MACHINE LEVEL REPRESENTATION OF DATA

Transcription

1 MACHINE LEVEL REPRESENTATION OF DATA CHAPTER 2 1

2 Objectives Understand how integers and fractional numbers are represented in binary Explore the relationship between decimal number system and number systems of other bases (binary, octal, hexadecimal) Understand signed numbers and unsigned numbers Understand number representations in two s complement Perform arithmetic operations in every base Understand the method of floating point number representation in IEEE format 2

3 Introduction NUMBERING SYSTEM : BASE DECIMAL BINARY OCTAL HEXA- DECIMAL 5 TH PLACE 4 TH PLACE 3 RD PLACE 2 ND PLACE 1 ST PLACE PLACE SINGLE UNIT 1 ST PLACE 2 ND PLACE RD PLACE 100,000 10,000 1, /10 1/100 1/ /2 1/4 1/ ,768 4, X /8 1/64 1/ ,048, X ,536 4, X /16 1/256 1/4096 3

4 Arithmetic Operations in Different Number Bases The following table shows the equivalent values for decimal numbers in binary, octal and hexadecimal: DECIMAL BINARY OCTAL HEXADECIMAL A B C D E F 4

5 Number Bases Radix : when referring to binary, octal, decimal, hexadecimal, a single lowercase letter appended to the end of each number to identify its type. E.g. hexadecimal 45 will be written as 45h octal 76 will be written as 76o or 76q binary will be written as b 5

6 Number Bases Decimal system: system of positional notation based on powers of 10. Binary system: system of positional notation based powers of 2 Octal system: system of positional notation based on powers of 8 Hexadecimal system: system of positional notation based on powers of 16 6

7 Why Binary? Early computer design was decimal Mark I and ENIAC John von Neumann proposed binary data processing (1945) Simplified computer design Used for both instructions and data Natural relationship between on/off switches and calculation using Boolean logic On Off True False Yes No 1 0 7

8 Binary Numbers A computer stores both instructions and data as individual electronic charges. represent these entities with numbers requires a system geared to the concept of on and off or true and false Binary is a base 2 numbering system each digit is either a 0 (off) or a 1 (on) Computers store all instructions and data as sequences of binary digit e.g = ABC 8

9 Binary Numbers Bit : each digit in a binary number Byte : consists of 8 bits which is the basic unit of storage on nearly all computers Each location in the computer s memory holds exactly 1 byte or 8 bits. A byte can hold a single instruction, a character or a number A Word : which is 16 bits or 2 bytes long : byte word byte A 16-bit computer : its instructions can operate on 16-bit quantities 9 bit

10 Numeric Data Representation FOR DECIMAL 43 = 4 x x s place 1 s place Place Value 10 1 Evaluate 4 x 10 3 x1 Sum

11 Numeric Data Representation FOR DECIMAL 527 = 5 x x x s place 10 s place 1 s place Place Value Evaluate 5 x x 10 7 x1 Sum

12 Numeric Data Representation = s place 8 s place 1 s place Place Value Evaluate 6 x 64 2 x 8 4 x 1 Sum for Base

13 Numeric Data Representation 6, = 26, ,096 s place 256 s place 16 s place 1 s place Place Value 4, Evaluate 6 x 4,096 7 x x 16 4 x 1 Sum for Base 10 24,576 1,

14 Numeric Data Representation = Place Value Evaluate 1 x x 64 0 x 32 1 x16 0 x 8 1 x 4 1 x 2 0 x 1 Sum for Base

15 Base or Radix Base: The number of different symbols required to represent any given number The larger the base, the more numerals are required Base 10 : 0,1, 2,3,4,5,6,7,8,9 Base 2 : 0,1 Base 8 : 0,1,2, 3,4,5,6,7 Base 16 : 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F 15

16 Number of Symbols vs Number of Digits For a given number, the larger the base the more symbols required but the fewer digits needed Example #1: Example #2: 11C

17 Numeric Conversion between Number Bases Decimal Other bases Other bases Decimal Binary Octal Binary Hexadecimal Octal Binary Hexadecimal Binary 17

18 CONVERSION : DECIMAL OTHER BASES 18

19 From Base 10 to Base 2 Base Remainder Quotient 2 ) 42 ( 0 Least significant bit 2 ) 21 ( 1 2 ) 10 ( 0 2 ) 5 ( 1 2 ) 2 ( 0 2 ) 1 ( 1 Most significant bit 0 Base

