1. Invitation to Computer Science 6th Edition
Chapter 4
The Building Blocks: Binary Numbers, Boolean Logic, and Gates

2. Introduction
Chapter 4 focuses on hardware design (also called logic design)
How to represent and store information inside a computer--The binary numbering system
How to use the principles of symbolic logic to design gates--Boolean logic and gates
How to use gates to construct circuits that perform operations such as adding and comparing numbers, and fetching instructions--Building computer circuits
Control circuits
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3. Introduction Computing agent
Abstract concept representing any object capable of understanding and executing our instructions
Fundamental building blocks of all computer systems
Binary representation
Boolean logic
Gates
Circuits
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4. The Binary Numbering System
Binary representation of numeric and textual information
Two types of information representation
External representation
Internal representation
Binary is a base-2 positional numbering system
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5. Figure 4.1 Distinction Between External Memory and Internal Representation of Information
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6. Binary Representation of Numeric and Textual Information
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7. Binary Representation of Numeric and Textual Information
Binary-to-decimal algorithm
Whenever there is a 1 in a column, add the positional value of that column to a running sum
Whenever there is a 0 in a column, add nothing
The final sum is the decimal value of this binary number
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8. Figure 4.2 Binary-to-Decimal Conversion Table
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9. Binary Representation of Numeric and Textual Information (continued)
To convert a decimal value into its binary equivalent
Use the decimal-to-binary algorithm
Maximum number of binary digits that can be used to store an integer: 16, 32, or 64 bits
Given k bits, the largest unsigned integer is 2k-1
Given 16 bits the largest is 216-1=65535
Arithmetic overflow
Operation that produces an unsigned value greater than 65,535
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10. Binary Representation of Numeric and Textual Information (continued)
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11. Binary Representation of Numeric and Textual Information (continued)
How to express signed integer?
Sign/Magnitude notation (old computer)
Two's complement notation (current computer)
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12. Sign/Magnitude notation
1(sign bit) (total: 7 bits, computer A)
When it is a signed value, it is -49
When it is a unsigned value, it is 113
1(sign bit) (total: 8 bits, computer B)
When it is a unsigned value, it is 177
You must tell the computer if it is signed or unsigned integer. The number of bit depends on computer (8bit, 16bit, 32bit, 64bit)
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14. Two's complement notation
Why we have this solution?
Because of two zeros problem in Sign/Magnitude notation : 10000, 00000
if (a = b)
do operation 1
else
do operation 2
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15. Two's complement notation
If A > 0
then do nothing
else
get complement value of each bit
a <- a+1
Eg: -3 (3bit): 3 -> 011 -> 100 -> 101
-3 (4bit): 3 -> > > 1101
The number of bit depends on computer (8bit, 16bit, 32bit, 64bit)
Easier method: Get complement value of each bit until first 1
Eg: -3 (3bit): 3-> 011 -> 101
-3 (4bit) : 3 -> > 1101
No substraction in two’s complement notation. Convert to complement value.
Eg 5 – 3 = 5 + (-3)
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16. Two's complement notation
Bit pattern Decimal Value
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17. Compare value range Suppose we have k bit.
Sign/Magnitude notation: -(2k-1 – 1) to (2k-1 – 1)
Eg 3bit: -3 ~ +3
Two's complement notation: -(2k-1) to (2k-1 – 1)
Eg 3bit: -4 ~ +3
Why different?
Because Sign/Magnitude notation has two zeros while two’s complement notation has only one zero.
Though two’s complement notation is difficult to human, it much clearer to computer.
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18. Invitation to Computer Science, 6th Edition

19. Binary <-> Decimal (Fractional number)
Decimal -> Binary
* 2 = Get 1
0.625 * 2 = Get 1
0.25 * = Get 0
0.5 * = Get 1
Final: (Attention: compare to integer conversion)

