Copyright (c) 2004 Professor Keith W. Noe Number System & Codes Number Conversions Part II.

Copyright (c) 2004 Professor Keith W. Noe

Number System & Codes

Number ConversionsPart II


Reading Assignment

Digital Design with CPLD Applications and VHDL, by Robert K. Dueck

Pages 6 through 17


Objectives

Convert decimal numbers to binary, octal, & hexadecimal.

Convert binary numbers to octal, & hexadecimal.

Convert octal numbers to binary. Convert hexadecimal numbers to binary.

Upon the successful completion of this lesson, you should be able to:


Number Conversions Number conversion from base 10 to

bases 2, 8, & 16 will be discussed first.

Next conversion from binary to bases 8 and 16 will be discussed.

Then we will discuss converting base 8 to binary.

Last, we will discuss converting base 16 numbers to binary.


Why so many number systems? Digital & microprocessor-based

electronic circuitry use the binary number system.

Man uses the decimal number system. Because the binary number system uses

only 0 and 1, it is hard for us to work with such huge binary numbers.

Example: 1001111101110111011011102

Errors are usually the rule when working only in the binary number system.


Why so many number systems? Technicians function better when

using the decimal number system. This is the number system we use

every day of our life. The octal number system (base 8)

closely resembles the decimal number system.


Why so many number systems? Another number system that resembles

the decimal number system is the hexadecimal number system.

The hexadecimal or base 16 number system has 16 symbols, some of which are the first 6 letters of the alphabet.

This number system is also used with digital systems.

This system is usually referred to as Hex (short for hexadecimal).


What does this mean for me?

Use all four number systems with relative ease.

Accurately convert numbers between all four number systems.

Computers and output circuits used by computers output codes using one of these four number systems.

As a technician working on digital-based circuits, you must be able to:


Decimal to

BinaryOctal

Hexadecimal

Number Conversions


Converting Decimal Numbers to Binary, Octal & Hex Converting decimal numbers to

any one of these number systems uses exactly the same process.

The process that you will use is repeated division with the integer portion of the decimal number being converted.


Converting Decimal Numbers to Binary, Octal & Hex Use the old style of division

9 5 = 1 R 4 It is important that you use this

process as shown above. The remainders form the number

in the new number system you are converting the decimal number to.


Number Conversion

Converting 2910 to Binary


Decimal to Binary Conversion To convert a decimal number to

binary, you will repeatedly divide the decimal number by 2 keeping track of the remainders.

Be sure to keep track of the remainders as the remainders are used to form the binary equivalent number.


Decimal to Binary Conversion

1 2 = 0 R 1 3 2 = 1 R 1 7 2 = 3 R 114 2 = 7 R 029 2 = 14 R 1

To read the binary equivalent of 2910, read the remainders from the top down: 11101


Practice SessionPractice Converting these decimal numbers to binary:

4410

11710

14210

25510


Practice Session

Answers:

4410 = 1011002

11710 = 11101012

14210 = 100011102

25510 = 111111112


Converting Decimal Numbers to Octal


Decimal to Octal Conversion Use the same process that you used

when converting decimal numbers to binary.

Use the repeated division process. When converting decimal numbers to

octal, divide the decimal number by 8.

Keep track of the remainders, The remainders form the octal equivalent.


Decimal to Octal Conversion

Convert 18310 to Octal.


Decimal to Octal Conversion

183 8 = 22 R 7 22 8 = 2 R 6 2 8 = 0 R 2

Use Repeated Division Dividing by 8

18310 = 2678


Practice SessionPractice converting these decimal numbers to octal.

7910

19410

20810

25510


Practice SessionAnswers

7910 = 1178

19410 = 3028

20810 = 3208

25510 = 3778


Converting Decimal Numbers to Hexadecimal


Conversion of Decimal Numbers to Hexadecimal The same process is used for

converting decimal numbers to hex that is for converting decimal numbers to binary and octal.

Use the process of repeated division keeping track of the remainders.

