One-two-many

Tribes in various parts of the world only have words for one, two, many - very primitive!

If you cannot form composite numbers like fifteen then we call this Systemless Counting.

Again by trade counting systems can be evolved from systemless counting.

Counting by Pairs

Many tribes when asked to count or give the number of as few as 3 grains of corn separate them into a pile of two and the remaining one.

Pure 2 count which develops from pairing forms numbers like this:

One
two
two - one
two - two
two - two - one etc.

A lot of tribes in Africa, Australia and South America use this - beyond 10 they are stuck!

Diffusion spread out the system? (as opposed to independent invention).

Seidenberg assumes that in ancient times 2 count was in use in much larger area then it is now and that it did spread by diffusion.

Neo - 2 - counting

In pure 2 count

6 =	2 + 2 + 2
8 =	2 + 2 + 2 + 2


A similar method is more efficient because it uses special words for 3 and 4 and pronounces 6 and 8 as twice 3 and twice 4.

Seems likely that in a lot of places original pure 2 count was replaced by neo - 2 - count. How did this happen?

Could be through tally-sticks. Since counting was in pairs their carried numerals would probably look like this:

111	1111	1111
111	111	1111
6	7	8


(Egyptians definitely did this 3000 BC)

Suppose this method of carving was used by a 2 counting population and that this method of noting numbers was taken over by neighbours who had numbers words for 1, 2, 3 and 4.

They might read the sign for 6 as 3 + 3, the sign for 7 as 4 + 3. In this way the passage from pure 2 count to neo 2 count might be explained.

4 count and 12 count

2 count is impractical. So further systems were developed. Counting in groups of 4 and counting in groups of 12.

eg.


1
2
3
4
5 = 4 + 1	)
6 = 4 + 2	)
7 = 4 + 3	)	Combination of both
8 = 4 x 2	)	found in Huku tribe
9 = 4 x 2 + 1	)	in South America
10 = 4 x 2 + 2	)
11 = 4 x 2 + 3	)
12
13 = 12 + 1
14 = 12 + 2


12 is a special number.

The 5 - 10 and 5 - 20 systems

5 - 10 has independent words up to 5 followed by composite number words from 6 to 9 and words for multiples of 10 are construed as such.

5 - 20 count extends 5 count principle to numbers up to 19.

eg.
Aztecs counted 1, 2, 3, 4, 5, 5 + 1, 5 + 2, 5 + 3, 5 + 4, 10,
10 + 2, 10 + 3 ... 20, 20 + 1, 1 x 20 + 10 ... 2 x 20 + 10.

We can represent these systems using staircases:

Aztecs and Mayas of Yucatan were chief users of this system.

The 10 - 60 system

Used by Sumerians (3000 BC)

Can have normal 10 system but then we come to 60 it is a new unit (gesh)

Remnants of 20 count in Western Europe

The French pronounce 80 as 4 x 20 and 90 as 4 x 20 + 10.

In English we use ' 3 score' - 60.

Danish language count in 20's

The Geographical Distribution of Counting Systems

The Development of Counting Systems and Notations

History of Mathematics Module

Links to other History of Mathematics sites

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These pages are maintained by M.I.Woodcock.