Goals for this Week. CSC 2400: Computer Systems. Bits, Bytes and Data Types. Binary number system. Finite representations of binary integers

Transcription

1 CSC 2400: Computer Systems Bits, Bytes and Data Types 1 Goals for this Week Binary number system Why binary? Converting between decimal and binary and octal and hexadecimal number systems Finite representations of binary integers Unsigned and signed integers Integer addition and subtraction Bitwise operators AND, OR, NOT, and XOR Shift-left and shift-right 2

2 Analog/Analogue Systems Analogue Systems V(t) can have any value between its minimum and maximum value V(t) Digital Systems Digital Systems V(t) takes a value selected from a set of values Binary digital systems form the basis of almost all hardware systems currently V(t)

3 Why Bits (Binary Digits)? Computers are built using digital circuits Inputs and outputs can have only two values True (high voltage) or false (low voltage) Represented as 1 and 0 Can represent many kinds of information Boolean (true or false) Numbers (23, 79, ) Characters ( a, z, ) Pixels, sounds Internet addresses Can manipulate in many ways Read and write Logical operations Arithmetic 5 Coding A single binary input can have two values: 1 or 0 More bits = more combinations How many values can you represent on 3 bits? What about n bits? 6

4 Memorize Powers of 2 Powers of = = = = = = = 512 Powers of = 1Kilo (1024) 10 3 = 1, = 1Mega ( = 1,048,576 ) 10 6 = 1,000, = 1Giga ( = 1,073,741,824) 10 9 = 1,000,000, = 1Tera ( = 1,099,511,627,776) = 1,000,000,000,000 7 Base 10 and Base 2 Decimal (base 10) Each digit represents a power of = 4 x x x x 10 0 Binary (base 2) Each bit represents a power of = 1 x x x x 2 0 =

5 Binary to Decimal Sum up the place values of all 1-bits: Place Value = 10? = 10? 9 Decimal to Binary Use the Placement Method Powers of = 2? 128 goes into 155 once leaving 27 to be placed 1??????? 64 and 32 are too big (make them 0) 16 goes in once leaving ???? and so on 10

6 You Try It 11 Working w/ Bits is Tedious for People Express large binary numbers in hex using ¼ fewer digits: Hex Binary Decimal Hex Binary Decimal A B C D E F This is the hexadecimal system. Memorize the tables above. 12

7 Exercises 1. Based on the lookup tables on the previous page, convert the hex value 0x7A8BF7D6 into its binary equivalent: 7 A 8 B F 7 D Based on the lookup tables on the previous page, convert the binary to hex: E 9 4 B 5 F Why hexadecimal? Widely used in assembly language programming Used in network programming and debugging Used in graphic design programs e.g., for the red, green and blue components of a color: FF0000 represents red, for example. How many bits are used to represent each color? How many different colors can be represented?

8 Decimal Addition From right to left, we add each pair of digits We write the sum, and add the carry to the next column Sum Carry Hex Addition Hex addition is similar to decimal addition except that each hex digit has a range of 0 to F instead of 0 to 9, and a carry out occurs when the sum of hex digits in a particular column exceeds F (15 decimal). Examples: 1 C 3 C A + E D E A F B 9 A + D 2 E E 8 0

9 Hex Subtraction Hex subtraction is similar to decimal subtraction except that if the subtrahend is greater than the minuend, we must borrow 16 (10 hex) from the previous digit. Examples: C A A 5 A 1 / B D C 1 2 / / 1 D 3 5 F B 6 8 F A 9 Recall: Decimal Addition From right to left, we add each pair of digits We write the sum, and add the carry to the next column Base 10 Base Sum Carry Sum Carry 18

10 Finite Representation of Integers Fixed number of bits in memory Usually 8, 16, or 32 bits (1, 2, or 4 bytes) Unsigned integer No sign bit Always 0 or a positive number All arithmetic is modulo 2 n Examples of unsigned integers Modulo Arithmetic Consider only numbers in a range E.g., five-digit car odometer: 0, 1,, E.g., eight-bit numbers 0, 1,, 255 Roll-over when you run out of space E.g., car odometer goes from to 0, 1, E.g., eight-bit number goes from 255 to 0, 1, Adding 2 n doesn t change the answer For eight-bit number, n=8 and 2 n =256 E.g., ( ) mod 256 is This can help us do subtraction Suppose you want to compute a b Note that this equals a + ( b)

11 One s and Two s Complement One s complement: flip every bit E.g., b is (i.e., 69 in decimal) One s complement is That s simply Subtracting from is easy (no carry needed!) b one s complement Two s complement Add 1 to the one s complement E.g., (255 69) Putting it All Together Computing a b Same as a b Same as a + (255 b) + 1 Same as a + onescomplement(b) + 1 Same as a + twoscomplement(b) Example: The original number 69: One s complement of 69: Two s complement of 69: Add to the number 172: The sum comes to: Equals: 103 in decimal 23