20. Fractional Numbers Fractional numbers (12.34 and –0.001275)
Can be represented in binary by using signed-integer techniques
Scientific notation
±M x B±E
M is the mantissa, B is the exponent base (usually 2), and E is the exponent
Normalize the number
First significant digit is immediately to the right of the binary point
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21. Binary Representation of Numeric and Textual Information (continued)
Representing real numbers
Real numbers should be put into binary scientific notation: a x 2b
Example: x 20
Number then normalized so that first significant digit is immediately to the right of the binary point
Example: x 23
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22. Normalization Example
Binary Normalization
5.75 = = * 23
Mantissa and exponent then stored
±M * B ±E
(8 bits for mantissa, 6 bits for exponent)
5.75 = = * 24 = * 22
* 24 and * 22 are not normalized number

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24. Textual Information Code mapping ASCII UNICODE
Assigning each printable letter or symbol in our alphabet a unique number
ASCII
International standard for representing textual information in the majority of computers
Uses 8 bits to represent each character
UNICODE
Uses a 16-bit representation for characters rather than the 8-bit format of ASCII
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25. Figure 4.3 ASCII Conversion Table
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26. Binary Representation of Sound and Images
Digital representation
Values for a given object are drawn from a finite set
Analog representation
Objects can take on any value
Figure 4.4
Amplitude of the wave: measure of its loudness
Period of the wave (T): time it takes for the wave to make one complete cycle (pitch—highness or lowness)
Frequency f: total number of cycles per unit time (cycles/seconds or hertz)
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27. Figure 4.4 Example of Sound Represented as a Waveform
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28. Binary Representation of Sound and Images (continued)
Digitizing means periodic sampling of amplitude values
Sampling rate
Measures how many times per second we sample the amplitude of the sound wave
Bit depth
Number of bits used to encode each sample
MP3
Most popular and widely used digital audio format
Mp3 44, bits
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29. Figure 4.5 Digitization of an Analog Signal
(a) Sampling the Original Signal
(b) Re-creating the Signal from the Sampled Values
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30. Binary Representation of Sound and Images (continued)
From samples, original sound can be approximated
To improve the approximation
Sample more frequently
Use more bits for each sample value
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31. Binary Representation of Sound and Images (continued)
Representing image data
Images are sampled by reading color and intensity values
Each sampled point is a pixel
Image quality depends on number of bits at each pixel
0.05~0.1mm
Scanning
Measuring the intensity values of distinct points located at regular intervals across the image’s surface
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32. Binary Representation of Sound and Images (continued)
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33. Invitation to Computer Science, 6th Edition

34. Data Compression Why we need data compression?
Because the original data need too much space.
Eg. 3,000,000 pixels/photograph * 24 bits/pixel = 72million bits.
Data compression algorithms
Attempt to represent information in ways that preserve accuracy while using significantly less space
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35. Data Compression Lossless compression schemes
No information is lost in the compression
It is possible to exactly reproduce the original data
Lossy compression schemes
Do not guarantee that all of the information in the original data can be fully and completely recreated
Compression ratio
Measures how much compression schemes reduce storage requirements of data
Compression rate = size of the uncompressed data / size of the compressed data
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36. Binary Representation of Sound and Images (continued)
Run-length encoding
Replaces a sequence of identical values v1, v2, . . ., vn by a pair of values (v, n)
for image compression
Eg. ( ), (255, 0, 0), (255, 255, 255), (255, 0, 0), (255, 255, 255) → (255, 4), (0, 2), (255, 4), (0, 2), (255, 3)
Variable length code sets
Often used to compress text
Can also be used with other forms of data
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37. Data Compression
Simple compression method - “variable length code sets” - for text compress
Letter 4 bit encoding Variable length encoding A I H W
H A W A I I → 0010, 0000, 0100, 0000, 0001, 0001 → 001, 00, 110, 00, 10, 10
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38. Figure 4.8 Using Variable Length Code Sets (a) Fixed Length
(b) Variable Length
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39. Invitation to Computer Science, 6th Edition

