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

2. Invitation to Computer Science, 6th Edition2 Objectives In this chapter, you will learn about: The binary numbering system Boolean logic and gates Building computer circuits Control circuits

3. Invitation to Computer Science, 6th Edition33 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

4. Invitation to Computer Science, 6th Edition44 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

5. Invitation to Computer Science, 6th Edition5 Figure 4.1 Distinction Between External Memory and Internal Representation of Information

6. Invitation to Computer Science, 6th Edition6 Binary Representation of Numeric and Textual Information Binary numbering system (Computer) –Base-2 –Built from ones and zeros –Each position is a power of 2 –1101 = 1 x 2 3 + 1 x 2 2 + 0 x 2 1 + 1 x 2 0 Decimal numbering system (Daily Life) –Base-10 –Each position is a power of 10 –3052 = 3 x 10 3 + 0 x 10 2 + 5 x 10 1 + 2 x 10 0

7. Invitation to Computer Science, 6th Edition7 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

8. Invitation to Computer Science, 6th Edition8 Figure 4.2 Binary-to-Decimal Conversion Table

9. Invitation to Computer Science, 6th Edition9 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 2 k -1 Arithmetic overflow –Operation that produces an unsigned value greater than 65,535

10. Invitation to Computer Science, 6th Edition10 Signed Numbers Sign/magnitude notation(old computer) –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(current computer) –Total number of values that can be represented with n bits is 2 n

11. Invitation to Computer Science, 6th Edition11 Signed Numbers 1(sign bit) 110001 (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) 0110001 (total: 8 bits, computer B) –When it is a signed value, it is -49 –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)

12. Invitation to Computer Science, 6th Edition12 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

13. Invitation to Computer Science, Java Version, Third Edition13 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 -> 0011 -> 1100 -> 1101 The number of bit depends on computer (8bit, 16bit, 32bit, 64bit) Easier method: Get complement value of each bit until before right most 1 Eg: -3 (3bit): 3-> 011 -> 101 -3 (4bit) : 3 -> 0011 -> 1101  No substraction in two’s complement notation. Convert to complement value.  Eg 5 – 3 = 5 + (-3)

14. Invitation to Computer Science, 6th Edition14 Two's complement notation Bit patternDecimal Value 001+1 010+2 011+3 100-4 101-3 110-2 111-1

15. Invitation to Computer Science, 6th Edition15 Compare value range Suppose we have k bit. Sign/Magnitude notation: -(2 k-1 – 1) to (2 k-1 – 1) –Eg. 3bit: -3 ~ +3 Two’s complement notation:-(2 k-1 ) to (2 k-1 – 1) –Eg 3bit: -4 ~ +3 Why different? –Eg. 3bit: -3 ~ +3 –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.

16. Invitation to Computer Science, 6th Edition16 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

17. Binary Decimal (Fractional number) Binary -> Decimal 0.1101 = 1*2 -1 + 1*2 -2 + 0*2 -3 + 1*2 -4 = 0.5 + 0.25 + 0 + 0.0625 = 0.8125 Decimal -> Binary 0.8125 0.8125 * 2 = 1.625 ------ Get 1 0.625 * 2 = 1.25 ------ Get 1 0.25 * 2 = 0.5 ------ Get 0 0.5 * 2 = 1------ Get 1 Final: 0.1101 (Attention: compare to integer conversion)

18. Invitation to Computer Science, 6th Edition18 Textual Information Code mapping –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 ( 256characters) UNICODE –Uses a 16-bit representation for characters rather than the 8-bit format of ASCII ( 655,36characters)

19. Invitation to Computer Science, 6th Edition19 Figure 4.3 ASCII Conversion Table

20. Invitation to Computer Science, 6th Edition20 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

21. Invitation to Computer Science, 6th Edition21 Figure 4.4 Example of Sound Represented as a Waveform

22. Invitation to Computer Science, 6th Edition22 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

23. Invitation to Computer Science, 6th Edition23 Figure 4.5 Digitization of an Analog Signal (a) Sampling the Original Signal (b) Re-creating the Signal from the Sampled Values

24. 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, 6th Edition24

25. Invitation to Computer Science, Java Version, Third Edition25 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. Simple compression method – “run – length encoding” for image compression –Replaces a sequence of identical values v 1, v 2,..., v n by a pair of values (v, n) –Eg. (255 255 255), (255, 0, 0), (255, 255, 255), (255, 0, 0), (255, 255, 255) → (255, 4), (0, 2), (255, 4), (0, 2), (255, 3)

26. Invitation to Computer Science, Java Version, Third Edition26 Data Compression Simple compression method - “variable length code sets” - for text compress Letter4 bit encodingVariable length encoding A000000 I000110 H0010010 W0100110 H A W A I I → 0010, 0000, 0100, 0000, 0001, 0001 → 001, 00, 110, 00, 10, 10

27. Invitation to Computer Science, 6th Edition27 Figure 4.8 Using Variable Length Code Sets (a) Fixed Length (b) Variable Length

