1. Invitation to Computer Science 5th Edition
Chapter 4
The Building Blocks: Binary Numbers, Boolean Logic, and Gates

2. Objectives In this chapter, you will learn about:
The binary numbering system
Boolean logic and gates
Building computer circuits
Control circuits
Invitation to Computer Science, 5th Edition

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
Invitation to Computer Science, 5th Edition

6. 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
Invitation to Computer Science, 5th Edition

7. Figure 4.2 Binary-to-Decimal Conversion Table
Invitation to Computer Science, 5th Edition

8. 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
Arithmetic overflow
Operation that produces an unsigned value greater than 65,535
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9. Signed Numbers Sign/magnitude notation Two’s complement representation
One of a number of different techniques for representing positive and negative whole numbers
Not used often in real computer systems
Two’s complement representation
Total number of values that can be represented with n bits is 2n
Invitation to Computer Science, 5th Edition

10. 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
Invitation to Computer Science, 5th Edition

11. 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
Invitation to Computer Science, 5th Edition

12. Figure 4.3 ASCII Conversion Table
Invitation to Computer Science, 5th Edition

13. 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
Frequency f: total number of cycles per unit time
Invitation to Computer Science, 5th Edition

14. Figure 4.4 Example of Sound Represented as a Waveform
Invitation to Computer Science, 5th Edition

15. Binary Representation of Sound and Images (continued)
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
Scanning
Measuring the intensity values of distinct points located at regular intervals across the image’s surface
Invitation to Computer Science, 5th Edition

16. Figure 4.5 Digitization of an Analog Signal
(a) Sampling the Original Signal
(b) Re-creating the Signal from the Sampled Values
Invitation to Computer Science, 5th Edition

17. Binary Representation of Sound and Images (continued)
Raster graphics
Each pixel is encoded as an unsigned binary value representing its gray scale intensity
RGB encoding scheme
Most common format for storing color images
True Color
24-bit color-encoding scheme
Data compression algorithms
Attempt to represent information in ways that preserve accuracy while using significantly less space
Invitation to Computer Science, 5th Edition

18. 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)
Compression ratio
Measures how much compression schemes reduce storage requirements of data
Variable length code sets
Often used to compress text
Can also be used with other forms of data
Invitation to Computer Science, 5th Edition

19. Figure 4.8 Using Variable Length Code Sets (a) Fixed Length
(b) Variable Length
Invitation to Computer Science, 5th Edition

20. Binary Representation of Sound and Images (continued)
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
Invitation to Computer Science, 5th Edition

21. 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
Invitation to Computer Science, 5th Edition

22. Binary Storage Devices
Magnetic cores
Used to construct computer memories
Core
Small, magnetizable, iron oxide-coated “doughnut,” about 1/50 of an inch in inner diameter, with wires strung through its center hole
Invitation to Computer Science, 5th Edition

23. Figure 4.9 Using Magnetic Cores to Represent Binary Values
Invitation to Computer Science, 5th Edition

24. Binary Storage Devices (continued)
Transistor
Solid-state device that has no mechanical or moving parts
Constructed from semiconductors
Can be printed photographically on a wafer of silicon to produce a device known as an integrated circuit
Circuit board
Interconnects all the different chips needed to run a computer system
Invitation to Computer Science, 5th Edition

25. Figure 4.10 Relationships Among Transistors, Chips, and Circuit Boards
Invitation to Computer Science, 5th Edition

26. Binary Storage Devices (continued)
Mask
Can be used to produce a virtually unlimited number of copies of a chip
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
Invitation to Computer Science, 5th Edition

27. Figure 4.11 Simplified Model of a Transistor
Invitation to Computer Science, 5th Edition

28. Boolean Logic Boolean logic
Construction of computer circuits is based on this
Boolean expression
Any expression that evaluates to either true or false
Truth table
Can express the idea that the AND operation produces the value true if and only if both of its components are true
Invitation to Computer Science, 5th Edition

