15 213 The Class That Gives CMU Its Zip Base 10 Number Representation That s why fingers are known as digits Natural representation for financial transactions Floating point number cannot exactly represent 1 20 Even carries through in scientific notation 1 5213 X 104 Bits and Bytes Aug 31 2000 Topics Why bits Representing information as bits Binary Hexadecimal Byte representations numbers characters and strings Instructions Bit level manipulations Boolean algebra Expressing in C Implementing Electronically Hard to store ENIAC First electronic computer used 10 vacuum tubes digit Hard to transmit Need high precision to encode 10 signal levels on single wire Messy to implement digital logic functions Addition multiplication etc class02 ppt CS 213 F 00 Binary Representations Base 2 Number Representation class02 ppt 2 CS 213 F 00 Byte Oriented Memory Organization Programs Refer to Virtual Addresses Represent 1521310 as 111011011011012 Represent 1 2010 as 1 0011001100110011 0011 2 Represent 1 5213 X 104 as 1 11011011011012 X 213 Electronic Implementation Easy to store with bistable elements Reliably transmitted on noisy and inaccurate wires 0 Why Don t Computers Use Base 10 1 0 Conceptually very large array of bytes Actually implemented with hierarchy of different memory types SRAM DRAM disk Only allocate for regions actually used by program In Unix and Windows NT address space private to particular process Program being executed Program can clobber its own data but not that of others Compiler Run Time System Control Allocation 3 3V Where different program objects should be stored Multiple mechanisms static stack and heap In any case all allocation within single virtual address space 2 8V 0 5V 0 0V Straightforward implementation of arithmetic functions class02 ppt 3 CS 213 F 00 class02 ppt 4 CS 213 F 00 Encoding Byte Values Byte 8 bits al im a r y c n H De B i 0 0 0000 1 1 0001 2 2 0010 3 3 0011 4 4 0100 5 5 0101 6 6 0110 7 7 0111 8 8 1000 9 9 1001 A 10 1010 B 11 1011 C 12 1100 D 13 1101 E 14 1110 F 15 1111 ex Binary 000000002 to 111111112 Decimal 010 to 25510 Hexadecimal 0016 to FF16 Base 16 number representation Use characters 0 to 9 and A to F Write FA1D37B16 in C as 0xFA1D37B Or 0xfa1d37b class02 ppt Machine Words 5 CS 213 F 00 Machine Has Word Size Nominal size of integer valued data Including addresses Most current machines are 32 bits 4 bytes Limits addresses to 4GB Becoming too small for memory intensive applications High end systems are 64 bits 8 bytes Potentially address 1 8 X 1019 bytes Machines support multiple data formats Fractions or multiples of word size Always integral number of bytes class02 ppt Word Oriented Memory Organization 32 bit 64 bit Words Words Addr 0000 Addresses Specify Byte Locations Address of first byte in word Addresses of successive words differ by 4 32 bit or 8 64 bit Addr 0000 Addr 0004 Addr 0008 Addr 0012 class02 ppt 7 Addr 0008 Bytes Addr 0000 0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 CS 213 F 00 6 CS 213 F 00 Data Representations Sizes of C Objects in Bytes C Data Type Compaq Alpha Typical 32 bit int 4 4 long int 8 4 char 1 1 short 2 2 float 4 4 double 8 8 long double 8 8 char 8 4 Or any other pointer class02 ppt 8 Intel IA32 4 4 1 2 4 8 10 12 4 CS 213 F 00 Byte Ordering Examining Data Representations Issue Code to Print Byte Representation of Data How should bytes within multi byte word be ordered in memory Conventions Casting pointer to unsigned char creates byte array Alphas PC s are Little Endian machines Least significant byte has lowest address Sun s Mac s are Big Endian machines Least significant byte has highest address typedef unsigned char pointer void show bytes pointer start int len int i for i 0 i len i printf 0x p t0x 2x n start i start i printf n Example Variable x has 4 byte representation 0x01234567 Address given by x is 0x100 Big Endian 0x100 0x101 0x102 0x103 01 Little Endian 23 45 67 Printf directives p Print pointer x Print Hexadecimal 0x100 0x101 0x102 0x103 67 45 23 01 9 class02 ppt CS 213 F 00 10 class02 ppt show bytes Execution Example Representing Integers int A 15213 int B 15213 long int C 15213 int a 15213 printf int a 15213 n show bytes pointer a sizeof int Result Decimal 15213 Binary 3 6 Sun A Alpha C Sun C 6D 3B 00 00 00 00 3B 6D 6D 3B 00 00 00 00 00 00 00 00 3B 6D 0x6d Alpha B Sun B 0x11ffffcb9 0x3b 0x11ffffcba 0x00 0x11ffffcbb 0x00 93 C4 FF FF FF FF C4 93 CS 213 F 00 B Alpha A 0x11ffffcb8 11 0011 1011 0110 1101 Hex int a 15213 class02 ppt CS 213 F 00 class02 ppt D Two s complement representation Covered next lecture 12 CS 213 F 00 Representing Pointers Representing Floats Alpha P int B 15213 int P B A0 FC FF FF 01 00 00 00 Alpha Address Hex 1 Binary F F F F F C A 0 0001 1111 1111 1111 1111 1111 1100 1010 0000 Sun F 00 B4 6D 46 46 6D B4 00 IEEE Single Precision Floating Point Representation Sun P EF FF FB 2C Alpha F Float F 15213 0 Hex Binary Sun Address Hex Binary E F F F F B 2 C 1110 1111 1111 1111 1111 1011 0010 1100 Different compilers machines assign different locations to objects 13 class02 ppt CS 213 F 00 Alpha S Sun S 31 35 32 31 33 00 31 35 32 31 33 00 Compatibility Byte ordering not an issue Data are single byte quantities Text files generally platform independent Except for different conventions of line termination character class02 ppt 15 1110 1101 1011 01 Not same as integer representation but consistent across machines 14 CS 213 F 00 Machine Level Code Representation char S 6 15213 Represented by array of characters Each character encoded in ASCII format Standard 7 bit encoding of character set Other encodings exist but uncommon Character 0 has code 0x30 Digit i has code 0x30 i String should be null terminated Final character 0 15213 class02 ppt Representing Strings Strings in C 4 6 6 D B 4 0 0 0100 0110 0110 1101 1011 0100 0000 0000 Encode Program as Sequence of Instructions Each simple operation Arithmetic operation Read or write memory Conditional branch Instructions encoded as bytes Alpha s Sun s Mac s use 4 byte instructions Reduced Instruction Set Computer RISC PC s use variable length instructions Complex Instruction Set Computer CISC Different instruction types and encodings for different machines Most code not binary compatible Programs are Byte Sequences Too CS 213 F 00 class02 ppt 16 CS 213 F 00 Representing Instructions Boolean Algebra Developed by George Boole in 19th Century Alpha sum int sum int x …
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