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Chapter 4

Systems of Numeration

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WHAT YOU WILL LEARN• Additive, multiplicative, and ciphered

systems of numeration

• Place-value systems of numeration

• Egyptian, Hindu-Arabic, Roman, Chinese, Ionic Greek, Babylonian, and Mayan numerals

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Section 1

Additive, Multiplicative, and Ciphered Systems of Numeration

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Systems of Numeration

A system of numeration consists of a set of numerals and a scheme or rule for combining the numerals to represent numbers

A number is a quantity. It answers the question “How many?”

A numeral is a symbol such as , 10 or used to represent the number (amount).

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Types Of Numeration Systems

Four types of systems used by different cultures will be discussed. They are:

Additive (or repetitive) Multiplicative Ciphered Place-value

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Additive Systems

An additive system is one in which the number represented by a set of numerals is simply the sum of the values of the numerals.

It is one of the oldest and most primitive types of systems.

Examples: Egyptian hieroglyphics and Roman numerals.

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Egyptian Hieroglyphics

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Roman Numerals

Roman Numerals Hindu-Arabic Numerals

I 1

V 5

X 10

L 50

C 100

D 500

M 1000

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Example

At the end of a movie the credits indicated a copyright date of MCMXCVII. How would you write this as a Hindu-Arabic numeral?

Solution: It’s an additive system so,

= M + CM + XC + V + I + I

= 1000 + (1000 – 100) + (100 – 10) + 5 + 1 + 1

= 1000 + 900 + 90 + 7

= 1997

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Multiplicative Systems

Multiplicative systems are more similar to the Hindu-Arabic system which we use today.

Example: Chinese numerals.

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Chinese Numerals

Written vertically Top numeral from 1 - 9 inclusive Multiply it by the power of 10 below it.

20 =

400 =

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Example

Write 538 as a Chinese numeral.

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Ciphered Systems

In this system, there are numerals for numbers up to and including the base and for multiples of the base.

The number (amount) represented by a specific set of numerals is the sum of the values of the numerals.

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Examples of Ciphered Systems:

Ionic Greek system (developed about 3000 B.C. and used letters of Greek alphabet as numerals).

Hebrew system Coptic system Hindu system Early Arabic systems

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Ionic Greek System

* Ancient Greek letters not used in the classic or modern Greek language.

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Ionic Greek System

* Ancient Greek letters not used in the classic or modern Greek language.

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Example

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Section 2

Place-Value or Positional-Value Numeration Systems

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Place-Value System(or Positional Value System)

The value of the symbol depends on its position in the representation of the number.

It is the most common type of numeration system in the world today.

The most common place-value system is the Hindu-Arabic numeration system. This is used in the United States.

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Place-Value System

A true positional-value system requires a base and a set of symbols, including a symbol for zero and one for each counting number less than the base.

The most common place-value system is the base 10 system. It is called the decimal number system.

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Hindu-Arabic System

Digits: In the Hindu-Arabic system, the digits are

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Positions: In the Hindu-Arabic system, the positional values or place values are:

… 105, 104, 103, 102, 10, 1.

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Expanded Form

To evaluate a number in this system, begin with the rightmost digit and multiply it by 1.

Multiply the second digit from the right by base 10.

Continue by taking the next digit to the left and multiplying by the next power of 10.

In general, we multiply the digit n places from the right by 10n–1 in order to show expanded form.

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Example: Expanded Form

Write the Hindu-Arabic numeral in expanded form.

a) 63 b) 3769Solution: 63 = (6 101 ) + (3 1 ) or (6 10) + 3

3769 = (3 1000) + (7 100) + (6 10) + 9 or (3 103 ) + (7 102 ) + (6 101 ) + (9 1 )