40. The Reliability of Binary Representation
Computers use binary representation for reasons of reliability
Building a base-10 “decimal computer”
Requires finding a device with 10 distinct and stable energy states that can be used to represent the 10 unique digits (0, 1, , 9) of the decimal system
Bistable environment
Only two (rather than 10) stable states separated by a huge energy barrier
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41. Binary Storage Devices
Magnetic core
‰Historic device for computer memory
‰Tiny magnetized rings; flow of current sets the direction of magnetic field direction of magnetic field
Binary values 0 and 1are represented using the direction of the magnetic field
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42. Figure 4.9 Using Magnetic Cores to Represent Binary Values
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43. Binary Storage Devices (continued)
Transistors
‰Solid-state switches; either permit or block current flow
‰A control input causes state change
‰Constructed from semiconductors
‰Can be printed photographically on a wafer of silicon to produce a device known as an integrated circuit to produce a device known as an integrated circuit
„Circuit board
‰Interconnects all the different chips needed to run a computer system
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44. Figure 4.10 Relationships Among Transistors, Chips, and Circuit Boards
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45. Binary Storage Devices (continued)
Figure 4.11
Control (base): used to open or close the switch inside the transistor
ON state: current coming from the In line (Collector) can flow directly to the Out line (Emitter), and the associated voltage can be detected by a measuring device
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46. Figure 4.11 Simplified Model of a Transistor
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47. Boolean Logic Boolean logic
Construction of computer circuits is based on this
Boolean expression
Any expression that evaluates to either true or false
Constructed by combining together Boolean operations
Example: (a AND b) OR ((NOT b) AND (NOT a))
Truth table
A column for each input plus the output
A row for each combination of input values
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48. Figure 4.12 Truth Table for the AND Operation
Truth Table: Can express the idea that the AND operation produces the value true if and only if both of its components are true
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49. Boolean Logic (continued)
Boolean operations
a AND b
True only when a is true and b is true
a OR b
True when a is true, b is true, or both are true
NOT a
True when a is false and vice versa
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50. Boolean Logic (continued)
Boolean operations
AND, OR, NOT
Binary operators
Require two operands
Unary operator
Requires only one operand
NOT operation
Reverses, or complements, the value of a Boolean expression
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51. Figure 4.13 Truth Table for the OR Operation
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52. Figure 4.14 Truth Table for the NOT Operation
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53. Boolean Logic (continued)
Example:
(a AND b) OR ((NOT b) and (NOT a)) a b
Value 1
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54. Invitation to Computer Science, 6th Edition

55. Invitation to Computer Science, 6th Edition

56. Gates
Gates
Hardware devices built from transistors to mimic Boolean logic
Transforms a set of (0 1) input values into a single ‰Transforms a set of (0,1) input values into a single (0,1) output value
AND gate
Two input lines, one output line
Outputs a 1 when both inputs are 1
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57. Gates (continued) OR gate NOT gate Two input lines, one output line
Outputs a 1 when either input is 1
NOT gate
One input line, one output line
Outputs a 1 when input is 0 and vice versa
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58. Figure 4.15 The Three Basic Gates and Their Symbols
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59. Figure 4.16 Construction of a NOT Gate
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60. Figure 4.17 Construction of NAND and AND Gates
(a) A Two-transistor NAND Gate (b) A Three-transistor AND Gate
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61. Gates (continued) NAND (acronym for NOT AND) To construct an OR gate
Produces the complement of the AND operation
To construct an OR gate
Start with two transistors
Transistors are connected in parallel
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62. Figure 4.18 Construction of NOR and OR Gates
(a) A Two-transistor NOR Gate
(b) A Three-transistor OR Gate
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63. Gates (continued) „Abstraction in hardware design
‰Map hardware devices to Boolean logic
‰Design more complex devices in terms of logic, not electronics
‰Conversion from logic to hardware design can be automated
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64. Building Computer Circuits
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65. Building Computer Circuits
Every Boolean expression:
Can be represented pictorially as a circuit diagram
Every output value in a circuit diagram:
Can be written as a Boolean expression
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66. Figure 4.19 Diagram of a Typical Computer Circuit
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67. A Circuit Construction Algorithm
Sum-of-products algorithm is one way to design circuits
Truth table to Boolean expression to gate layout
Step 1: Truth Table Construction
Determine how the circuit should behave under all possible circumstances
If a circuit has N input lines and if each input line can be either a 0 or a 1, then:
There are 2N combinations of input values, and the truth table has 2N rows
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68. A Truth Table for a Circuit with 8 Input Combinations
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69. A Circuit Construction Algorithm (continued)
Step 2: Subexpression Construction Using AND and NOT Gates
Choose any one output column of the truth table built in step 1, and scan down that column
Every place that you find a 1 in that output column, you build a Boolean subexpression that produces the value 1 for exactly that combination of input values and no other
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70. Output Column Labeled Output-1 from the Previous Truth Table
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71. Taking Snapshots Step 3: Subexpression Combination Using OR Gates
Take each of the subexpressions produced in step 2 and combine them, two at a time, using OR gates
Step 4: Circuit Diagram Production
Construct the final circuit diagram
Algorithms for circuit optimization
Reduce the number of gates needed to implement a circuit
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72. Figure 4.20 Circuit Diagram for the Output Labeled Output-1
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73. Figure 4.21 The Sum-of-Products Circuit Construction Algorithm
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74. Invitation to Computer Science, 6th Edition

