Chapter 2_Number Systems and Codes

Lecture: DIGITAL SYSTEMS Nguyen Thanh Hai, PhD Chapter 2: Number Systems and Codes Number Systems and Codes University of Technical Education Faculty of Electrical & Electronic Engineeri

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Lecture:

DIGITAL SYSTEMS

Nguyen Thanh Hai, PhD

Chapter 2:

Number Systems and Codes

University of Technical Education

Faculty of Electrical & Electronic Engineering

2

2.1 Binary to Decimal Conversions 2.2 Decimal to Binary Conversions 2.3 Octal Number System

2.4 Hexadecimal Number System 2.5 BCD Code

2.6 Gray code

2.7 Putting It All Together

2.8 Applications

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Basic Concepts

Bit: a binary digit composed of 0 or 1

Byte: composed of 8 bits

Word: composed of 4 bytes and equal to 32 bits

Base 2 (binary digits): numbers composed of bits

Base 10 (decimal digit): numbers composed of the digits from 0-9 Base 16 (hexa digit or hexadecimal): numbers composed of the digits from 0-9 and letters A-F

Base 8 (octal digit): numbers composed of the digits from 0-7

4

Binary digits:

MSB (Most Significant Bit)

LSB (Least Significant Bit)

bit

1 Byte = 8 bits

bit bit

Kb

Binary system: 0 and 1

Binary digits

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Ex 1: 11010101 2 =

= (1x 2 7 ) + (1x 2 6 ) + (0x2 5 ) + (1x2 4 ) + (0x2 3 ) + (1x2 2 ) + (0x 2 1 ) + (1x 2 0 )

= 128 + 64 + 0x32 + 16 + 0x8 + 4 + 0x2 + 1

= 213 10

2.1 Binary to Decimal Conversion

Binary digits

Weights

MSD: Most Significant Digit

6

Ex 2: 1101.101 2 =

= (1x 2 3 ) + (1x 2 2 ) + (0x2 1 ) + (1x2 0 ) + (1x2 -1 ) + (0x2 -2 ) + (1x 2 -3 )

= 8 + 4 + 0 + 1 + 0.5 + 0 + 0.125

= 13.625 10

point

LSB

Binary digits

Weights

2.1 Binary to Decimal Conversion (Cont.)

Decimal point

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25

2 = 12 + remainder of 1

12

6

3

1

2510= 1 1 0 0 12

LSB

MSB

EX 3:

45 10 =

= 1 0 1 1 0 1 2

Convert number 25:

2.2 Decimal to Binary Conversion

8

0.25x2 = 0.50 + remainder of 0

0.25 10 = 0 1 0 2

= 0.010 2

MSB

LSB

Convert number 0.25:

0.5x2 = 1.0 + remainder of 1

0x2 = 0 + remainder of 0

0.010= 0x2 -1 + 1x2 -2 + 0x2 -3

= 0 + 0.25 + 0=0.25

Notice:

2.2 Decimal to Binary Conversion

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Ex 4: 372 8 =

= 3x(8 2 ) + 7x(8 1 ) + 2x(8 0 )

= 3x64 + 7x8 + 2x1

= 250 10

2.3 Octal Number System

Ex 5: 24.6 8 =

= 2x(8 1 ) + 4x(8 0 ) + 6x(8 -1 )

= 2x8 + 4x1 + 6/8

= 20.75 10

Octal to Decimal Conversion

10

Decimal to Octal Conversion

266

33

4

26610= 4 1 28

LSD

MSD

EX 6: 14510 =

145

18

8 = 2 + remainder of 2 2

14510= 2 2 18

LSD

MSD

Convert number 266:

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Ý nghĩa một số nhị phân: 0 1 2 3 4 5 6 7

000 001 010 011 100 101 110 111

Octal to Binary Conversion

Octal digit

Binary

equivalent

Ex 7:

472 8 =

4 7 2

100 111 010

12

000 001 010 011 100 101 110 111

Binary to Octal Conversion

Octal

equivalent

Binary digit

Ex 8:

111 101 110

7 5 6 8

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Ex 9:

