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Section 4.1

Additive, Multiplicative, and Ciphered

Systems of Numeration

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What You Will Learn

Additive, multiplicative, and

ciphered systems of numeration

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Systems of Numeration

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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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.

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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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Example 1: From Egyptian to Hindu-Arabic NumeralsWrite the following numeral as a Hindu-Arabic numeral.

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Solution10,000 + 10,000 + 10,000 + 100 + 100 + 100 + 10 + 1 + 1 + 1= 30,134

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Try This

Write the following numeral as a Hindu-Arabic Numberal

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Example 2: From Hindu-Arabic to Egyptian Numerals

Write 43,628 as an Egyptian numeral.

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Solution43,628 = 40,000 + 3000 + 600 + 20 + 8

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Try This

Write 1,234,527 as an Egyptian numeral

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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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Roman Numerals Two advantages over Egyptian system:

Uses the subtraction principle as well as addition principleDC = 500 + 100 = 600CD = 500 – 100 = 400

Uses the multiplication principle for numerals greater than 1000 V 5 1000 5000

CD 400 1000 400,0004.1-13

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Example 4: From Roman to Hindu-Arabic NumeralsWrite CMLXIV as a Hindu-Arabic numeral.SolutionIt’s an additive system so,= CM + L + X + IV= (1000 – 100) + 50 + 10 + (5 – 1)= 900 + 50 + 10 + 4= 964

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Try THis

Write MCMLXXI as a Hindu-Arabic numeral

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Example 5: Writing a Roman Numeral

Write 439 as a Roman numeral.

Solution439 = 400 + 30 + 9= (500 – 100) + 10 + 10 + 10 + (10 – 1)= CDXXXIX

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Try This

Write 3794 as a Roman Numeral

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

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Chinese Numerals

Written verticallyTop numeral from 1 - 9 inclusiveMultiply it by the power of 10 below it.

20 =

400 =

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Example 7: A Traditional Chinese NumeralWrite 538 as a Chinese numeral.Solution:

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Try This

Write 3,271 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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Ciphered Systems

Ciphered numeration systems require the memorization of many different symbols but have the advantage that numbers can be written in a compact form.

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Examples of Ciphered SystemsWe discuss in detail the Ionic Greek systemdeveloped about 3000 B.C.used letters of Greek alphabet as numeralsBase is 10An iota, ι, placed to the left and above a numeral represents the numeral multiplied by 1000

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Examples of Ciphered SystemsHebrewCopticHinduBrahminSyrianEgyptian Hieraticearly Arabic

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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 9

Write as a Hindu-Arabic numeral.Solution

The sum is 839.

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800, 30, 9

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Example 10

Write 1654 as an Ionic Greek numeral.Solution1654 = 1000 + 600 + 50 + 4

= (1 × 1000) + 600 + 50 + 4

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ι ι

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Homework

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