Chapter 4 Section 3 - Slide 1Copyright © 2009 Pearson Education, Inc.

AND

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Chapter 4

Systems of Numeration

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WHAT YOU WILL LEARN• Converting base 10 numerals to

numerals in other bases

• Converting numerals in other bases to base 10 numerals

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Section 3

Other Bases

Chapter 4 Section 3 - Slide 5Copyright © 2009 Pearson Education, Inc.

Bases

Any counting number greater than 1 may be used as a base for a positional-value numeration system.

If a positional-value system has a base b,

then its positional values will be

… b4, b3, b2, b, 1. Hindu-Arabic system uses base 10.

Chapter 4 Section 3 - Slide 6Copyright © 2009 Pearson Education, Inc.

Example: Converting from Base 8 to Base 10

Convert 45368 to base 10.

Solution:

3 284536 4 8 5 8 3 8 6 1

4 512 5 64 3 8 6 1

2048 320 24 6

2398

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Example: Converting from Base 5 to Base 10

Convert 425 to base 10.

Solution:

1542 4 5 2 1

20 2

22

Chapter 4 Section 3 - Slide 8Copyright © 2009 Pearson Education, Inc.

Example: Convert to Base 3

Convert 342 to base 3.

Solution:

The place values in the base 3 system are

…, 36, 35, 34, 33, 32, 3, 1 or

…, 729, 243, 81, 27, 9, 3, 1.

The highest power of the base that is less than or equal to 342 is 243.

Chapter 4 Section 3 - Slide 9Copyright © 2009 Pearson Education, Inc.

Example: Convert to Base 3 (continued)

Successive division by the powers of the base gives the following result.

342 243 1 with remainder 99

99 811 with remainder 18

18 27 0 with remainder 18

18 9 2 with remainder 0

0 3 0 with remainder 0

Chapter 4 Section 3 - Slide 10Copyright © 2009 Pearson Education, Inc.

Example: Convert to Base 3 (continued)

The remainder, 0, is less than the base, 3, so no further division is necessary.

5 4 3 2

3

342 1 243 1 81 0 27 2 9 0 3 0 1

(1 3 ) 1 3 0 3 2 3 0 3 0 1

110200

Chapter 4 Section 3 - Slide 11Copyright © 2009 Pearson Education, Inc.

Computers

Computers make use of three numeration systems

Binary Octal Hexadecimal

Chapter 4 Section 3 - Slide 12Copyright © 2009 Pearson Education, Inc.

Numeration Systems

Binary system Base 2 It is very important because it is the

international language of the computer. Computers use a two-digit “alphabet” that

consists of numerals 0 and 1. Octal system

Base 8 Hexadecimal system

Base 16