20 From Base 10 to Base 2 Convert to binary = 18 balance 1 (LSB) 18 2 = 9 balance = 4 balance = 2 balance = 1 balance = 0 balance 1 (MSB) Therefore, = * MSB (most-significant-bit) : left most bit LSB (least-significant-bit) : right most bit 20

21 From Base 10 (decimal point) to Base 2 What is the value of in binary? Steps : 1. Convert the integer to binary by using method shown in previous slide. 2. Convert the decimal point to binary by using the following method X 2 (MSB) X X X 2 (LSB) The 1 is saved as result, then dropped and the process repeated So, (0.6875) 10 = (0.1011) 2 ; Therefore, =

22 From Base 10 to Base 8 Base Remainder Quotient 8) 135 ( 7 Least significant bit 8) 16 ( 0 8) 2 ( 2 Most significant bit 0 Base

23 From Base 10 (decimal point) to Base 8 Convert to octal x Therefore, =

24 From Base 10 (decimal point) to Base 8 Convert to octal. (OTHER METHOD) 21 2 = 10 balance 1 (LSB) 10 2 = 5 balance = 2 balance = 1 balance = 0 balance 1 (MSB) So, = Therefore, Refer to conversion of binary to hexadecimal = = Now, 0.25 X 2 (MSB) X 2 (LSB) So, =

25 From Base 10 to Base 16 Base 10 5,735 Remainder Quotient 16 ) 5,735 ( 7 Least significant bit 16 ) 358 ( 6 16 ) 22 ( 6 16 ) 1 ( 1 Most significant bit 0 Base

26 From Base 10 to Base 16 Base 10 8,039 Remainder Quotient 16 ) 8,039 ( 7 Least significant bit 16 ) 502 ( 6 16 ) 31 ( ) 1 ( 1 Most significant bit 0 Base 16 1F67 26

27 From Base 10 to Base 16 Convert to hexadecimal = 1 balance 5 (LSB) 1 16 = 0 balance 1 (MSB) Therefore, =

28 From Base 10 (decimal point) to Base 16 Convert to hexadecimal x Therefore, =

29 From Base 10 (decimal point) to Base 16 Convert to hexadecimal. (OTHER METHOD) 21 2 = 10 balance 1 (LSB) 10 2 = 5 balance = 2 balance = 1 balance = 0 balance 1 (MSB) So, = Therefore, Refer to conversion of binary to hexadecimal = = Now, 0.25 X 2 (MSB) X 2 (LSB) So, =

30 CONVERSION : OTHER BASES DECIMAL 30

31 From Base 2 to Base 10 Convert to decimal *32 1*4 1* = = (1 X 2 5 ) + (0 X 2 4 ) +(0 X 2 3 ) +(1 X 2 2 ) + (1 X 2 1 ) + (0 X 2 0 ) = = Therefore, =

32 From Base 16 to Base 10 Convert E5 16 to decimal. E *16 5* = E5 16 = (E X 16 1 ) + (5 X 16 0 ) = (14 X 16) + (5 X 1) = = Therefore, E5 16 =

33 From Base 16 (decimal point) to Base 10 Convert E5.A8 16 to decimal. E 5. A *16 5*1. 10* * =

34 From Base 16 (decimal point) to Base 10 Convert E5.A8 16 to decimal. (OTHER METHOD) 1. Firstly convert the number into binary, Hex E 5. A 8 Binary From Example 15: E5 16 = Now, ( ) 2 = (1 X 2-1 ) + (0 X 2-2 ) + (1 X 2-3 ) + (0 X 2-4 ) + (1 X 2-5 ) + (0 X 2-6 ) + (0 X 2-7 ) + (0 X 2-8 ) = 1/2 + 1/ /2 5 = 1/2 + 1/8 + 1/32 = = ( ) 10 Therefore, E5.A8 16 =

35 From Base 8 to Base 10 Convert to decimal *512 2*64 6*8 3* =

36 From Base 8 to Base = 3, (OTHER METHOD) 7 x = 58 x = 470 x = 3,763 36

37 From Base 8 (decimal point) to Base 10 Convert to decimal *8 6*1. 3* * =

38 CONVERSION : BINARY HEXADECIMAL BINARY OCTAL 38

39 From Base 16 to Base 2 The nibble approach Hex easier to read and write than binary Why hexadecimal? Modern computer operating systems and networks present variety of troubleshooting data in hex format The relationship between BINARY and HEXADECIMAL 1 digit hexadecimal is EQUIVALENT to 4 bits binary Base 16 1 F 6 7 Base

40 From Base 16 to Base 2 Method: Get the binary representation for the hexadecimal number. Example: Convert F1 16 to binary. F1 16 = F F1 16 =

