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Introduction to Numbering System

Introduction to Numbering System. What is a numbering system ? A method to count; know quantity Many ways to represent numbers For humans we use decimal numbering system 0, 1, 2, ….9, pattern keeps repeating, till infinity Why 10 digits? Not sure….. Take a guess

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Introduction to Numbering System

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  1. Introduction to Numbering System • What is a numbering system ? • A method to count; know quantity • Many ways to represent numbers • For humans we use decimal numbering system • 0, 1, 2, ….9, pattern keeps repeating, till infinity • Why 10 digits? Not sure….. Take a guess • Computers use binary numbering system Computers only understand zeros and ones • Computers are made of electronic circuit which has current flowing through it. Voltage makes the current flow • Only two levels of voltage (there are always exceptions) Zero volt = ZERO = ground = OFF +5 volt (or 3.5V) = ONE = ON

  2. Why? • Why do we have numbering systems? • So we can count • Why so many numbering systems? • Computers can only understand zeros and ones • Decimal numbering systems started long before the dawn of computes and is easy for humans to use • Not easy to use zeros and ones for daily use, takes to may bits to represent a small amount • Example • 65536 in decimal takes five digits • 65536 in binary takes 16 bits

  3. Introduction to Numbering System • Decimal • Hexadecimal • Octal • Binary

  4. Decimal Most commonly used numbering system Base 10, there are ten unique units (digits) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Pattern starts at 0 and ends with base – 1 then keeps repeating, tens, hundreds, thousands,… 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 . . . . .. . . . . . . . . . . . . . 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109

  5. Hexadecimal • Used extensively in computer programming Base 16, 16 digits, From 0 to f (0 to base –1) 0 1 2 3 4 5 6 7 8 9 a b c d e f 10 11 12 13 14 15 16 17 18 19 1a 1b 1c 1d 1e 1f 20 21 22 23 24 25 26 27 28 29 2a 2b 2c 2d 2e 2f 30 31 32 33 34 35 36 37 38 39 3a 3b 3c 3d 3e 3f  0 to F  10 to FF  100 to FFF  1000 to FFFF int x = 0xFF; // 255 in decimal int x = 0xFFFF; // 65536 in decimal int x = 0x7FFFFFFF; //2,147,483,647 in decimal

  6. Octal • WAS used extensively in computer programming a while ago, replaced by Hex • Still used in debuggers and assembly language programs • Base 8, 8 digits from 0 to 7 ( 0 to base-1) • 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 21 22 23 24 25 26 27 int y = 011; // 9 in decimal

  7. Binary • Language of computers • Computer only understand 0s and 1s Base 2, 2 variations of 1 bit0 and 1 (0 to base –1) 0 1 one bit = 2 unique values 00 01 10 11 two bits = 4 unique values 100 101 110 111 three bits = 8 unique values 1000 1001 1010 1011 1100 1101 1110 1111

  8. Bits, Bytes and Words 1 byte has 8 bits 1 word has 16 bits 1 double word has 32 bits • Byte is the smallest unit of memory allocated • An int is 32 bits, one double word • 00000001 00000010 00000011 0000010000000101 00000110 00000111 0000100000001001 00001100 00001011 00001100 example: 00000001 in binary = 1 in decimal00000011 in binary = 3 in decimal

  9. Binary 2 bits can represent 4 (22) unique combinations From 00 01 10 11 (0, 1, 2, 3) 4 bits can represent 16 (24) unique combinations From 0000 (0) to 1111 (0..15) 8 bits can represent 256 (28) unique combinations From 00000000 (0) to 11111111 (0..255) ) 8 bits = 1 byte = 28 = 256 unique combinations 16 bits = 2 bytes = 216 = 65536 unique combinations 32 bits = 4 bytes = 232 = 4,294,967,296 (~ 4.2 billion) 0000 0 0001 10010 20011 30100 40101 50110 60111 7 1000 81001 91010 101011 111100 121101 131110 141111 15

  10. Binary MSB <- <- <- LSB128 64 32 16 8 4 2 1 2726252423222120--------------------------------------------- 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 = 1 0 0 0 0 0 0 1 0 = 2 0 0 0 0 0 0 11 = 3 (2+1) 0 0 0 0 1 1 1 1 =15(8+4+2+1) 0 1 1 1 0 0 0 0 = 112 (64+32+16) 1 0 0 0 0 0 0 0 = 128 128 or -1 1 1 1 1 1 1 1 1=255 255 or -128 8 bits = 1 byte in binary system, can represent 256 numbers 2 8 bits can represent 256 numbers 2 16 bits can represent 65536 numbers