29. Figure 4.12 Truth Table for the AND Operation
Invitation to Computer Science, 5th Edition

30. 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
Invitation to Computer Science, 5th Edition

31. Figure 4.13 Truth Table for the OR Operation
Invitation to Computer Science, 5th Edition

32. Figure 4.14 Truth Table for the NOT Operation
Invitation to Computer Science, 5th Edition

33. Gates Gate NOT gate To construct an AND gate
Electronic device that operates on a collection of binary inputs to produce a binary output
Transforms a set of (0,1) input values into a single (0,1) output value
NOT gate
Can be constructed from a single transistor
To construct an AND gate
Connect two transistors in series with the collector line of transistor 1 connected to the power supply (logical-1) and the emitter line of transistor 2 connected to ground (logical-0)
Invitation to Computer Science, 5th Edition

34. Figure 4.15 The Three Basic Gates and Their Symbols
Invitation to Computer Science, 5th Edition

35. Figure 4.16 Construction of a NOT Gate
Invitation to Computer Science, 5th Edition

36. Figure 4.17 Construction of NAND and AND Gates
(a) A Two-transistor NAND Gate (b) A Three-transistor AND Gate
Invitation to Computer Science, 5th Edition

37. 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
Invitation to Computer Science, 5th Edition

38. Figure 4.18 Construction of NOR and OR Gates
(a) A Two-transistor NOR Gate
(b) A Three-transistor OR Gate
Invitation to Computer Science, 5th Edition

39. Building Computer Circuits
Introduction
Circuit: collection of logic gates that transforms a set of binary inputs into a set of binary outputs
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
Invitation to Computer Science, 5th Edition

40. Figure 4.19 Diagram of a Typical Computer Circuit
Invitation to Computer Science, 5th Edition

41. A Circuit Construction Algorithm
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
Invitation to Computer Science, 5th Edition

42. A Truth Table for a Circuit with 8 Input Combinations
Invitation to Computer Science, 5th Edition

43. 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
Invitation to Computer Science, 5th Edition

44. Output Column Labeled Output-1 from the Previous Truth Table
Invitation to Computer Science, 5th Edition

45. 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
Invitation to Computer Science, 5th Edition

46. Figure 4.20 Circuit Diagram for the Output Labeled Output-1
Invitation to Computer Science, 5th Edition

47. Figure 4.21 The Sum-of-Products Circuit Construction Algorithm
Invitation to Computer Science, 5th Edition

48. 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
Invitation to Computer Science, 5th Edition

49. Figure 4.22 One-Bit Compare for Equality Circuit
Invitation to Computer Science, 5th Edition

50. Figure 4.23 N-Bit Compare for Equality Circuit
Invitation to Computer Science, 5th Edition

51. 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
Invitation to Computer Science, 5th Edition

52. Figure 4.24 The 1-ADD Circuit and Truth Table
Invitation to Computer Science, 5th Edition

53. Figure 4.25 Sum Output for the 1-ADD Circuit
Invitation to Computer Science, 5th Edition

54. Figure 4.26 Complete 1-ADD Circuit for 1-Bit Binary Addition
Invitation to Computer Science, 5th Edition

55. Figure 4.27 The Complete Full Adder ADD Circuit
Invitation to Computer Science, 5th Edition

56. 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
Invitation to Computer Science, 5th Edition

57. Figure 4.28 A Two-Input Multiplexor Circuit
Invitation to Computer Science, 5th Edition

58. 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
Invitation to Computer Science, 5th Edition

59. Figure 4.29 A 2-to-4 Decoder Circuit
Invitation to Computer Science, 5th Edition

60. Figure 4.30 Example of the Use of a Decoder Circuit
Invitation to Computer Science, 5th Edition

61. Figure 4.31 Example of the Use of a Multiplexor Circuit
Invitation to Computer Science, 5th Edition

62. 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
Invitation to Computer Science, 5th Edition

63. Summary (continued) Circuits Computational and control circuits
Constructed using Boolean expressions as an abstraction
Computational and control circuits
Can be built from Boolean gates
Invitation to Computer Science, 5th Edition