41 From Base 16 to Base 2 (decimal point) Method: Get the binary representation for the hexadecimal number. Example: Convert 2A.5C 16 to binary. 2A.5C 16 = 2 A. 5 C A.5C 16 =

42 From Base 2 to Base 16 Method: Find the hexadecimal representation for the GROUP of 4 binary numbers. Example: Convert to hexadecimal number = =

43 From Base 2 to Base 16 (decimal point) Method: Find the hexadecimal representation for the GROUP of 4 binary numbers. Use the decimal point as your STARTING point. Example: Convert to hexadecimal number = E = 3E

44 From Base 8 to Base 2 The relationship between BINARY and OCTAL 1 digit octal is EQUIVALENT to 3 bits binary Base Base

45 From Base 8 to Base 2 Method: Get the binary representation for every digit in the octal number. Example 17: Convert 53 8 to binary = =

46 From Base 8 to Base 2 (decimal point) Method: Get the binary representation for every digit in the octal number. Example 17: Convert to binary = =

47 From Base 2 to Base 8 Method: Find the octal representation for the GROUP of 3 binary numbers. Example: Convert to octal number = =

48 From Base 2 to Base 8 (decimal point) Method: Find the octal representation for the GROUP of 3 binary numbers. Use the decimal point as your STARTING point. Example: Convert to octal number = =

49 ADDITION AND SUBTRACTION OF DIFFERENT BASES 49

50 Addition Base Problem Largest Single Digit Decimal Octal Hexadecimal Binary F 1 50

51 Addition and Subtraction of Binary Numbers Example ADD Example SUB

52 Addition of Octal Numbers Example ADD = 9 8 = = 11 8 = = 8 8 = = 1 52

53 Subtraction of Octal Numbers Example SUB = = = 1 53

54 Addition of Hexadecimal Numbers Example ADD 1 1 A = 11 = B + 7 C = B = = = 9 54

55 Subtraction of Hexadecimal Numbers Example SUB E F 7 D = 15 = F - 9 E = 13 = D 2 E D F E 0 = E 2 0 = 2 55

56 THE ALPHANUMERIC REPRESENTATION 56

57 The Alphanumeric Representation The data entered as characters, number digits, and punctuation are known as alphanumeric data. 3 alphanumeric codes are in common use. ASCII (American Standard Code for Information Interchange) Unicode EBCDIC (Extended Binary Coded Decimal Interchange Code). 57

58 The Alphanumeric Representation The following tables show the comparisons between the ASCII code in binary and hexadecimal for the given characters. Character Binary Hex A B C D E F G H I J A K B Space Full stop E ( B $

59 THE DECIMAL REPRESENTATION 59

60 The Decimal Representation BCD (Binary Coded Decimal) is often used to represent decimal number in binary. Decimal BCD

61 SIGNED AND UNSIGNED NUMBERS 61

62 Signed and Unsigned Numbers Binary numbers may either signed or unsigned Oddly, CPU performs arithmetic and comparison operations for both type equally well, without knowing which type it s operating on. An unsigned numbers : numbers with only positive values for 8-bit storage location : store unsigned integer value between for 16-bit storage location : store unsigned integer value between Unsigned number can be converted directly to binary numbers and processed without any special care 62

63 Signed and Unsigned Numbers For negative numbers, there are several ways used to represent it in binary form, depending on the process take place : i. Sign-and-magnitude representation ii. 1 s complement representation iii. 2 s complement representation 63

64 Sign-and-Magnitude In daily usage, signed integers are represented by a plus or minus sign and a value. In the computer, the uses of 0 s and 1 s take place. 0 : plus ( positive) 1 : minus (negative) The leftmost bit in a binary number is considered the sign bit. The remaining (n-1) bits are used for magnitude. 64

65 Sign-and-Magnitude Example 1 : (+37) (-37) (+1) (-1) (+32767) (-32767) 65

66 Sign-and-Magnitude Example 2 : What is the sign-and-magnitude representation of the decimal numbers 31 and +31 if the basic unit is a byte? = Unit is a byte = 8 bits +31 = =

67 Sign-and-Magnitude Example 3 : What is the sign-and-magnitude representation of the decimal numbers 31 and +31 if the basic unit is a word? = Unit is a word= 16 bits +31 = =

68 Sign-and-Magnitude Example 4 : What is the decimal equivalent value of the sign-and -magnitude binary sequence ? Sign is negative = =

69 Sign-and-Magnitude Addition of 2 numbers in sign-and-magnitude : using the usual conventions of binary arithmetic if both have same sign : magnitude are added and the same sign copied if the sign different : number that has smaller magnitude is subtracted from the larger one. The sign is copied from the larger magnitude. 69