  11. Conversion 8 bits = 1 byte = 256 unique combinations Binary Octal Decimal Hex 012345671011121314151617-- 377 00000000000000010000001000000011000001000000010100000110000001110000100000001001000010100000101100001100000011010000111000001111 --------11111111 0123456789101112131415 ---255 0123456789ABCDEF --FF

  12. Conversion – bin to hex, octal Binary Number = Octal = Hexadecimal 100011010001 4321 8D1 • To convert a binary number to Octal, split binary number into pairs of three (need three binary digits to represent an octal) 100 011 010 001 4 3 2 1 (bin 001 = octal 1, bin 100 = octal 4) • To convert a binary number to hex, split binary number into pairs of four (need four binary digits to represent a hex num) 1000 1101 0001 8 D 1 (binary 1000 = hex 8 binary 1101 = decimal 13 = hex D)

  13. Conversion – hex, octal to bin • Hexadecimal to Binary 8D1 8 D 1 1000 1101 0001 = 100011010001 • Octal to Binary 4321 4 3 2 1 100 011 010 001 = 100011010001

  14. Conversion to Decimal • Binary to Decimal Pos Val 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1 Bin Val 1 1 0 1 0 0 1 0 (1*128) + (1*64) + (0*32) + (1*16) + (0*8) + (0*4) + (1*2) + (0*1) 128 + 64 + 0 + 16 + 0 + 0 + 2 + 0 Dec Val 210

  15. Conversion – Octal to Decimal • Octal to Decimal Positional Val 83828180 512 64 8 1 Octal Value7 6 1 4 (7 * 512) + (6 * 64) + (1* 8) + (4*1) 3584 + 384 + 8 + 4 Octal Value was 7614 Decimal Value 3980

  16. Conversion – Hex to Decimal • Hex to Decimal Positional Val 163162161160 4096 256 16 1 Hex ValueA D 3 B (A * 4096) + (D *256) + (3 * 16) + (B*1) 40960 + 3328 + 48 + 11 Hex Value was AD3B Decimal Value is 44347

  17. Conversion – Decimal to Bin • Decimal to Binary • Convert decimal 13 to Binary Positional Val 25 24 23 22 21 20 32 16 8 4 2 1 0 0 1 1 0 1 first find (divisor and Modulus) operator. Is 8 > 13, no, it is OK, is 16 > 13, yes, NOT OK. Therefore first operator is 8 (13 / 8) = 1, (13 Mod 8) = 5 (5 / 4) = 1, (5 mod 4) = 1 (1 / 2) = 0, (1 mod 2) = 1 (1 / 1) = 1, (1 mod 1) = 0 Result = 1 1 0 1 in binary is decimal 13

  18. Conversion – Decimal to Hex • Decimal to Hex • Convert decimal 44347 to Hex Positional Val 163162161160 4096 256 16 1 (44347 / 4096) = 10, (44347 Mod 4096) = 3387 (3387 / 256) = 13, (3387 mod 256) = 59 (59 / 16) = 3, (59 /16) = 11 (11 / 1) = 11 Result = 10, 13, 3, 11 = Hex AD3B

  19. Conversion – compliment • Ones compliment (~) or NOT • Consider a 32 bit value “36” 00000000 00000000 00000000 00100100 ~ 11111111 11111111 11111111 11011011 Simply invert the bits • Twos compliment (~ + 1) or NOT + 1 • Consider a 32 bit value “36” 00000000 00000000 00000000 00100100 ~ 11111111 11111111 11111111 11011011 + 1 00000000 00000000 00000000 00000001 ------------------------------------------- = -36 11111111 11111111 11111111 11011100

  20. Conversion – compliment -3611111111 11111111 11111111 11011100 +36 00000000 00000000 00000000 00100100 ----------------------------------------- 0 00000000 00000000 00000000 00000000 -36 + 36 = 0

  21. End – Back to Lecture 2 http://www.geocities.com/msaleemyusuf/lecture_2.htm

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