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Chapter 4 Section 1- Slide 1Copyright © 2009 Pearson Education, Inc.
AND
Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 1- Slide 2
Chapter 4
Systems of Numeration
Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 1- Slide 3
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
Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 1- Slide 4
Section 1
Additive, Multiplicative, and Ciphered Systems of Numeration
Chapter 4 Section 1- Slide 5Copyright © 2009 Pearson Education, Inc.
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).
Chapter 4 Section 1- Slide 6Copyright © 2009 Pearson Education, Inc.
Types Of Numeration Systems
Four types of systems used by different cultures will be discussed. They are:
Additive (or repetitive) Multiplicative Ciphered Place-value
Chapter 4 Section 1- Slide 7Copyright © 2009 Pearson Education, Inc.
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.
Chapter 4 Section 1- Slide 8Copyright © 2009 Pearson Education, Inc.
Egyptian Hieroglyphics
Chapter 4 Section 1- Slide 9Copyright © 2009 Pearson Education, Inc.
Roman Numerals
Roman Numerals Hindu-Arabic Numerals
I 1
V 5
X 10
L 50
C 100
D 500
M 1000
Chapter 4 Section 1- Slide 10Copyright © 2009 Pearson Education, Inc.
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
Chapter 4 Section 1- Slide 11Copyright © 2009 Pearson Education, Inc.
Multiplicative Systems
Multiplicative systems are more similar to the Hindu-Arabic system which we use today.
Example: Chinese numerals.
Chapter 4 Section 1- Slide 12Copyright © 2009 Pearson Education, Inc.
Chinese Numerals
Written vertically Top numeral from 1 - 9 inclusive Multiply it by the power of 10 below it.
20 =
400 =
Chapter 4 Section 1- Slide 13Copyright © 2009 Pearson Education, Inc.
Example
Write 538 as a Chinese numeral.
Chapter 4 Section 1- Slide 14Copyright © 2009 Pearson Education, Inc.
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.
Chapter 4 Section 1- Slide 15Copyright © 2009 Pearson Education, Inc.
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
Chapter 4 Section 1- Slide 16Copyright © 2009 Pearson Education, Inc.
Ionic Greek System
* Ancient Greek letters not used in the classic or modern Greek language.
Chapter 4 Section 1- Slide 17Copyright © 2009 Pearson Education, Inc.
Ionic Greek System
* Ancient Greek letters not used in the classic or modern Greek language.
Chapter 4 Section 1- Slide 18Copyright © 2009 Pearson Education, Inc.
Example
Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 1- Slide 19
Section 2
Place-Value or Positional-Value Numeration Systems
Chapter 4 Section 1- Slide 20Copyright © 2009 Pearson Education, Inc.
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.
Chapter 4 Section 1- Slide 21Copyright © 2009 Pearson Education, Inc.
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.
Chapter 4 Section 1- Slide 22Copyright © 2009 Pearson Education, Inc.
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.
Chapter 4 Section 1- Slide 23Copyright © 2009 Pearson Education, Inc.
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.
Chapter 4 Section 1- Slide 24Copyright © 2009 Pearson Education, Inc.
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 )
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