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Early Numeration Systems

All earlier civilizations had some form of counting system that was agreed upon whether it was verbal, tally marks on a tree or rocks piled up on the ground, etc..  The Alaskan's counted using a base five system with words that stood for repeated fingers hands and feet.  They essentially counted as one finger, two fingers, … four fingers, one hand, one hand and one finger, … one hand and four fingers, two hands, two hands and one finger… two hands and four fingers, one foot, one foot, … one foot and four fingers and instead of two feet their word for our twenty meant "one man ended." The Egyptians' hieroglyphics or picture symbols were first carved into stone and later written on a type of reed paper after the process for making the paper was discovered.  The Chinese and Japanese had carefully drawn characters, which they too wrote on paper.  Many African civilizations used a base two counting system, which is the foundation language of our modern calculator and computer systems, one for on and zero for off.

Our calculators and computers of today are just the latest tools mankind uses to count, add and subtract etc.  The Chinese and Japanese were able to count fantastically fast using an abacus and the Peruvians had their knotted cords. To get some appreciation of the diversity of tools used over the years, investigate the differences between the Chinese abacus and the Japanese abacus. There are, of course, many other systems not mentioned here. The following systems are those for which you are responsible.

Egyptian: 

   Numerals:      Symbols called hieroglyphics       (Please see your text for pics; symbols shown here are the best font representations I could find.)

|    One, Tally Mark Ten, Heel Bone  ¶    Hundred, Scroll

Thousand, Lotus Flower   Y   One Million,  (said to be a surprised man or a man holding up the universe in some texts.)

   System Properties:  Additive, no base or place value, no zero, little orientation with symbols written both horizontally and vertically but tending toward decreasing values left to right.

Babylonian:

   Numerals: Symbols  for tens and ones by . (Again, please see your text for symbols)

   System Properties: Additive with base sixty-place value written horizontally, 60n, …   602, 601, 600  or units.  But has no place holder numeral like zero.  Spaces are meant to separate place values, which leads to the problem of knowing “How much space is enough space? Too much space?”  Does the numeral    , represent one plus one which is two or the sum of sixty plus one or sixty-one?  Some texts say that latter generations of Babylonians used a vertical symbol resembling two triangles one on top of the other as a placeholder to alleviate this problem.

Mayan:

Numerals:            Mayan place values are written vertically as follows:

One is represented by a dot    Five is represented by a bar   Zero looks elliptical like a football

   System Properties: Additive, almost base twenty place value with a symbol serving as a placeholder in the same manner as our number zero. The Mayans wrote their numbers vertically.  The bottom or first row goes from zero  to nineteen   (three fives plus four ones); the second row goes from twenty,  (zero ones plus one twenty) to three hundred fifty-nine,   [3*5+4 ones plus 3*5+2 twenties]. The Mayans were very advanced astronomically and three hundred sixty is closer to the number of days in a year than is four hundred. So their third tier starts at 18*20=360 in lieu of 20*20=20 =400.   All rows starting with the third row and above have the value eighteen times consecutively increasing powers of twenty 18*20 , n≥3.

Roman:

Numerals:

I      One V     Five

X     Ten L      Fifty

C      One Hundred D      Five    Hundred

M      One Thousand      Four Thousand, etc.

System Properties:  Additive with some subtraction and some multiplication features. No base or place value and written horizontally according to a very structured set of rules.  (To use Roman numerals to multiply two times thirty-nine you almost need to know the answer before you can write it out using the symbols that represent the product. Try adding XLVIII+XClX  (48+99) in Roman numerals and see how quickly you get the answer,CXLVll (147). How about multiplying the two numbers?  You quickly come to the conclusion that the Roman system is rather complex when compared to the rote simplicity in the algorithms of our decimal system.

      While the overall structure of the Roman is essentially additive, it contains some subtraction and some multiplication features.  For example, four and six are respectfully, lV for one less than and VI for one more than five.  Fifteen is written as XV for ten plus five but VX has no meaning.  The Roman one, l can be subtracted only from V and X; X can only be subtracted from L and C; C can be subtracted only from D and M. The symbols V, L and D have no meaning preceding the representation of any higher valued number. Because of the very structured rules of the Roman numeration system, there was no need for a placeholder such as zero.

Roman numerals can be repeated consecutively a max number of three times, for example thirty-five, thirty-nine and forty are written XXXVlll, XXXlX and forty XL. Numerals larger than three thousand ninety-nine or MMMCMlX are written using a bar above a valid symbol meaning that the number represented should be multiplied by one thousand.  For example, four thousand has a single bar over the numeral four , …  is a thousand times a thousand,  is a thousand times four thousand, etc. FYI: No text I have ever read gave a limit as to how many bars could be repeated and I have not tried to search the Internet or elsewhere for it.  But because so many texts leave students wondering just what how thirty-nine (39) or the last millennium one thousand ninety-nine (1999) should be written.  Perhaps the following table will helping  you read the next production or release date you see on a motion picture screen.

I	II	III	IV	V	VI	VII	VIII	lX	X
Xl	XII	XIII	XIV	XV	XVI	XVII	XVIII	XlX	XX
XXl	XXII	XXIII	XXIV	XXV	.	etc		XXlX	XXX
repeat as above to		XL forty	XLl	…	XlX forty-nine	fifty L	ninety XC	XClX	one hundred C
C|	CXL	CL	CDXC|X	D five hundred		DCCC	CM	one thousand M
M|	MCD	MD	MDCCC	MCM	MCMXC|X	MM	MMM	, , , , , etc.

      Discussion: Note that zero is a very real number.  Our use of zero today tells us how many items are in an empty box.  Technically, 0=n( ).  That is zero stands for the number of elements in the empty set, that is the set that has no elements in it, { }. The symbol used to denote the empty set is not but looks like the Greek letter phi, { }= .   While elementary textbooks use the symbol  for the empty set, I encourage any teacher just introducing the concept of the empty set to any age group to use the {} notation exclusively.  A set of brackets with no elements inside is readily accepted as a set that is empty.  The task of teaching or learning  (accepting) the empty set as a valid entity is only exasperated when an unknown symbol is used to represent this set. Be sure your students are familiar with the concept of the empty set and that zero is the number that represents the quantity of elements in the empty set before using an abstract symbol for it.

      Zero is important in our number system in its use as a placeholder.  The fact that its existence was not introduced into Western culture until ~900AD attests to the difficulty mankind had in being able to conceive the abstract concept of "having something that means nothing." Zero is not "nothing".  Zero is the number that says you have" no quantity of" something.  Consider the following garden problem:  How many rose bushes do you need to buy if you are going to plant a rose at the ends of and one foot apart from each other in a 1' by 5' area?  If you said, "Five," then you forgot about the concept of zero.  You need six bushes, one for the zero position and five more if you are going to have the rose bushes one foot apart, two for the end plants and four more in between.

Use the information above to complete the following numeration system property table. 

Property System	Numerals	Additive Structure	Orienta-tion	Base Structure	Place Value	Zero
Egyptian	See above	YES	Not Really	NO	NO	NO
Babylonian		YES	Horizontal	Base 60	YES	NO
Mayan		YES	Vertical	~Base 20	YES	YES
Roman		YES	Has Rules	NO	NO	NO