Chapter 4: The Building Blocks: Binary Numbers, Boolean Logic, and Gates

1
Chapter 4 The Building Blocks Binary Numbers,
Boolean Logic, and Gates

• Invitation to Computer Science,
• C Version, Third Edition
• Spring 2005 Additions by S. Steinfadt

2
Objectives

• In this chapter, you will learn about
• The binary numbering system
• Boolean logic and gates
• Building computer circuits
• Control circuits

3
Introduction

• Chapter 4 focuses on hardware design (also called
  logic design)
• How to represent and store information inside a
  computer
• How to use the principles of symbolic logic to
  design gates
• How to use gates to construct circuits that
  perform operations such as adding and comparing
  numbers, and fetching instructions

4
The Binary Numbering System

• A computers internal storage techniques are
  different from the way people represent
  information in daily lives
• Information inside a digital computer is stored
  as a collection of binary data

5
Binary Representation of Numeric and Textual
Information

• Binary numbering system
• Base-2
• Built from ones and zeros
• Each position is a power of 2
• 1101 1 x 23 1 x 22 0 x 21 1 x 20
• Decimal numbering system
• Base-10
• Each position is a power of 10
• 3052 3 x 103 0 x 102 5 x 101 2 x 100

6

• Figure 4.2
• Binary-to-Decimal
• Conversion Table

7
Binary Representation of Numeric and Textual
Information (continued)

• Representing integers
• Decimal integers are converted to binary integers
• Given k bits, the largest unsigned integer is 2k
  - 1
• Given 4 bits, the largest is 24-1 15
• Signed integers must also represent the sign
  (positive or negative) - Sign/Magnitude notation

8
Binary Representation of Numeric and Textual
Information (continued)

• Representing real numbers
• Real numbers may be put into binary scientific
  notation a x 2b (or M x BE)
• Example 101.11 x 20
• Number then normalized so that first significant
  digit is immediately to the right of the binary
  point
• Example .10111 x 23
• Mantissa and exponent then stored

9
Binary Representation of Numeric and Textual
Information (continued)

• Characters are mapped onto binary numbers
• ASCII code set
• 8 bits per character 256 character codes
• UNICODE code set
• 16 bits per character 65,536 character codes
• Text strings are sequences of characters in some
  encoding

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Binary Representation of Textual Information
(contd)
Decimal Binary Val.
48 00110000 0
49 00110001 1
50 00110010 2
51 00110011 3
52 00110100 4
53 00110101 5
54 00110110 6
55 00110111 7
56 00111000 8
57 00111001 9
58 00111010
59 00111011
60 00111100 lt
61 00111101
62 00111110 gt
63 00111111 ?
64 01000000 _at_
65 01000001 A
66 01000010 B
Dec. Unicode Charac.
0x30 0x0030 0
0x31 0x0031 1
0x32 0x0032 2
0x33 0x0033 3
0x34 0x0034 4
0x35 0x0035 5
0x36 0x0036 6
0x37 0x0037 7
0x38 0x0038 8
0x39 0x0039 9
0x3A 0x003A
0x3B 0x003B
0x3C 0x003C lt
0x3D 0x003D
0x3E 0x003E gt
0x3F 0x003F ?
0x40 0x0040 _at_
0x41 0x0041 A
0x42 0x0042 B
Unicode 16 bits long
ASCII 8 bits long
Partial listings only!
11
Binary Representation of Sound and Images

• Multimedia data is sampled to store a digital
  form, with or without detectable differences
• Representing sound data
• Sound data must be digitized for storage in a
  computer
• Digitizing means periodic sampling of amplitude
  values

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Binary Representation of Sound and Images
(continued)

• From samples, original sound may be approximated
• To improve the approximation
• Sample more frequently (increase sampling rate)
• Use more bits for each sample value (? bit depth)

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• Figure 4.5
• Digitization of an Analog Signal
• (a) Sampling the Original
• Signal
• (b) Recreating the
• Signal from the Sampled
• Values

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Binary Representation of Sound (contd)

• MP3 format discussed in text, AAC format here
• AAC (Advanced Audio Coding) advantages over MP3
• Improved compression provides higher-quality
  results with smaller file sizes
• Higher resolution audio, yielding sampling rates
  up to 96 kHz
• Improved decoding efficiency, requiring less
  processing power for decode
• http//www.apple.com/mpeg4/aac/
• http//www.aac-audio.com/

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Binary Representation of Sound and Images
(continued)

• Representing image data
• Images are sampled by reading color and intensity
  values at even intervals across the image
• Each sampled point is a pixel
• Image quality depends on number of bits at each
  pixel

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Binary Representation of Images (contd)

• Representing image data
• Images are sampled by reading color and intensity
  values at even intervals across the image
• Each sampled point is a pixel
• Image quality depends on number of bits at each
  pixel
• More image information http//cat.xula.edu/tutori
  als/imaging/grayscale.php

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The Reliability of Binary Representation

• Electronic devices are most reliable in a
  bistable environment
• Bistable environment
• Distinguishing only two electronic states
• Current flowing or not
• Direction of flow
• Computers are bistable hence binary
  representations

18
Binary Storage Devices

• Magnetic core
• Historic device for computer memory
• Tiny magnetized rings flow of current sets the
  direction of magnetic field
• Binary values 0 and 1 are represented using the
  direction of the magnetic field

19

• Figure 4.9
• Using Magnetic Cores to Represent Binary Values

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Binary Storage Devices (continued)

• Transistors
• Solid-state switches either permits or blocks
  current flow
• A control input causes state change
• Constructed from semiconductors

21

• Figure 4.11
• Simplified Model of a Transistor