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1 컴퓨터구성 Lecture Series #4 Chapter 3: Data Representation Spring, 2013 컴퓨터구성 : Spring, 2013: No. 4-1

2 Data Types Introduction This chapter presents data types used in computers for representing diverse numbers and characters Binary Information Data Information Numbers & other binary-coded info. to be used in computation Control Information Sequence of command signals needed for data manipulation such as programmed instructions Data Types Numbers used in arithmetic computations Alphabet letters in data processing Other discrete symbols for specific purposes Grey level data, etc. 컴퓨터구성 : Spring, 2013: No. 4-2

3 Data Types Number System The number system of base, or radix, r is a system that uses r distinct symbols for r digits Example 1: Decimal System (Radix r = 10) 10 distinct symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Quantity of 724.5: 7 x x x x 10-1 Example 2: Binary System (Radix r = 2) 2 distinct symbols: 0, 1 Quantity of 101.1: 1 x x x x 2-1 컴퓨터구성 : Spring, 2013: No. 4-3

4 Data Types Number System Octal System (Radix r = 8) 8 distinct symbols: 0, 1, 2, 3, 4, 5, 6, 7 Hexa-Decimal System (Radix r = 16) 16 distinct symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Quantity of 4F3 4 x F x x 16 0 = (1267) A F 5 Octal Binary Hexa Conversion between number systems is your job!! 컴퓨터구성 : Spring, 2013: No. 4-4

5 Data Types Examples: Octal & Hexa Octal Number Binary-coded Octal Decimal HexadecimalBinary-coded Number hexadecimal Decimal Code for one octal digit A B C D E F 14 F Code for one hexadecimal digit 컴퓨터구성 : Spring, 2013: No. 4-5

6 Data Types Decimal Representation BCD (Binary-Coded Decimal) Only 0-9 are represented with 4 bits To make conversion to decimal numbers easy after computation 16 distinct binary combinations are unassigned (not used) Wastes 6 combinations Decimal Number Binary-coded Decimal (BCD) Code for one decimal digit 컴퓨터구성 : Spring, 2013: No. 4-6

7 Data Types Many applications require handling of not only numbers but also alphabet letters and special characters ASCII Code 8 bits (including a parity bit) Differences b/w Lower case & Upper case alphabets are 32 BCD with leading 011 are ASCII Alphanumeric Representation Char. Binary code Char. Binary code A B C D E F G H I J K L M SPACE N O ( P Q $ R * S ) T U / V , W = X Y Z 컴퓨터구성 : Spring, 2013: No. 4-7

8 Complements (r-1) s Complement Used for simplifying subtraction & logical manipulations (r-1) s Complement Given a number N with n digits, its (r-1) s complement is (r n -1) - N Decimal (r = 10) 9 s complement Example: N = 3450 r n -1 = = = 6549 Binary (r = 2) 1 s complement Example: N = r n -1 = = = Change 0 1 & 1 0 컴퓨터구성 : Spring, 2013: No. 4-8

9 Complements r s Complement r s Complement Given a number N with n digits, its r s complement is (r n -N) (r n -N) = (r n - 1) - N + 1 = (r -1) s complement + 1 Decimal (r = 10) 10 s complement Example: N = 3450 r n = 10 4 = = = = 6550 Binary (r = 2) 2 s complement Example: N = r n = 2 7 = = = Leave all lower order 0 s before the first 1 and the first 1 unchanged and complement the remaining upper binary numbers 컴퓨터구성 : Spring, 2013: No. 4-9

10 Complements Notes N O T E S No differences even when radix point is present Complement of a complement results in the original Three ways to subtract depending on how negative numbers are represented (will be shown shortly) 컴퓨터구성 : Spring, 2013: No. 4-10

11 Complements Subtraction of Unsigned Numbers What if numbers are represented as positive numbers? Subtraction of 2 n-digit Unsigned Numbers: M - N (N 0) Find r s complement of N & add to M M + (r n - N) = M - N + r n If M N ( M - N 0 ), sum will produce an end carry r n & discard this (M - N + r n - r n ) = M - N This is the answer If M N ( M - N 0 ), sum will not produce an end carry r n M + (r n -N) = { r n -(N-M)} = r s complement of (N - M) For answer, take r s complement and regard as negative 컴퓨터구성 : Spring, 2013: No. 4-11

12 Complements Subtraction of Unsigned Numbers Ex 1.1: = Ex 1.2: = M = M = s complement of N = s complement of N = Sum = Sum = Discard 10 5 = s complement of Sum = Answer = Answer = Ex 2.1: = Ex 2.2: = X = Y = s complement of Y = s complement of X = Sum = Sum = Discard 2 8 = s complement of Sum = Answer = Answer = 컴퓨터구성 : Spring, 2013: No. 4-12

13 Fixed-Point Representation Introduction Negative Values Sign Bit at the leftmost bit position 0 for positive & 1 for negative Binary Point Fixed Point At extreme left or right of the stored number in a register We assume the latter the number is integer Floating Point Will be discussed in the next section Most of computer systems use floating-point for real numbers 컴퓨터구성 : Spring, 2013: No. 4-13

