## Types of Numeration Systems |

The source for this information is

A Survey of Mathematics with Applicationsby Angel and Porter, pp. 138-151 (see bibliography)

Theadditivesystem is one of the oldest and the most primitive numeration systems dating from about 3000 B.C. One of the earliest was the Egyptian system were a different symbol for 1, 10, 100, 1,000, 10,000, 100,000, and 1,000,000 and then the number was formed by simply adding them together. Because there are symbols, order or placement of the symbols doesnotmatter. The main problem with an additive system is that some numbers must be expressed using many many symbols. Also, although addition and subtraction are easy, multiplication and division are not!

ValueEgyptian Numeral

Description

1

Staff or Vertical Stroke

10

Heel Bone or Arch

100

Scroll or Coiled Rope

1000

Lotus Flower

10,000

Pointed Finger

100,000

Whale or Tadpole

1,000,000

Astonished Person

SO with an additive

system --->>>## Or to write

43,628 as an Egyptian numeral -->>Egyptian graphics in the last 3 examples are from: http://www.eyelid.co.uk/numbers.htm

The Roman system is an example of a multiplicative system. Its symbols are:

## Roman Numerals

IVXLCDM## Hindu-Arabic Numerals

1510501005001000Similar to the Egyptian system,

MC = 1000 + 100 = 1100.However, placement is important in this system. Addition works from the right, but subtraction works from the left. Therefore,MC = 1000, butCM = 1000 - 100 = 900.The Roman system had an advantage in that it made use of the multiplicative principle for numbers

onlyover 1000. Placing a bar over the symbol(s) had the effect of multiplying the number by 1000. For example:But again, the number above: 43,628, would have to be written as: .

Ciphered numeration systems are based on many different symbols, and the best example of that is the Ionic Greek systems. Other ciphered systems include the Hebrew, the Coptic, the Syrian, and early Arabic. The Ionic Greek system used letters of their alphabets for numerals and was developed as early as the Egyptian system around 3000 B.C.The Greek alphabet had only 24 letters in it, but it borrowed from the Phoenicians the 3 symbols listed as obsolete. A number was very similar to the additive system. For example:

would equal 300 + 40 + 4 = 344.

To multiply, the system of an apostrophe ( ' ) was placed above a number which multiplied that number by 1000. So, for example:

would equal 8 x 1000 = 8000.

The advantage of a ciphered system is that the numbers are compactly written, but it does mean memorizing many different symbols!

The place-value system is the most commonly used numeration system today. The Hindu-Arabic numeration system is the one used throughout much of the Western Hemisphere. The oldest place-value system was the Babylonian system which was based on 60 instead of 10. Below are the comparable positional values for the Hindu-Arabic system and the Babylonian system:We retain the Babylonian system in the recording of time and the position as in degrees in a circle.

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