Base Valued Numbers

Any use of numbers implies the use of a base value for the numbers. The simplest base value to use in a numbering scheme is '1'. In this scheme the number '2' is two things, or two groups of ones. The number '7' is seven things or seven groups of ones.  Evidence of numbering in this fashion has been found in archaeological evidence dating as far back as 37,000 years.

• Tally Sticks -

  • Bones have been found in Africa dating to 37,000 years ago with 29 orderly notches on them.  This number most probably represents the number of days in a moon cycle, a readily observable occurrence.

  

  • In a hunter/gatherer culture a member of the group may keep a tally of how many days are spent in a geographic location by making notches on a bone. These notches would represent a numbering scheme of base 1 with each notch representing a day or a moon cycle. Traditional wisdom may include how many units (notches) to stay in a location, matching the abundance of game in a location.

Numbers are a tool that was honed and forged from the interaction of our ancestor's cultural and social behavior with our ancestor's ability to conceptualize.  From our ancestor's various uses of number emerge various ways to group numbers that make their use easier, more efficient, more consistent or more meaningful. Some uses of numbers are:

• Trading + numbers = (grouping and dividing quantities) Trading and Numbers

• Building + numbers = (measuring distance or length)  Building, Measuring and Numbers

• Writing + numbers = (record keeping) Writing and Numbers

• Tracking Time + numbers = (measuring temporal quantities) Time and Numbers

• Symbolism + numbers = (relating diverse ideas to properties or spatial relations of numbers) Symbolism and Numbers

• Abstract thinking + numbers = (more numbers and relations between numbers) Abstract Thinking and Numbers

The basic rules for a formalized base numbering system involve ordering items, grouping ordered items and then expressing the groups and items in a consistent way.  The way it represents the different groups gives the numbering system an order of magnitude.  This can be expressed in several ways.

• A special symbol represents a specified grouping value.  For example a (picture of a hand) represents 5.  The Roman numerals and the Egyptian numbering represent each order of magnitude with a special symbol.  Note that this will in some cases limit how high a numbering system may count because a new symbol needs to be developed for each successive grouping.  For example, the number 1,475,268 is represented in the Egyptian numbering system as follows:

.

Each of the magnitudes of 10 were represented in the above number, for example the 4 frogs represent 4 hundred thousands and the 5 lotus flowers represent 5 thousands, etc.  In this numbering system, only the magnitudes of 10 that are used are expressed in the written number.   The number 5,060 is thus represented as:

Press here for Active Translation of Egyptian numbers

• Grouping values can be represented by the place they hold in the representation of the number.  Our Hindu/Arabic numbering system uses this method, as we have a place value for 1's, 10's, 100's, etc.  Since no new symbols are used (0-9), the numbering system can continue to incredibly large values.   From the use of this method a representation of 0 emerges so that there is a way to represent 0 of a group.

• The orders of magnitude may represent a consistent gradation of value where each successive order of magnitude will be 'n' times the last order of magnitude.  In the decimal system, each successive place value is 10 times the last place value. 

Some examples of number base systems, some more formalized than others, are:

• Binary (base 2) Base 2 (Binary)

• Hand (base 5) Base 5 (Hand)

• Octal (base 8) Base 8 (Octal)

• Decimal (base 10) Base 10 (Decimal)

• Groupings using 12  (base 12) Base 12 (duodecimal)

• Hexadecimal (base 16) Base 16 (Hexadecimal)

• Mayan (base 20) Base 20 (Vigesimal)

• Time and Ancient Sumerian (base 60) Base 60 (Sexagesimal)

K-12 Adding has some exercises involving base valued arithmetic.

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Bibliography:

The basic picture's for the Egyptian Hieroglyphics used are used with permission from the following site. http://eyelid.ukonline.co.uk/ancient/numbers.htm:

"A Cultural History of Numbers" - Karl Menninger

"The Story of Numbers" - John McLeish

"A Beginner's Guide to Construction the Universe" - Michael S. Schneider

"Empires of Time" -- Anthony Aveni

"About Time: -- Paul Davies

"Timelines of the Ancient World"  -- Chris Scarre

"Elements" -- Euclid

Last update on 10/11/98

Copyright 1997 Poseidon Software and Invention