14 Fixed-Point Representation Integer Representation Positive Integer: Sign bit and magnitude: Negative Integer: 14 = Signed-magnitude: -14 = Used in ordinary arithmetic; but awkward in computers Signed-1 s complement: -14 = Difficult since it has two zeros (+0 (0000) & -0 (1111)) Used only in old computers for numerical arithmetic Useful in logic operations (complementation) Signed-2 s complement: -14 = Primarily used in most computers as well as in my lecture 컴퓨터구성 : Spring, 2013: No. 4-14

15 Data Representation Summary n- bit Numbers Unsigned Numbers = Unsigned Magnitude Signed Magnitude Signed 1 s complement Signed 2 s complement Max Min Zero 2 n (2 n-1-1) -(2 n-1-1) + 0, , (2 n-1-1) -(2 n-1-1) + 0, , (2 n-1-1) -2 n Complement 컴퓨터구성 : Spring, 2013: No. 4-15

16 Fixed-Point Representation Arithmetic Addition Signed-Magnitude: Same Signs Add two magnitudes and put the sign bit Different Signs Subtract smaller magnitude from larger one Put the sign of data with larger magnitude Need comparison, addition, subtraction Q: Think about the rule for adding two numbers in signed- 1 s complement format Signed-2 s Complement: Rule Add two numbers including sign bits Discard carry out of sign bit, if any Need addition, only Negative result is in 2 s complement form 컴퓨터구성 : Spring, 2013: No. 4-16

17 Fixed-Point Representation Arithmetic Subtraction Signed-Magnitude: Rule Change the sign of subtrahend Perform Signed-Magnitude Addition Signed-2 s Complement: Rule Take 2 s complement of subtrahend Add it to minuend Discard carry out of sign bit, if any Need addition, only Negative result is in 2 s complement form Q: Think about the rule for adding two numbers in signed- 1 s complement format 컴퓨터구성 : Spring, 2013: No. 4-17

18 Fixed-Point Representation Overflow When two numbers of n digits with the same sign are added & the sum occupies n+1 digits, we say that an overflow occurred Overflow Flip-Flop: A flip-flop to store the (n+1) th bit of the result When is overflow? Q: Think about the case two numbers in signed-1 s complement format Addition of 2 Unsigned Numbers End carry out of MSB Addition of 2 Signed-2 s Complement Numbers End carry out of sign bit does not imply the overflow End carry out of sign bit and the carry into sign bit are different Carries: 0 1 Carries: 컴퓨터구성 : Spring, 2013: No. 4-18

19 Fixed-Point Representation Decimal Representation Disadvantages: Wasting storage space Numeric computations & their circuits are complicated ( ) BCD ( ) BCD ( ) BCD Advantages: No need to convert between decimals & binaries Largely used in data processing where computations are rare For BCD adder see Fig. in your textbook 컴퓨터구성 : Spring, 2013: No. 4-19

20 Floating-Point Representation Composition: Mantissa (m) Signed fixed-point number Exponent (e) Position of binary point Interpretation m 2 e Normalization: MSB of the mantissa is nonzero Example (in Textbook): Fraction Exponent m x 2 e = +( ) 2 x 2 +4 컴퓨터구성 : Spring, 2013: No. 4-20

21 Floating-Point Representation IEEE 754 Standard Example (in Real World): Sign bit Exponent (e) 8 bits in Excess-127 Mantissa (m) 23 bits Interpretation N = (-1) s 2 e-127 (1.m) Exponent Mantissa 8 bits 23 bits Sign bit 64-bit Extention Sign bit Exponent (e) 11 bits coded into Excess-1023 Mantissa (m) 52 bits Interpretation N = (-1) s 2 e-1023 (1.m) 컴퓨터구성 : Spring, 2013: No. 4-21

22 Error Detection Codes Binary information transmitted thru communication medium is subject to external noise. Parity Bit: Error Detection Code An extra bit to make number of 1 s in a message either even or odd Even Parity: P(even) What if the message contains all zeros? Odd Parity : P(odd) So, this is generally used. Source x y z Destination x y z Error Indication Parity Generator Parity Checker 컴퓨터구성 : Spring, 2013: No. 4-22

23 Homework Exercise: 3-7, 3-8, 3-15, 3-16, 3-17, 3-19, 3-26 Hamming Code: Have you ever heard about the Hamming Code? I don t think so Find a book that describes the Hamming Code and make 1 or 2 page hand-written summary as a homework. Some of you will be chosen to present the idea during the class. 컴퓨터구성 : Spring, 2013: No. 4-23

step 1-1

step 1-1 Written by Dr. In Ku Kim-Marshall STEP BY STEP Korean 1 through 15 Action Verbs Table of Contents Unit 1 The Korean Alphabet, hangeul Unit 2 Korean Sentences with 15 Action Verbs Introduction Review Exercises