D.2. Abbreviating Binary Numbers as Octal and Hexadecimal Numbers

The main use for octal and hexadecimal numbers in computing is for abbreviating lengthy binary representations. Figure D.7 highlights the fact that lengthy binary numbers can be expressed concisely in number systems with higher bases than the binary number system.

Figure D.7. Decimal, binary, octal and hexadecimal equivalents.

Decimal number

Binary representation

Octal representation

Hexadecimal representation

0

0

0

0

1

1

1

1

2

10

2

2

3

11

3

3

4

100

4

4

5

101

5

5

6

110

6

6

7

111

7

7

8

1000

10

8

9

1001

11

9

10

1010

12

A

11

1011

13

B

12

1100

14

C

13

1101

15

D

14

1110

16

E

15

1111

17

F

16

10000

20

10

A particularly important relationship that both the octal number system and the hexadecimal number system have to the binary system is that the bases of octal and hexadecimal (8 and 16 respectively) are powers of the base of the binary number system (base 2). Consider the following 12-digit binary number and its octal and hexadecimal equivalents. See if you can determine how this relationship makes it convenient to abbreviate binary numbers in octal or hexadecimal. The answer follows the numbers.


Binary number

Octal equivalent

Hexadecimal equivalent

100011010001

4321

8D1

To see how the binary number converts easily to octal, simply break the 12-digit binary number into groups of three consecutive bits each, starting from the right, and write those groups over the corresponding digits of the octal number as follows:

100 011 010 001
4 3 2 1

Note that the octal digit you have written under each group of thee bits corresponds precisely to the octal equivalent of that 3-digit binary number, as shown in Fig. D.7.

The same kind of relationship can be observed in converting from binary to hexadecimal. Break the 12-digit binary number into groups of four consecutive bits each, starting from the right, and write those groups over the corresponding digits of the hexadecimal number as follows:

1000 1101 0001
8 D 1

Notice that the hexadecimal digit you wrote under each group of four bits corresponds precisely to the hexadecimal equivalent of that 4-digit binary number as shown in Fig. D.7.

Introduction to Computers, the Internet and World Wide Web

Introduction to C++ Programming

Introduction to Classes and Objects

Control Statements: Part 1

Control Statements: Part 2

Functions and an Introduction to Recursion

Arrays and Vectors

Pointers and Pointer-Based Strings

Classes: A Deeper Look, Part 1

Classes: A Deeper Look, Part 2

Operator Overloading; String and Array Objects

Object-Oriented Programming: Inheritance

Object-Oriented Programming: Polymorphism

Templates

Stream Input/Output

Exception Handling

File Processing

Class string and String Stream Processing

Web Programming

Searching and Sorting

Data Structures

Bits, Characters, C-Strings and structs

Standard Template Library (STL)

Other Topics

Appendix A. Operator Precedence and Associativity Chart

Appendix B. ASCII Character Set

Appendix C. Fundamental Types

Appendix D. Number Systems

Appendix E. C Legacy Code Topics

Appendix F. Preprocessor

Appendix G. ATM Case Study Code

Appendix H. UML 2: Additional Diagram Types

Appendix I. C++ Internet and Web Resources

Appendix J. Introduction to XHTML

Appendix K. XHTML Special Characters

Appendix L. Using the Visual Studio .NET Debugger

Appendix M. Using the GNU C++ Debugger

Bibliography



C++ How to Program
C++ How to Program (5th Edition)
ISBN: 0131857576
EAN: 2147483647
Year: 2004
Pages: 627

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