the first level of abstraction: the representation of a discrete object through the use of a symbol.

Although the tokens were far more durable than grains they were also quite heavy in bulk. So sometime around 3000 B.C. the tokens began to be replaced by clay tablets on which images of tokens were impressed. A different symbol was used for each of the token types. At this point the "Number System" was still a qualitative tally system, in which a symbol represented the tally of a particular type of object.

The next stage may seem minor but it is truly crucial: the separation of quantity from quality. One advantage of having a numeric tally mark, instead of simply using different symbols for each type being counted is that the numeric tally mark may be significantly simplified, typically to a single stroke. When written in front of a "type symbol" one still retains the ability to specify "how much" and of "what kind". To illustrate the advantage this can have in terms of space consider the following image:

As you can see, the drawing out of all the "Oil Jars" takes up a much greater space than the use of the tally marks in the second example. This would have been the motivation for going one more level of abstraction to a true tally system (a purely quantitative one). Unfortunately the Sumerian Number system wasn't quite this simple. Different Number Systems were used depending on the type of object involved. None the less it is at this stage that an important development takes place : the first occurrence of group symbols.

The most common of these number systems (which was used for most discrete objects) begins with the tally mark:

This represented a single unit. Repetitions of this symbol could be used to specify repetitions of the unit, just as with any ordinary tally system. The crucial step however was to create a "new symbol" that would represent a specific number of repetitions of the tally mark. The first of these, which we will call the "tens-mark" looked like a circle. By definition the circle would be equal to the count of all fingers on both hands:

The advantage of this is immediately obvious from the image above. The circle takes up less space and is easier to draw than drawing out the full number of tally marks. With the addition of this new symbol, the ability to express large numbers increases dramatically.

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The Babylonian Number System

The earliest known numeration system after tally marks is that of the Babylonians. Tablets dating back as far as 3400 B.C. reveal the Babylonian number system, thus making it more than 5000 years old! The system the Babylonians used is remarkably sophisticated for the first numerical notation in that it is quite economical. The system only requires 2 number symbols and 1 place holder symbol.

As we saw in the previous article, the most primitive forms of numeration involved direct representation, where a mark or symbol would represent a single unit. The Babylonian system begins in the same way. To write their numerals the Babylonians used moist clay tablets that could be pressed into using a special stylus. The stylus was used to make various kinds of wedge shapes which represented the simplest elements of their form of writing.

The Babylonians chose to use a down-pointing wedge to represent a count of one:

To continue, more such wedges were added. These were added in rows of up to three. When a row filled up, more wedges could be added below in the next row. Lines were sometimes used to separate rows. This continued until one reached three rows of three:

What the Babylonians decided to do next was innovative. Rather than continue by starting a new "block", or simply adding another unit wedge somewhere, they invent a new symbol. This symbol stands not for a unit, but for ten wedges. It's a symbol to represent a group of symbols! The symbol they chose to use looks something like a left pointing arrow head:

To continue, one simply added more unit wedges ...

Once the block of unit wedges was completely filled, you could continue just by adding another 10-wedge. This pattern of unit-wedges and 10-wedges could be continued until you reached 5 10-wedges and 9 unit-wedges. In other words, this would continue until you reached 59:

The next thing they decided to do is surprisingly modern. Rather than come up with some new symbol to represent "60", they come up with a way to say "one set of 60", and use a single down wedge to represent this. Unfortunately this leads to some ambiguity as this is also the symbol used for "1", but it could be differentiated based on context.

To continue, you could simply write any of the numbers from 1 to 59 to the right of the single wedge representing the count of 60:

Citations:

http://it.stlawu.edu/~dmelvill/mesomath/sumerian.html

http://www.fsmitha.com/h1/ch01.htm (The Sumerians)

http://www.storyofmathematics.com/sumerian.html (The Story of Mathematics)

http://www.math.tamu.edu/~dallen/masters/egypt_babylon/babylon.pdf (Babylonian Mathematics)