75. Invitation to Computer Science, 6th Edition

76. Examples of Circuit Design and Construction
A Compare-For-Equality Circuit
Tests two unsigned binary numbers for exact equality
Produces the value 1 (true) if the two numbers are equal and the value 0 ( false) if they are not
Built by combining together 1-bit comparison circuits (1-CE)
Integers are equal if corresponding bits are equal (AND together 1-CE circuits for each pair of bits)
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77. A Compare-for-Equality Circuit (continued)
1-CE circuit truth table a b
Output 1
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78. A Compare-for-Equality Circuit (continued)
1-CE Boolean expression
First case: (NOT a) AND (NOT b)
Second case: a AND b
Combined:
((NOT a) AND (NOT b)) OR (a AND b)
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79. Figure 4.22 One-Bit Compare for Equality Circuit
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80. Figure 4.23 N-Bit Compare for Equality Circuit
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81. An Addition Circuit Full adder Figure 4.27 Addition circuits
Performs binary addition on two unsigned N-bit integers
Figure 4.27
Shows the complete full adder circuit called ADD
Addition circuits
Found in every computer, workstation, and handheld calculator in the marketplace
Built from 1-bit adders (1-ADD)
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82. Figure 4.24 The 1-ADD Circuit and Truth Table
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83. Figure 4.25 Sum Output for the 1-ADD Circuit
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84. An Addition Circuit (continued)
Building the full adder
Put rightmost bits into 1-ADD, with zero for the input carry
Send 1-ADD’s output value to output, and put its carry value as input to 1-ADD for next bits to left
Repeat process for all bits
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85. Figure 4.26 Complete 1-ADD Circuit for 1-Bit Binary Addition
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86. Figure 4.27 The Complete Full Adder ADD Circuit
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87. Control Circuits Used to: Multiplexor
Determine the order in which operations are carried out
Select the correct data values to be processed
Multiplexor
Circuit that has 2N input lines and 1 output line
Function: to select exactly one of its 2N input lines and copy the binary value on that input line onto its single output line
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88. Figure 4.28 A Two-Input Multiplexor Circuit
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89. Figure 4.31 Example of the Use of a Multiplexor Circuit
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90. Control Circuits (continued)
Decoder
Has N input lines numbered 0, 1, 2, , N – 1 and 2N output lines numbered 0, 1, 2, 3, , 2N – 1
Determines the value represented on its N input lines and then sends a signal (1) on the single output line that has that identification number
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91. Figure 4.29 A 2-to-4 Decoder Circuit
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92. Figure 4.30 Example of the Use of a Decoder Circuit
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93. Summary Digital computers Binary values Boolean logic
Use binary representations of data: numbers, text, multimedia
Binary values
Create a bistable environment, making computers reliable
Boolean logic
Maps easily onto electronic hardware
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94. Summary (continued) Circuits Computational and control circuits
Constructed using Boolean expressions as an abstraction
Computational and control circuits
Can be built from Boolean gates
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95. Test 2 Chapter 3 & 4 18 questions: Multiple choice, Essay questions
Time efficiency analysis
Binary search algorithm
Binary, Decimal, Hexadecimal number conversion
Unsigned number
Boolean expression
Circuit construction
Multiple choice, Essay questions
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