70 Sign-and-Magnitude Example 5 : What is the decimal value of the sum of the binary numbers and if they are represented in sign-and-magnitude? Assume that the basic unit is the byte. Same signs : magnitude are added Sign is positive = =

71 Sign-and-Magnitude Example 6 : What is the decimal value of the sum of the binary numbers and if they are represented in sign-and-magnitude? Assume that the basic unit is the byte. Different signs : Larger magnitude - smaller magnitude Larger magnitude : Smaller magnitude : Sign is negative = =

72 1 S Complement Convention In base 2, the largest digit is 1. The 1 s complement is performed simply by changing: every 0 1 and every 1 0. Also known as inversion 72

73 2 S Complement Convention Most popular among computer manufacturers since it does not present any of the problems of the sign-and-magnitude or 1 s complement. Positive numbers : using similar procedure as sign-and-magnitude Given n bits, the range of numbers that can be represented in 2 s complement is ( (2 n )) to (2 n-1 1) (127) (0) (-1) (-128) ( 2 n-1 1) - (2 n ) Notice that the range of negative numbers is one larger than the range of the positive values 73

74 2 S Complement Convention To represent a negative number in this convention, follow the 3 steps process below: Step 1 : Express the absolute value of the number in binary Step 2 : Change all 0s to 1s and vise versa 1 s complement Step 3 : Add 1 to the binary number of step 2 Example 1 : What is the 2 s complement representation of -23? Example 2 : What is the decimal positive value of the 2 s complement number ? 74

75 2 S Complement Convention Example 1 : What is the 2 s complement representation of -23? Step: 23 in binary s (inverse) s s complement representation

76 2 S Complement Convention Example 2 : What is the decimal positive value of the 2 s complement number ? Step: 2 s complement representation s (inverse) s + 1 Decimal positive value

77 The IEEE Floating Point Representation 2 problems with integers; They can t express fractions, AND The range number is limited to the number of bits used. An efficient way of storing fractions floating point method involves splitting the fraction into two parts, an exponent and a mantissa Computer industry agreed upon a standard for the storage of floating point numbers the IEEE 754 standard; uses 32 bits of memory (single precision) or 64 bits (double precision) 77

78 The IEEE Floating Point Representation IEEE short real : 32 bits (Exponent system = excess 127) Total number of bits : Sign Biased exponent Mantissa (fraction) IEEE short real : 64 bits (Exponent system = excess 1023) Total number of bits : Sign Biased exponent Mantissa (fraction) 78

79 The IEEE Floating Point Representation Convert to the IEEE floating point format. Step 1 : convert into binary Step 2 : put into 1.xxxx X 2 y format Step 3 : get the biased exponent Step 4 : get the sign (from the question) Step 5 : identify significand and mantissa Step 6 : put into the IEEE single precision format Step 7 : convert into hexadecimal 79

80 The IEEE Floating Point Representation Convert to binary = Put into 1.xxxx X 2 y = x 2 7 Biased exponent Exponent = 7 Biased exponent = y = = 134 = (8 bits) Sign Sign = +ve (0) Significand Mantissa Significand = Mantissa = (23bits) IEEE format : IEEE format Sign Biased Exponent Mantissa (23) Hexadecimal C = 4319 C000 h 80

81 Eg: Convert -29 / The IEEE Floating Point Representation Convert to binary 29 / 128 = / 2 7 = x 2-7 Put into 1.xxxx X 2 y = x 2 4 x 2-7 = x 2-3 Biased exponent Exponent = -3 Biased exponent = y = (-3) = 124 = (8 bits) Sign Sign = -ve = 1 Significand Mantissa Significand = Mantissa = (23bits) IEEE format : IEEE format Sign Biased Exponent Mantissa (23) Hexadecimal B E = BE h 81

82 The IEEE Floating Point Representation Convert C2F in IEEE floating point format to the decimal number. Step 1 : in hexadecimal Step 2 : convert into binary Step 3 : put into the IEEE single precision format Step 4 : get biased exponent Step 5 : get the sign Step 6 : identify significand and mantissa Step 7 :put into 1.xxxx X 2 y Step 8 : identify actual number 82

83 The IEEE Floating Point Representation In hexadecimal C2F Convert to binary C 2 F Put into IEEE format Get the biased exponent Sign Mantissa Significand Put into 1.xxxx X 2 y IEEE format : Sign Biased Exponent Mantissa (23) Biased Exponent:: = 133 Exponent : = 6 1 = -ve x 2 6 = = The actual number