dce 2015 COMPUTER ARCHITECTURE CSE Fall 2014 Faculty of Computer Science and Engineering Department of Computer Engineering BK TP.HCM Vo Tan Phuong http://www.cse.hcmut.edu.vn/~vtphuong dce 2015 Chapter Data Representation Computer Architecture – Chapter © Spring 2015, CE dce 2015 Presentation Outline • • • • • • • • Positional Number Systems Binary and Hexadecimal Numbers Base Conversions Binary and Hexadecimal Addition Binary and Hexadecimal subtraction Carry and Overflow Character Storage Floating Point Number Computer Architecture – Chapter © Spring 2015, CE dce 2015 Positional Number Systems Different Representations of Natural Numbers XXVII 27 110112 Roman numerals (not positional) Radix-10 or decimal number (positional) Radix-2 or binary number (also positional) Fixed-radix positional representation with k digits Number N in radix r = (dk–1dk–2 d1d0)r Value = dk–1×r k–1 + dk–2×r k–2 + … + d1×r + d0 Examples: (11011)2 = 1×24 + 1×23 + 0×22 + 1×2 + = 27 (2103)4 = 2×43 + 1×42 + 0×4 + = 147 Computer Architecture – Chapter © Spring 2015, CE dce 2015 Binary Numbers • Each binary digit (called bit) is either or • Bits have no inherent meaning, can represent – Unsigned and signed integers – Characters – Floating-point numbers – Images, sound, etc • Bit Numbering Most Significant Bit Least Significant Bit 1 0 1 1 27 26 25 24 23 22 21 20 – Least significant bit (LSB) is rightmost (bit 0) – Most significant bit (MSB) is leftmost (bit in an 8-bit number) Computer Architecture – Chapter © Spring 2015, CE dce Converting Binary to Decimal 2015 • Each bit represents a power of • Every binary number is a sum of powers of • Decimal Value = (dn-1  2n-1) + + (d1  21) + (d0  20) • Binary (10011101)2 = 27 + 24 + 23 + 22 + = 157 1 0 1 1 27 26 25 24 23 22 21 20 Some common powers of Computer Architecture – Chapter © Spring 2015, CE dce 2015 Convert Unsigned Decimal to Binary • Repeatedly divide the decimal integer by • Each remainder is a binary digit in the translated value least significant bit 37 = (100101)2 most significant bit stop when quotient is zero Computer Architecture – Chapter © Spring 2015, CE dce 2015 Hexadecimal Integers • 16 Hexadecimal Digits: – 9, A – F • More convenient to use than binary numbers Binary, Decimal, and Hexadecimal Equivalents Computer Architecture – Chapter © Spring 2015, CE dce 2015 Converting Binary to Hexadecimal  Each hexadecimal digit corresponds to binary bits  Example: Convert the 32-bit binary number to hexadecimal 1110 1011 0001 0110 1010 0111 1001 0100  Solution: E 1110 B 1011 0001 0110 Computer Architecture – Chapter A 1010 0111 1001 0100 © Spring 2015, CE dce 2015 Converting Hexadecimal to Decimal • Multiply each digit by its corresponding power of 16 Value = (dn-1  16n-1) + (dn-2  16n-2) + + (d1  16) + d0 • Examples: (1234)16 = (1  163) + (2  162) + (3  16) + = Decimal Value 4660 (3BA4)16 = (3  163) + (11  162) + (10  16) + = Decimal Value 15268 Computer Architecture – Chapter © Spring 2015, CE 10 dce 2015 Carry and Overflow Examples • We can have carry without overflow and vice-versa • Four cases are possible (Examples are 8-bit numbers) 0 0 1 1 15 + 1 1 0 0 1 1 15 + 0 0 0 1 1 0 248 (-8) 0 1 1 23 0 0 1 Carry = Overflow = Carry = 1 1 0 1 79 + Overflow = 1 1 1 218 (-38) + 0 0 0 64 0 1 1 157 (-99) 0 1 1 143 (-113) 1 1 1 Carry = Overflow = Computer Architecture – Chapter Carry = 119 Overflow = © Spring 2015, CE 25 dce 2015 Range, Carry, Borrow, and Overflow • Unsigned Integers: n-bit representation Numbers < Numbers > max Borrow = Subtraction Finite Set of Unsigned Integers Carry = Addition max = 2n–1 = • Signed Integers: n-bit 2's complement representation Numbers < Numbers > max Negative Overflow Finite Set of Signed Integers n-1 = -2 Computer Architecture – Chapter Positive Overflow max = 2n-1–1 © Spring 2015, CE 26 dce 2015 Character Storage • Character sets – – – – Standard ASCII: 7-bit character codes (0 – 127) Extended ASCII: 8-bit character codes (0 – 255) Unicode: 16-bit character codes (0 – 65,535) Unicode standard represents a universal character set • Defines codes for characters used in all major languages • Used in Windows-XP: each character is encoded as 16 bits – UTF-8: variable-length encoding used in HTML • Encodes all Unicode characters • Uses byte for ASCII, but multiple bytes for other characters • Null-terminated String – Array of characters followed by a NULL character Computer Architecture – Chapter © Spring 2015, CE 27 dce 2015 Printable ASCII Codes ! " # A B C D E F $ % & ' ( ) * + , - / 3 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~ space DEL  Examples:  ASCII code for space character = 20 (hex) = 32 (decimal)  ASCII code for 'L' = 4C (hex) = 76 (decimal)  ASCII code for 'a' = 61 (hex) = 97 (decimal) Computer Architecture – Chapter © Spring 2015, CE 28 dce 2015 Control Characters • The first 32 characters of ASCII table are used for control • Control character codes = 00 to 1F (hexadecimal) – Not shown in previous slide • Examples of Control Characters – Character is the NULL character  used to terminate a string – Character is the Horizontal Tab (HT) character – Character 0A (hex) = 10 (decimal) is the Line Feed (LF) – Character 0D (hex) = 13 (decimal) is the Carriage Return (CR) – The LF and CR characters are used together • They advance the cursor to the beginning of next line • One control character appears at end of ASCII table – Character 7F (hex) is the Delete (DEL) character Computer Architecture – Chapter © Spring 2015, CE 29 dce 2015 Representing Big (and Small) Numbers • What if we want to encode the approx age of the earth? 4,600,000,000 or 4.6 x 109 or the weight in kg of one a.m.u (atomic mass unit) 0.0000000000000000000000000166 or 1.6 x 10-27 There is no way we can encode either of the above in a 32-bit integer • Floating point representation (-1)sign x F x 2E – Still have to fit everything in 32 bits (single precision) s E (exponent) bit bits F (fraction) 23 bits – The base (2, not 10) is hardwired in the design of the FPALU – More bits in the fraction (F) or the exponent (E) is a trade-off between precision (accuracy of the number) and range (size of the number) Computer Architecture – Chapter © Spring 2015, CE 30 dce 2015 Real Numbers • Conversion from real binary to real decimal • – 1101.10112 = – 13.687510 • since: • 11012 = 23 + 22 + 20 = 1310 and • 0.10112 = 2-1 + 2-3 + 2-4 = 0.5 + 0.125 + 0.0625 = 0.687510 • Conversion from real decimal to real binary: • +927.4510 = + 1110011111.01 1100 1100 1100 … • 927/2 = 463 + 1/2