1. Invitation to Computer Science, C++ Version, Third Edition
Chapter 4: The Building Blocks: Binary Numbers, Boolean Logic, and Gates
Invitation to Computer Science,
C++ Version, Third Edition

2. Objectives In this chapter, you will learn about:
Boolean logic and gates
The binary numbering system
Building computer circuits
Control circuits
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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
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4. The Reliability of Binary Representation
Sec 4.2.3
Electronic devices are most reliable in a bistable environment
Bistable environment
Distinguishes only two electronic states
Current flowing or not
Direction of flow
Easiest way to make computers reliable was to use a two-state design. Hence they are designed to store and process binary information.
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5. Binary Storage Devices
Sec 4.2.4
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
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6. Using Magnetic Cores to Represent Binary Values
Sec 4.2.4
Figure 4.9
Using Magnetic Cores to Represent Binary Values
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7. Binary Storage Devices (continued)
Sec 4.2.4
Transistors
Solid-state switches: either permits or blocks current flow
A control input causes state change
Constructed from semiconductors
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8. Simplified Model of a Transistor
Sec 4.2.4
Figure 4.11
Simplified Model of a Transistor
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9. Sec 4.2.4
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10. Sec 4.2.4
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11. Boolean Logic and Gates: Boolean Logic
Sec 4.3.1
Boolean logic is a branch of mathematics which describes operations on objects with two possible values, True or False
True/False values map easily onto a two-state electronic environment.
Boolean logic operations may be performed on electronic signals using devices built out of transistors and other electronic devices.
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12. Boolean Logic (continued)
Sec 4.3.1
Let a and b represent objects with two possible values, true and false
Boolean logic operations
a AND b
True only when a is true and b is true
a OR b
True when either a is true or b is true, or both are true
NOT a
True when a is false, and vice versa
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13. Boolean Logic (continued)
Sec 4.3.1
Boolean expressions
Constructed by combining together Boolean operations
Example: (a AND b) OR ((NOT b) AND (NOT a))
Truth tables capture the output/value of a Boolean expression
A column for each input plus the output
A row for each combination of input values
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14. Boolean Logic (continued)
Sec 4.3.1
Example: (a AND b) OR ((NOT b) and (NOT a)) _ _ alternate expression (a ∙ b) + ( b ∙ a ) a b
Value 1
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15. Gates
Sec 4.3.2
Gates
Hardware devices built from transistors to mimic Boolean logical operations on digital electrical signals
AND gate
Two input lines, one output line
Outputs a 1 when both inputs are 1
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16. Gates (continued) OR gate NOT gate Two input lines, one output line
Sec 4.3.2
OR 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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17. The Three Basic Gates and Their Symbols
Sec 4.3.1
Figure 4.15
The Three Basic Gates and Their Symbols
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18. Sec 4.3.2
NAND gate AND gate
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19. Sec 4.3.2
NOR gate
OR gate
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20. Gates (continued) Abstraction in hardware design
Sec 4.3.2
Abstraction in hardware design
Transform hardware devices to Boolean logical descriptions
Design more complex devices in terms of logic, not electronics
Conversion from logic to hardware design may be automated
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21. Diagram of a Typical Computer Circuit
Sec 4.4.1
Figure 4.19
Diagram of a Typical Computer Circuit
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22. Building Computer Circuits: Introduction
Sec 4.4.1
A circuit is a collection of logic gates:
Transforms a set of binary inputs into a set of binary outputs
Values of the outputs depend only on the current values of the inputs
Combinational circuits have no cycles in them (no outputs feed back into their own inputs)
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24. Invitation to Computer Science, C++ Version, Third Edition

25. Building Computer Circuits: Intro Cont’d.
Sec 4.4.1
inputs
output a b c 1
We want a 1 only when both inputs are 0.
Is there a gate which will do this?
An AND gate gives a 1 only when both inputs are 1.
An OR gate gives a 1 whenever at least one input is 1
Could we use an AND gate or an OR gate somehow?