When converting decimal numbers to hex, divide the decimal number by 16.


Conversion of Decimal Numbers to Hexadecimal Do not forget, when remainders

are 10 or higher, convert the remainder to the appropriate letter of the alphabet.

10 = A, 11 = B, 12 = C, 13 = D, 14 = E, and F = 15.


Decimal-to-Hexadecimal Conversion

Convert 19510 to Hexadecimal.


Decimal-to-Hexadecimal Conversion

195 16 = 12 R 3 12 16 = 0 R 12 (C)

19510 = C316


Practice SessionPractice converting the following decimal numbers to hexadecimal.

5710

13810

21710

25510


Practice SessionSolutions

5710 = 3916

13810 = 8A16

21710 = D916

25510 = FF16


Other Number System Conversion Methods


Other Number Conversions There are times when it is

necessary to convert binary numbers to octal & vice versa;

Between the binary and hexadecimal numbers systems.

These conversions basically do not require math such as multiplication or division.


Binary-to-Octal Conversion


Binary-to-Octal Conversion It is a simple process. Begin by dividing the binary

number into groups of three bits each beginning on the right.

Convert each group of three bits into its equivalent octal number from 0 to 7.


Binary-to-Octal Equivalency

000 = 0

001 = 1

010 = 2

011 = 3

100 = 4

101 = 5

110 = 6

111 = 7



Convert 101110012 to Octal.



10111001

10 | 111 | 001 2 7 1

101110012 = 2718


Practice Session

Convert the following binary numbers to octal.

011011002

101100112

001010012

111111112


Practice Session

Answers

011011002 = 1548

101100112 = 2638

001010012 = 0518

111111112 = 3778


Octal-to-Binary Conversion


Octal-to-Binary Conversion The method for converting octal

numbers to binary is similar to the method used for converting binary numbers to octal.

First, separate the octal digits. Second, write the binary

equivalent for each of the octal digits.



Convert 2538 to Binary



2 5 38

2 | 5 | 3

10 101 011

2538 = 101010112


Practice Session

Convert the following octal numbers to binary. 0378

1168

1458

2738


Practice Session

Answers

0378 = 001111112

1168 = 010011102

1458 = 011001012

2738 = 101110112


Binary-to-Hexadecimal Conversion


Binary-to-Hexadecimal Conversion Converting binary numbers to

hexadecimal is similar to the process used for converting binary numbers to octal.

When converting binary numbers to hexadecimal, divide the 8-bit binary number in-half.


Binary-to-Hexadecimal Conversion After dividing the binary number in

half, write the hexadecimal equivalent for each 4-bit group.



Binary to Hex Equivalency

0000 = 0 1000 = 8

0001 = 1 1001 = 9

0010 = 2 1010 = A

0011 = 3 1011 = B

0100 = 4 1100 = C

0101 = 5 1101 = D

0110 = 6 1110 = E

0111 = 7 1111 = F



Convert 100111002 to Hexadecimal



10011100

1001 | 1100 9 C

100111002 = 9C16


Practice Session

Convert the following binary numbers to hexadecimal.

000010102

100111102

111100112

011111012


Practice Session

Solutions

000010102 = 0A16

100111102 = 9E16

111100112 = F316

011111012 = 7D16


Hexadecimal-to-Binary Conversion


Hexadecimal-to-Binary Conversion Converting a hexadecimal number

to binary is similar to converting an octal number to binary.

Divide the hexadecimal number into its individual numbers (symbols).

Write the binary equivalent for each symbol.



Convert 5C16 to binary.



5 C 5 | C 0101 1100

5C16 = 010111002


Practice Session

Convert the following hexadecimal numbers to binary.

1B16

9416

A516

FB16


Practice Session

Answers

1B16 = 000110112

9416 = 100101002

A516 = 101001012

FB16 = 111110112


Copyright (c) 2004 Professor Keith W. Noe Number System & Codes Number Conversions Part II.

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