12 Reading Two s Complement Patterns When a two s complement number has a highest bit 1, it indicates that the number is negative. To find the value, perform the same steps: Unknown value: One s complement: Add 1 (two s complement): We get a value of 69, so the original pattern must have been -69. Practice: Ch. 2, Exercise 2 24 Signed Integers Sign-magnitude representation Use one bit to store the sign Zero for positive number One for negative number Examples E.g., Hard to do arithmetic this way, so it is rarely used Complement representation One s complement Flip every bit E.g., Two s complement Flip every bit, then add 1 E.g.,

13 Fill in the Table Bit Pattern Value (Sign Magnitude) Value (One s Complement) Value (Two s Complement) Question What value does represent? [Ex. 6, page 69] 27

14 Overflow: Running Out of Room Adding two large integers together Sum might be too large to store in the number of bits available What happens? Unsigned integers All arithmetic is modulo arithmetic Sum would just wrap around Signed integers Can get nonsense values Example with 16-bit integers Sum: Result: Exercise Assume only four bits are available for representing integers, and signed integers are represented in 2 s complement. Compute the value of the expression

15 Bitwise Operations 30 Networking 31

16 Encryption/ Decryption 32 Compression 33

17 Bitwise Operators: AND and OR Bitwise AND (&) Bitwise OR ( ) & & Bitwise Operators: Not and XOR One s complement (~) Turns 0 to 1, and 1 to 0 E.g., set last three bits to 0 x = x & XOR (^) 0 if both bits are the same 1 if the two bits are different ^ Practice: Ch. 2, Exercise 9 35

18 Bitwise Operators: Shift Left/Right Shift left (<<): Multiply by powers of 2 Shift some # of bits to the left, filling the blanks with <<2 Shift right (>>): Divide by powers of 2 Shift some # of bits to the right. Fill in blanks with sign bit >>2-75>>2 sign extension sign extension Practice: Ch. 2, Exercises 10, 12, 13, Bitmasks Used to change or query one or more bits in a variable. The bitmask indicates which bits are to be affected. Common operations: Operation Expression C expression Set N th bit of x x = x OR 2 N x = x (1 << N); Clear N th bit of x x = x x = ; Read N th bit of x??? = x??? = x ; Can extend to groups of bits 37

19 Exercise Let x be an 8-bit integer. Set the 5 th least significant bit of x: Clear the 5 th least significant bit of x: 38 Bitmask Example /* This program demonstrates setting a bit, clearing a bit, and ** reading a bit. (pg 64) */ #include <stdio.h> main() { char a; int i; a=17; a=a (1 << 3); /* set 3rd bit */ printf("%d\n",a); a=a & (~(1<<4)); /* clear 4th bit */ printf("%d\n",a); } for (i=7; i>=0; i--) printf("%d ",(a&(1<<i)) >> i); /* read i'th bit */ printf("\n"); 39

20 Exercise #1 Write a small program that reads in an integer and prints out its binary representation. Sample output: Please enter an integer: 1025 The binary representation of 1025 is Exercise #2 Write a small program that reads in an integer prints out hexadecimal value of its bytes, separated by spaces, starting with the most significant byte. Sample output: Please enter an integer: 1025 The four bytes (in hexadecimal) are:

21 Example: Counting the 1 s How many 1 bits in a number? E.g., how many 1 bits in the binary representation of 53? Four 1 bits How to count them? Counting the Number of 1 Bits #include <stdio.h> #include <stdlib.h> int main(void) { unsigned n, count; printf( Number: "); if (scanf("%u", &n)!= 1) { fprintf(stderr, "Error: Expect number.\n"); exit(exit_failure); } /* Enter code to count the 1 bits */ } printf( Number of 1 bits: %u\n, count); return 0; 45

22 Practice Write a function getbyte that extracts a byte from a word. /* * getbyte(x, n) - Extract byte n from word x * Bytes numbered from 0 (LSB) to 3 (MSB) * Examples: getbyte(0x ,1) = 0x56 * Legal operations: ~ & ^ + << >> */ int getbyte(int x, int n) { /* Add code here */ } 46 Summary Computer represents everything in binary Integers, floating-point numbers, characters, addresses, Pixels, sounds, colors, etc. Binary arithmetic through logic operations Sum (XOR) and Carry (AND) Two s complement for subtraction Binary operations in C AND, OR, NOT, XOR, shift left and shift right Useful for efficient and concise code, though sometimes cryptic 47

23 Required Reading Textbook, Chapter 2 48