Types of Numeration Systems


Additive | Multiplicative | Ciphered | Place-Value

The source for this information is A Survey of Mathematics with Applications by Angel and Porter, pp. 138-151 (see bibliography)

Additive Systems

The additive system 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 does not matter. 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!

Egyptian Numeral



Staff or Vertical Stroke


Heel Bone or Arch


Scroll or Coiled Rope


Lotus Flower


Pointed Finger


Whale or Tadpole


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

Multiplicative Systems

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

Roman Numerals


Hindu-Arabic Numerals


Similar 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, but CM = 1000 - 100 = 900.

The Roman system had an advantage in that it made use of the multiplicative principle for numbers only over 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 Systems

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!

Place-Value or Positional-Value Systems

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.

Week 3