Data Representa5on. CSC 2400: Computer Systems. What kinds of data do we need to represent?
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1 CSC 2400: Computer Systems Data Representa5on What kinds of data do we need to represent? - Numbers signed, unsigned, integers, floating point, complex, rational, irrational, - Text characters, strings, - Images pixels, colors, shapes, - Sound - Logical true, false - Instructions - q Data type: - representation and operations within the computer 1
2 Word-Oriented Memory Organization q Addresses Specify Byte Locations - Address of first byte in word - Addresses of successive words differ by 4 (32-bit) or 8 (64-bit) 32-bit 64-bit Words Words Bytes Addr. Addr = 0000?? Addr = 0004?? Addr = 0008?? Addr = 0012?? Addr = 0000?? Addr = 0008?? Data Representations q Sizes of C Objects (in Bytes) C Data Type Sparc/Unix Typical 32-bit Intel IA32 int long int char short float double long double /12 char * (or any other pointer) 2
3 The sizeof Operator Category sizeof Operators sizeof(type) sizeof(expr) q Unique among operators: evaluated at compile- 6me q Evaluates to type size_t; on tanner, same as unsigned int q Examples int i = 10; double d = 100.0; sizeof(int) /* On tanner, evaluates to 4 */ sizeof(i) /* On tanner, evaluates to 4 */ sizeof(double) /* On tanner, evaluates to 8 */ sizeof(d) /* On tanner, evaluates to 8 */ sizeof(d ) /* On tanner, evaluates to 8 */ Determining Data Sizes q To determine data sizes on your computer #include <stdio.h> int main() { printf("char: %d\n", (int)sizeof(char)); printf("short: %d\n", (int)sizeof(short)); printf("int: %d\n", (int)sizeof(int)); printf("long: %d\n", (int)sizeof(long)); printf("float: %d\n", ); printf("double: %d\n", ); printf("long double: %d\n", ); return 0; } q Output on tanner char: 1 short: 2 int: 4 long: 4 float: 4 double: 8 long double: 16 3
4 Byte Ordering q How should bytes within multi-byte word be ordered in memory? q Conventions - Sun s, Mac s are Big Endian machines o Least significant byte has highest address - Alphas, PC s are Little Endian machines o Least significant byte has lowest address Byte Ordering Example q Big Endian - Least significant byte has highest address q Little Endian - Least significant byte has lowest address q Example - Variable x has 4-byte representation 0x Address given by &x is 0x100 Big Endian Little Endian 0x100 0x101 0x102 0x x100 0x101 0x102 0x
5 Integers Unsigned Integers q An n-bit unsigned integer represents 2 n values: from 0 to 2 n
6 Signed Integers q How do computers differen6ate between posi6ve and nega6ve integers? - Posi6ve integers have most significant bit 0 - Nega6ve integers have most significant bit 1 q Nega6ve integer representa6ons: 1. Sign Magnitude 2. One s Complement 3. Two s Complement 1. Sign Magnitude q Use the lermost bit to store the sign - Zero for posi6ve number - One for nega6ve number q Examples è è -44 Sign Magnitude q Hard to do arithme6c this way, so it is rarely used - What is the result of 44 44? 6
7 1. Sign Magnitude (contd.) è è -44 Sign Magnitude q For numbers represented on n bits: - Range of posi6ve integers: from 0 to (2 n-1 1) - Range of nega6ve integers: from (2 n-1 1) to 1 2. One s Complement q LeRmost bit is 0 for posi6ve numbers è 44 q To obtain the corresponding nega6ve number (- 44), flip every bit: è -44 q So - 44 is the one s complement of 44. 7
8 2. One s Complement (contd.) q What is the result of 44 44? ( 44) (-44) q Issue: two different representa6ons for zero 3. Two s Complement q LeRmost bit is 0 for posi6ve numbers è 44 q To obtain the corresponding nega6ve number - 44, add 1 to the one s complement of 44: è one s complement è two s complement 8
9 3. Two s Complement (contd.) q What is the result of 44 44? ( 44) (-44) q Used by most computer systems q For numbers represented on n bits: - Range of posi6ve integers: - Range of nega6ve integers: from 0 to (2 n-1 1) from 2 n-1 to 1 Converting Two s Complement to Decimal 1. If leading bit is one, take two s complement to get a positive number. 2. Add powers of 2 that have 1 in the corresponding bit positions. 3. If original number was negative, add a minus sign. X = two = = = 104 ten Assuming 8-bit 2 s complement numbers. n 2 n
10 More Examples X = two = = = 39 ten X = two -X = = = = 26 ten X = -26 ten n 2 n Assuming 8-bit 2 s complement numbers. Two s Complement Signed Integers q Most significant bit is sign bit and has weight 2 n-1. q Range of an n-bit number: -2 n-1 through 2 n The most negative number (-2 n-1 ) has no positive counterpart
