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

Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 3 - Slide 2 Chapter 4 Systems of Numeration

Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 3 - Slide 3 WHAT YOU WILL LEARN Converting base 10 numerals to numerals in other bases Converting numerals in other bases to base 10 numerals

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

Chapter 4 Section 3 - Slide 5 Copyright © 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 … b 4, b 3, b 2, b, 1. Hindu-Arabic system uses base 10.

Chapter 4 Section 3 - Slide 6 Copyright © 2009 Pearson Education, Inc. Example: Converting from Base 8 to Base 10 Convert to base 10. Solution:

Chapter 4 Section 3 - Slide 7 Copyright © 2009 Pearson Education, Inc. Example: Converting from Base 5 to Base 10 Convert 42 5 to base 10. Solution:

Chapter 4 Section 3 - Slide 8 Copyright © 2009 Pearson Education, Inc. Example: Convert to Base 3 Convert 342 to base 3. Solution: The place values in the base 3 system are …, 3 6, 3 5, 3 4, 3 3, 3 2, 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 9 Copyright © 2009 Pearson Education, Inc. Example: Convert to Base 3 (continued) Successive division by the powers of the base gives the following result.

Chapter 4 Section 3 - Slide 10 Copyright © 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.

Chapter 4 Section 3 - Slide 11 Copyright © 2009 Pearson Education, Inc. Computers Computers make use of three numeration systems  Binary  Octal  Hexadecimal

Chapter 4 Section 3 - Slide 12 Copyright © 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