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26. A Circuit Construction Algorithm
Sec 4.4.2
Sum-of-products algorithm is one way to design circuits:
Truth table to Boolean expression to gate layout
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27. The Sum-of-Products Circuit Construction Algorithm
Sec 4.4.2
Figure 4.21
The Sum-of-Products Circuit Construction Algorithm
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28. A Circuit Construction Algorithm (continued)
Sec 4.4.2
Sum-of-products algorithm
Truth table captures every input/output possible for circuit
Repeat process for each output line
Build a Boolean expression using AND and NOT for each 1 of the output line
Combine together all the expressions with ORs
Build circuit from whole Boolean expression
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29. The Binary Numbering System
A computer’s 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
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30. 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 x x x 20
Decimal numbering system
Base-10
Each position is a power of 10
3052 = 3 x x x x 100
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31. Figure 4.2 Binary-to-Decimal Conversion Table
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32. Taxonomy of Integers
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33. 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)
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34. Binary to Decimal Conversion
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35. Decimal to Binary Conversion
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36. Two’s Complement Representation of Signed Integers
Used by almost all computers today
All places hold the value they would in binary except for the leftmost place which is negative
8 bit integer:
Range: [-128,127]
If last bit (leftmost)
is 0, then positive  looks just as before
is 1, then negative add values for first 7 digits and then subtract 128
= = -115
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37. Two’s Complement Representation of Signed Integers
Converting from decimal to 2’s complement
For positive numbers: find the binary representation
For negative numbers:
Find the binary representation for its positive equivalent
Flip all bits
Add a 1 43
43 =
-43
43 =  
= = -43
We can use the usual laws of binary addition
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38. Two’s Complement Representation
= 106
= 106 !
What happened to the one that we carried at the end?
Got lost but still we got the right answer!
We carried into and carried out of the leftmost column
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39. Binary Representation of Numeric and Textual Information (continued)
Representing real numbers
Real numbers may 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
Mantissa and exponent then stored
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40. Binary Representation of Numeric and Textual Information (continued)
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41. Converting Decimal Fractions To Binary
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42. Converting Decimal Fractions To Binary
Note that the binary representation of is a repeating base two fraction,
This just says that 0.4 cannot be exactly represented as a sum of inverse powers of two.
.875 =
.400 =
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43. Binary Representation of Textual Information
Many transformations exist and all are arbitrary
Two popular
EBCDIC (Extended Binary Coded Decimal Interchange Code) by IBM
ASCII (American Standard Code for Information Interchange) by American National Standards Institute (ANSI)
Most widely used
Every letter/symbol is represented by 7 bits
How many letters/symbols do we have in total?
A-Z (26) , a-z (26) , 0-9 (10), symbols (33), control characters (33)
If using 1 byte/character  we have one extra bit
Extended ASCII-8 (more mathematical and graphical symbols or phonetic marks)
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44. 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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46. Binary Representation of Images
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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47. Binary Representation of Images
Divide the screen into a grid of cells each referred to as a pixel 512*256 image  grid has 512 columns and 256 rows
Pixel values and sizes depend on the type of the image
For black & white images, we can use 1 bit for every pixel such that 1  black and 0  white
For grayscale images, we use 1 byte where 255  black and 0  white and anything in between is gray (higher/lower values are closer to black/white)
For color images, we need three values per pixel (depends on the color scheme used):
Red/Green/Blue
We need 1 byte per value  3 bytes per pixel
For an 512x256 image
512x256x3 = 384K bytes
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51. Binary Representation of Movies
Image movies are built from a number of images (or frames) that are displayed in a certain sequence at high speeds
30 frames per second
2-hr movie needs (assume previous image used)
384K * 30 * 60 * 120 = 83 GB (billion) bytes!