28. Invitation to Computer Science, Java Version, Third Edition28 Data Compression Compression rate = size of the uncompressed data / size of the compressed data –Measures how much compression schemes reduce storage requirements of data

29. Invitation to Computer Science, 6th Edition29 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

30. Invitation to Computer Science, 6th Edition30 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

31. Invitation to Computer Science, 6th Edition31 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

32. Invitation to Computer Science, 6th Edition32 Figure 4.9 Using Magnetic Cores to Represent Binary Values

33. Invitation to Computer Science, 6th Edition33 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

34. Invitation to Computer Science, 6th Edition34 Figure 4.10 Relationships Among Transistors, Chips, and Circuit Boards

35. Invitation to Computer Science, 6th Edition35 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

36. Invitation to Computer Science, 6th Edition36 Figure 4.11 Simplified Model of a Transistor

37. Invitation to Computer Science, 6th Edition37 Boolean Logic Boolean logic –Construction of computer circuits is based on this –Boolean expression Constructed by combining together Boolean operations Example: (a AND b) OR ((NOT b) AND (NOT a)) –Truth table capture the output/value of a Boolean expression A column for each input plus the output A row for each combination of input values

38. Invitation to Computer Science, 6th Edition38 Figure 4.12 Truth Table for the AND Operation

39. Invitation to Computer Science, 6th Edition39 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

40. Invitation to Computer Science, 6th Edition40 Figure 4.13 Truth Table for the OR Operation

41. Invitation to Computer Science, 6th Edition41 Figure 4.14 Truth Table for the NOT Operation

42. Gates Gate –Hardware devices built from transistors to mimic Boolean logic AND gate –Two input lines, one output line –Outputs a 1 when both inputs are 1 Invitation to Computer Science, 6th Edition42

43. Gates(continued) 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 Invitation to Computer Science, 6th Edition43

44. Invitation to Computer Science, 6th Edition44 Figure 4.15 The Three Basic Gates and Their Symbols

45. 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 Invitation to Computer Science, 6th Edition45

46. Invitation to Computer Science, 6th Edition46 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

47. Invitation to Computer Science, 6th Edition47 Figure 4.19 Diagram of a Typical Computer Circuit

48. Invitation to Computer Science, 6th Edition48 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 2 N combinations of input values, and the truth table has 2 N rows

49. Invitation to Computer Science, 6th Edition49 A Truth Table for a Circuit with 8 Input Combinations

50. Invitation to Computer Science, 6th Edition50 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

51. Invitation to Computer Science, 6th Edition51 Output Column Labeled Output-1 from the Previous Truth Table

52. Invitation to Computer Science, 6th Edition52 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

53. Invitation to Computer Science, 6th Edition53 Figure 4.20 Circuit Diagram for the Output Labeled Output-1

54. Invitation to Computer Science, 6th Edition54 Figure 4.21 The Sum-of-Products Circuit Construction Algorithm

55. Invitation to Computer Science, 6th Edition55 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

56. Invitation to Computer Science, 6th Edition56 Figure 4.22 One-Bit Compare for Equality Circuit

57. Invitation to Computer Science, 6th Edition57 Figure 4.23 N-Bit Compare for Equality Circuit

58. Invitation to Computer Science, 6th Edition58 An Addition Circuit Full adder –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

59. Invitation to Computer Science, 6th Edition59 Figure 4.24 The 1-ADD Circuit and Truth Table

60. Invitation to Computer Science, 6th Edition60 Figure 4.25 Sum Output for the 1-ADD Circuit

61. Invitation to Computer Science, 6th Edition61 Figure 4.26 Complete 1-ADD Circuit for 1-Bit Binary Addition

62. Invitation to Computer Science, 6th Edition62 Figure 4.27 The Complete Full Adder ADD Circuit

63. Invitation to Computer Science, 6th Edition63 Control Circuits Used to: –Determine the order in which operations are carried out –Select the correct data values to be processed Multiplexor –Circuit that has 2 N input lines and 1 output line –Function: to select exactly one of its 2 N input lines and copy the binary value on that input line onto its single output line

64. Invitation to Computer Science, 6th Edition64 Figure 4.28 A Two-Input Multiplexor Circuit

65. Invitation to Computer Science, 6th Edition65 Control Circuits (continued) Decoder –Has N input lines numbered 0, 1, 2,..., N – 1 and 2 N output lines numbered 0, 1, 2, 3,..., 2 N – 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

66. Invitation to Computer Science, 6th Edition66 Figure 4.29 A 2-to-4 Decoder Circuit

67. Invitation to Computer Science, 6th Edition67 Figure 4.30 Example of the Use of a Decoder Circuit

68. Invitation to Computer Science, 6th Edition68 Figure 4.31 Example of the Use of a Multiplexor Circuit

69. Invitation to Computer Science, 6th Edition69 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

70. Invitation to Computer Science, 6th Edition70 Summary (continued) Circuits –Constructed using Boolean expressions as an abstraction Computational and control circuits –Can be built from Boolean gates