356 16 =

= 3x(16 2 ) + 5x(16 1 ) + 6x(16 0 )

= 3x256 + 5x15 + 6x1

= 768 + 80 + 6

= 854 10

2.4 Hexadecimal Number System

Ex 10: 2AF 16 =

= 2x(16 2 ) + 10x(16 1 ) + 15x(16 0 )

= 512 + 160 + 15

= 687 10

Ex 11:

2B.E 16 =

= 2x(16 1 ) + 11x(16 0 ) + 14x(16 -1 )

= 32 + 11 + 0.875

= 43.875 10

Hex to Decimal Conversion

14

Decimal to Hex Conversion

423

26

1

42310= 1 A 716

LSD

MSD

EX 12: 14510 = 145

9

14510= 9 116

LSD MSD

Convert number 423:

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Hex to Binary Conversion

9F2 16 = 9 F 2

= 1001 1111 0010

= 100111110010 2

1110100110 2 = 0011 1010 0110

= 3 A 6

= 3A6 16

Binary to Hex Conversion

EX 13: 1AF 16 =

EX 14:

11010011001 2 =

16

Hex to Octal Conversion

B2F16 = 1011 0010 1111

= 101 100 101 111

= 5 4 5 78

2.5 BCD Code ( Binary-Coded-Decimal Code)

8 7 4 (decimal)

1000 0111 0100 (BCD)

EX 15: 1BC16 =

EX 16: 195BCD =

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• Comparison of BCD and Binary

It is important to realize that BCD is not another

number system like binary, decimal, and

hexadecimal

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2.6 The Gray Code

The unique aspect of the Gray code is that only one bit ever changes between two successive numbers in the sequence.

18

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Converting binary to Gray

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Converting Gray to binary.

20

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Decimal Binary Octal Hex BCD Gray

2.7 Putting It All Together

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Example 1:

Example 2:

Solution:

1 Convert to decimal

2 11000 2

3 2 12 - 1

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Application 1:

A typical CD-ROM can store 650 megabytes of digital data Since 1 mega = 2 20 , how many bits of data can a CD-ROM hold?

Solution:

650 x 2 20 x 8 = 5,452,595,200 bits

2.8 Applications

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Application 2:

A small process-control computer

uses octal codes to represent its

12-bit memory addresses.

a How many octal digits are

required?

b What is the range of addresses

in octal?

c How many memory locations are

there?

2.8 Applications

Solution:

a 12/3 = 4

b 0000 8 to 7777 8

c With 4 octal digit,

8 4 = 4096

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Application 3:

A typical PC uses a 20-bit

address code for its memory

locations.

a How many hex digits are

needed to represent a

memory address?

b What is the range of

addresses?

c What is the total number of

memory locations?

c With 5 hex digits, 16 5 =

1,048,576

b 00000 16 to FFFFF 16

Solution:

a 20/4 = 5 hex digits

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Application 4:

Most calculators use BCD to

store the decimal values as

they are entered into the

keyboard and to drive the digit

displays.

a If a calculator is designed to

handle 8-digit decimal

numbers, how many bits does

this require?

b What bits are stored when the

number 375 is entered into the

calculators?

b 375 10 converts to

0011 0111 0101 (BCD) Solution:

a 8 x 4 = 32 bits

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Open and closed

switches

representing 0 and

1, respectively of holes in paper tape Absence or presence

representing 0 and 1, respectively

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Typical voltage

assignments in digital

system

Typical digital system timing diagram

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A digital circuit

responds to an

input’s binary

level (0 or 1) and

not to its actual

voltage

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Application 5:

Solution:

1 False

2 Yes

3 Logic

4 Timing diagram

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Parallel transmission between two

computers with 8 bits

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Series transmission between two

computers

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Functional Diagram

of a digital

computer

Comparison between non-memory and memory operation

34

-Take a look Examples

- Answer Review questions at pages 20

- Homework at pages 21-22

[1] Ronald J Tocci, Neal S Widmer, Digital

Systems: Principles and Applications, 8th

Ed Prentice Hall, 2007.

[2] N D Phu, N T Duy Giao Trinh: Ky

Thuat So, 2013.

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