People use compression to reduce large movie sizes
Usually the change between two consecutive images is small  store only difference between frames (Temporal compression)
Large areas with the same color can be stored by saving the boundary pixels only (everything within the boundaries has the same value) (Spatial compression)
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52. Binary Representation of Sound
Sound/Audio Data
An object produces sound when it vibrates in matter (e.g. air)
A vibrating object sends a wave of pressure fluctuations through the atmosphere
We hear different sounds from different vibrating objects because of variations in the sound wave frequency
Higher wave frequency means air pressure fluctuation switches back and forth more quickly during a period of time
We hear this as a higher pitch (tone)
When there are fewer fluctuations in a period of time, the pitch is lower
The level of air pressure in each fluctuation, the wave's amplitude or height, determines how loud the sound is
Cycle
TIME
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53. Binary Representation of Sound
Numbers used to represent the amplitude of sound wave
Analog is continuous and we need digital
Digitize the sound signal
Measure the voltage of the signal at certain intervals (40,000 per sec)
Reconstruct wave
Compression can also be used for audio files
MP3: reduces size to 1/10th
 faster transfer over the Internet
Digitizing a sound wave
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54. Digitization of an Analog Signal
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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55. Binary Representation of Sound and Images
Storage required
Text – 300 page novel
100,000 words x 5 char/word x 8 bits/char = 4 million bits
Music – WAV
44,100 samp/sec x 16bits/samp x 60 sec/min = 42 Mbits / min
For stereo there are left and right channels => 84 Mbits / min
Digital Photo – 3 Megapixel camera, 24 bit color
≈ 3,000,000 pixels/photo x 24 bits/pixel = 72 Mbits = 7 Mbytes
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56. Binary Representation of Sound and Images
Data Compression
Lossless
Run-length Encoding (RLE)
Variable length code sets (Huffman code)
Lossy
JPEG – pictures
MP3, ATRAC – audio
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57. Binary Representation of Sound and Images
Digital image and audio have a lot of advantages over non-digital ones
Can easily be modified by changing the bit pattern
Image enhancement, noise/distortion removal, etc …
Superimpose one sound on another or image on another results in newer ones
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58. Back To Circuit Design And Construction
Compare-for-equality circuit
Addition circuit
Both circuits can be built using the sum-of- products algorithm
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59. A Compare-for-equality Circuit
CE compares two unsigned binary integers for equality
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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60. A Compare-for-equality Circuit (continued)
1-CE circuit truth table a b
Output 1
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61. 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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62. One-Bit Compare for Equality Circuit
Figure 4.22
One-Bit Compare for Equality Circuit
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63. N-bit Compare for Equality Circuit
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64. Binary Addition
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65. An Addition Circuit Addition circuit
Adds two unsigned binary integers, setting output bits and an overflow
Built from 1-bit adders (1-ADD)
Starting with rightmost bits, each pair produces
A value for that order
A carry bit for next place to the left
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66. An Addition Circuit (cont’d)
Full 4-bit Adder Circuit
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67. An Addition Circuit (continued)
1-ADD truth table
Input
One bit from each input integer
One carry bit (always zero for rightmost bit)
Output
One bit for output place value
One “carry” bit
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68. An Addition Circuit (continued)ai bi ci
Ci+1
Sumi 1
Truth Table
Full 1-bit Adder
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69. The 1-ADD Circuit and Truth Table
Figure 4.24
The 1-ADD Circuit and Truth Table
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72. 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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74. Circuit Analysis Introduction
To analyze what a circuit does there are three approaches.
Convert the circuit into its corresponding truth table by hand
Simulate the circuit and generate its truth table from observations
Build the circuit and study its behavior fro various inputs
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75. Circuit Analysis Intro. Cont’d.
Generate truth table by hand  Remove Internal Lines
inputs
internal lines
Output a b c d e z 1
inputs
output a b z 1
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76. Control Circuits Do not perform computations
Choose order of operations or select among data values
Major types of controls circuits
Multiplexors
Select one of inputs to send to output
Decoders
Sends a 1 on one output line, based on what input line indicates
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77. Control Circuits Cont’d.
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78. Control Circuits (continued)
Decoder
Form
N input lines
2N output lines
N input lines indicate a binary number, which is used to select one of the output lines
Selected output sends a 1, all others send 0
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79. Control Circuits – Decoders
A 1-to-2 Decoder
Input
Out 0
Out 1 1
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80. Control Circuits – Decoders
Input Lines
Output Lines
in1
in2
out0
out1
out2
out3 1
A 2-to-4 Decoder
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81. Figure 4.29 A 2-to-4 Decoder Circuit
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84. Control Circuits (continued)
Decoder purpose
Given a number code for some operation, trigger just that operation to take place
Numbers might be codes for arithmetic: add, subtract, etc.
Decoder signals which operation takes place next
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85. Control Circuits (continued)
Multiplexor form
2N regular input lines
N selector input lines
1 output line
Multiplexor purpose
Given a code number for some input, selects that input to pass along to its output
Used to choose the right input value to send to a computational circuit
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86. Control Circuits (continued)
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87. A Two-Input Multiplexor Circuit
Figure 4.28
A Two-Input Multiplexor Circuit
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88. A Two-Input Multiplexor Circuit
Using A 1-to-2 Decoder
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89. A 4-Input Multiplexor Circuit
Using A 2-to-4 Decoder
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90. Decoder Example 1
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91. Multiplexor Example 1
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92. Summary
Digital computers use binary representations of data: numbers, text, multimedia
Binary values create a bistable environment, making computers reliable
Boolean logic maps easily onto electronic hardware
Circuits are constructed using Boolean expressions as an abstraction
Computational and control circuits may be built from Boolean gates
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