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
4.1-2
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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).
4.1-3
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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.
4.1-4
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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
4.1-5
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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.
4.1-6
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Egyptian Hieroglyphics
4.1-7
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Example 1: From Egyptian to
Hindu-Arabic Numerals
Write the following numeral as a HinduArabic numeral.
Solution
10,000 + 10,000 + 10,000 + 100 +
100 + 100 + 10 + 1 + 1 + 1
= 30,134
4.1-8
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Example 2: From Hindu-Arabic to
Egyptian Numerals
Write 43,628 as an Egyptian numeral.
Solution
43,628 = 40,000 + 3000 + 600 + 20 + 8
4.1-9
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Roman Numerals
Roman Numerals
I
V
X
L
C
D
M
4.1-10
Hindu-Arabic Numerals
1
5
10
50
100
500
1000
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Roman Numerals
Two advantages over Egyptian system:
Uses the subtraction principle as well as
addition principle
DC = 500 + 100 = 600
CD = 500 – 100 = 400
Uses the multiplication principle for
numerals greater than 1000
V = 5 × 1000 = 5000
CD = 400 × 1000 = 400,000
4.1-11
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Example 4: From Roman to
Hindu-Arabic Numerals
Write CMLXIV as a Hindu-Arabic
numeral.
Solution
It’s an additive system so,
=
CM
+ L + X+
IV
= (1000 – 100) + 50 + 10 + (5 – 1)
= 900 + 50 + 10 + 4
= 964
4.1-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 5: Writing a Roman
Numeral
Write 439 as a Roman numeral.
Solution
439 = 400 + 30 + 9
= (500 – 100) + 10 + 10 + 10 + (10 – 1)
= CDXXXIX
4.1-13
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Multiplicative Systems
Multiplicative systems are more similar
to the Hindu-Arabic system which we
use today.
Chinese numerals
4.1-14
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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 =
4.1-15
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Example 7: A Traditional Chinese
Numeral
Write 538 as a Chinese numeral.
Solution:
4.1-16
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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.
4.1-17
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
4.1-18
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Examples of Ciphered Systems
We discuss in detail the Ionic Greek
system
developed about 3000 B.C.
used letters of Greek alphabet as
numerals
Base is 10
An iota, ι , placed to the left and above
a numeral represents the numeral
multiplied by 1000
4.1-19
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Examples of Ciphered Systems
Hebrew
Coptic
Hindu
Brahmin
Syrian
Egyptian Hieratic
early Arabic
4.1-20
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Ionic Greek System
* Ancient Greek letters not used in the classic or modern Greek language.
4.1-21
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Ionic Greek System
* Ancient Greek letters not used in the classic or modern Greek language.
4.1-22
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Example 9
Write ωλθ
numeral.
Solution
as a Hindu-Arabic
ω = 800, λ = 30, θ = 9
The sum is 839.
4.1-23
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 10
Write 1654 as an Ionic Greek numeral.
Solution
1654 = 1000 + 600 + 50 + 4
= (1 × 1000) + 600 + 50 + 4
ι
= α
χ
ν δ
ι
= αχνδ
4.1-24
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