Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 4.1 Additive, Multiplicative, and...

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 4.1

Additive, Multiplicative, and Ciphered

Systems of Numeration


What You Will Learn

Additive, multiplicative, and

ciphered systems of numeration

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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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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


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


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


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


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


Homework

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 4.1 Additive, Multiplicative, and...

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