11 Number Representa5ons Review q LeRmost bit zero indicates q Sign Magnitude: - nega6ve values q One s Complement: - nega6ve values q Two s complement: - nega6ve values positive number most significant bit 1 has weight 0 are the 1 s complement of positive values (flip every bit) are the 2 s complement of positive values most significant bit 1 has weight 2 n-1 Fill in the Table Bit Pattern Value (Sign Magnitude) Value (One s Complement) Value (Two s Complement)
12 Ques5on q What value does represent? ASCII 12
13 The ASCII Code American Standard Code for Information Interchange NUL SOH STX ETX EOT ENQ ACK BEL BS HT LF VT FF CR SO SI 16 DLE DC1 DC2 DC3 DC4 NAK SYN ETB CAN EM SUB ESC FS GS RS US 32 SP! " # $ % & ' ( ) * +, -. / : ; < = >? A B C D E F G H I J K L M N O 80 P Q R S T U V W X Y Z [ \ ] ^ _ 96 ` a b c d e f g h i j k l m n o 112 p q r s t u v w x y z { } ~ DEL Lower case: and upper case: E.g., a is 97 and A is 65 (i.e., 32 apart) char Constants q C has char constants (sort of) q Examples Constant Binary Representation (assuming ASCII) Note 'a' letter '0' digit '\x61' hexadecimal form Use single quotes for char constant Use double quotes for string constant * Technically 'a' is of type int; automatically truncated to type char when appropriate 13
14 More char Constants Escape characters Constant Binary Representation (assuming ASCII) Note '\a' alert (bell) '\b' backspace '\f' form feed '\n' newline '\r' carriage return '\t' horizontal tab '\v' vertical tab '\\' backslash '\?' question mark '\'' single quote '\"' double quote '\0' null Used often Interesting Properties of ASCII Code q What is relationship between a decimal digit ('0', '1', ) and its ASCII code? q What is the difference between an upper-case letter ('A', 'B', ) and its lower-case equivalent ('a', 'b', )? q Given two ASCII characters, how do we tell which comes first in alphabetical order? q Are 128 characters enough? ( 14
15 Other Data Types q Floating Points - IEEE representation, to be covered later q Text strings - sequence of characters, terminated with NULL (0) q Image - array of pixels o monochrome: one bit (1/0 = black/white) o color: red, green, blue (RGB) components (e.g., 8 bits each) o other properties: transparency q Sound - sequence of fixed-point numbers Limits of the Machine: Overflow 15
16 Review: Memory Space for Data q Sizes of C Objects (in Bytes) C Data Type Sparc/Unix Typical 32-bit Intel IA32 int long int char short float double long double /12 char * (or any other pointer) Overflow: Running Out of Room q Adding two large integers together - Sum might be too large to store in available bits - What happens? (8) (-8) (9) (-9) (-15) (+15) Assuming 5-bit 2 s complement numbers. q We have overflow if: - signs of both operands are the same, and - sign of sum is different. 16
17 Overflow q Unsigned integers - All arithme6c is modulo arithme6c - Sum would just wrap around q Signed integers - Can get nonsense values - Example with 16- bit integers o Sum: o Result: Try It Out q Write a program that computes the sum Use only short int variables in your code: short int a = 10000; short int b = 20000; short int c = 20000; short int sum = a + b + c; printf("sum = %d\n", sum); 17
18 Exercise q Assume only four bits are available for represen6ng integers, and signed integers are represented in 2 s complement. q Compute the value of the expression Casting Signed to Unsigned q C Allows Conversions from Signed to Unsigned short int x = 15213; unsigned short int ux = (unsigned short) x; short int y = ; unsigned short int uy = (unsigned short) y; q Resulting Value - No change in bit representation - Nonnegative values unchanged o ux = Negative values change into (large) positive values o uy =
19 Try It Out q C code: char a = 0xFF; unsigned char b = 0xFF; printf("a = %d\n", a); printf("b = %d\n", b); Int to Char? Try It Out #include <stdio.h> int main() { char c = 0x81; int i; } i = c; printf(" integer = %x\n character = %x\n", i, c); i = 0x ; c = i; printf(" integer = %d\n character code = %d\n", i, c); return 0; 19
20 C vs. Java: Cast Conversions q Java: demo6ons are not automa6c C: demo6ons are automa6c int i; char c; i = c; /* Implicit promotion */ /* Sign extension in Java and C */ c = i; /* Implicit demotion */ /* Java: Compiletime error */ /* C: OK; truncation */ c = (char)i; /* Explicit demotion */ /* Truncation in Java and C */ What did we learn? q Computer represents everything in binary - Integers, floa6ng- point numbers, characters, - Pixels, sounds, colors, etc. q Memory is bytes, words, endian, two s complement, ASCII, conver6ng between hex- binary- decimal, limits of machine, overflow, cast conversions 20
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