I Love the Many Ways
The process of counting is as unique as the language from which is comes....
Types of Counting:
Some historians believe that ancient cultures in many parts of the world had only words for one, two, and many. Although this is primitive, we teach a similar pattern in algebra by recognizing expressions as: monomials, binomials, trinomials, and polynomials.
Counting by Pairs and Neo-2 Counting:
Counting by pairs was typically used with very small quantities and hints of a place-value system. Pair counting would be expressed as follows:
one = one
two = two
three = two and one
four = two and two
five = two and two and one
Neo-2 counting was a bit more efficient because it used special words for 3 and 4. It then followed that numbers like six and eight were represented as twice 3 and twice 4.
Place Value Systems vs. Non-Place Value Systems:
Place value systems of course represent numbers as increasing powers of the base in which we are operating. For instance, we operate in base 10 which means we have 10 unique symbols (0-9) and each position in a given number is represented as power of the base.
i.e. 573 = (5 x 100) + (7 x 10) + (3 x 1)
We also rely heavily on other bases in our culture, i.e. binary for computers, etc.
Non-place value systems like Roman numerals rely on several distinct symbols each representing a sum or difference depending on how they were arranged. Largest value symbols placed in front of smaller ones represented a sum, where smaller values placed in front of larger ones represented a difference. There was no zero represented in this system.
The true difference in the systems lies in the arithmetic operations
The Mayan System:
The Mayans of ancient Mexico used a numeration system called quinquavigesimal (if you can pronounce this without practice you can pass Dr. Ostler's class). This was an odd place value system based on 5 and 20, which included a zero and was written vertically with the highest place values at the top.
The ancient Mesopotamians used a base-60 numbering system because the number 60 has 12 unique factors where as our base-10 system has only 4. This meant that mathematical operations came out evenly more often which was important because they were avid astronomers and mathematicians. Some historians also suppose that 60 was chosen because of its fit to calculations relative to the Earth's rotation around the sun.
The Far East:
The Soroban or Abacus as it is now called is a combination of place value and multiples of 1, 10, 100, etc. on one side, and 5, 50, 500, etc. on the other side.
Places to Look for More Information: