The architecture of computer hardware and systems software an information technology approach ch02

CHAPTER 2: Number Systems The Architecture of Computer Hardware and Systems Software: An Information Technology Approach 3rd Edition, Irv Englander John Wiley and Sons 2003 Why Binary?  Early computer design was decimal   John von Neumann proposed binary data processing (1945)    Mark I and ENIAC Simplified computer design Used for both instructions and data Natural relationship between on/off switches and calculation using Boolean logic Chapter Number Systems On Off True False Yes No 2-2 Counting and Arithmetic       Decimal or base 10 number system   Origin: counting on the fingers “Digit” from the Latin word digitus meaning “finger” Base: the number of different digits including zero in the number system  Example: Base 10 has 10 digits, through Binary or base Bit (binary digit): digits, and Octal or base 8: digits, through Hexadecimal or base 16: 16 digits, through F  Examples: 1010 = A16; 1110 = B16 Chapter Number Systems 2-3 Keeping Track of the Bits  Bits commonly stored and manipulated in groups    bits = byte bytes = word (in many systems) Number of bits used in calculations   Affects accuracy of results Limits size of numbers manipulated by the computer Chapter Number Systems 2-4 Numbers: Physical Representation   Different numerals, same number of oranges    Cave dweller: IIIII Roman: V Arabic: Different bases, same number of oranges    510 1012 123 Chapter Number Systems 2-5 Number System   Roman: position independent Modern: based on positional notation (place value)     Decimal system: system of positional notation based on powers of 10 Binary system: system of positional notation based powers of Octal system: system of positional notation based on powers of Hexadecimal system: system of positional notation based powers of 16 Chapter Number Systems 2-6 Positional Notation: Base 10 43 = x 101 + x 100 10’s place 1’s place Place 101 100 Value 10 x 10 x1 40 Evaluate Sum Chapter Number Systems 2-7 Positional Notation: Base 10 527 = x 102 + x 101 + x 100 100’s place 1’s place 10’s place Place 102 101 100 Value 100 10 x 100 x 10 x1 500 20 Evaluate Sum Chapter Number Systems 2-8 Positional Notation: Octal 6248 = 40410 64’s place 8’s place 1’s place Place 82 81 80 Value 64 Evaluate x 64 2x8 4x1 Sum for Base 10 384 16 Chapter Number Systems 2-9 Positional Notation: Hexadecimal 6,70416 = 26,37210 4,096’s place 256’s place 16’s place Place 163 162 161 160 Value 4,096 256 16 6x x 256 x 16 4x1 1,792 Evaluate 1’s place 4,096 Sum for Base 10 24,576 Chapter Number Systems 2-10 Binary Multiplication x 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 2’s place (bits shifted to line up with 2’s place of multiplier) 4’s place 32’s place 1 Result (AND) Note the at the end, since the 1’s place is not brought down Note: multiple carries are possible Chapter Number Systems 2-29 Converting from Base 10  Powers Table Power Base 256 128 64 32 16 512 64 16 16 Chapter Number Systems 32,768 4,096 65,536 4,096 256 2-30 From Base 10 to Base 4210 = 1010102 Power Base 2 64 32 16 1 1 Integer 42/32 =1 10/16 =0 10/8 =1 2/4 =0 2/2 =1 0/1 =0 Remainder 10 0 Chapter Number Systems 10 2-31 From Base 10 to Base Base 10 42 Quotient Remainde r ) 42 ( Least significant bit ) 21 ( ) 10 ( 2) 2) 2) Base Chapter Number Systems (1 (0 Most significant bit 101010 2-32 From Base 10 to Base 16 5,73510 = 166716 Power 65,536 4,096 256 16 1 6 Base 16 Integer 5,735 /4,096 =1 Remainder 5,735 - 4,096 1,639 –1,536 103 – 96 = 1,639 = 103 =7 Chapter Number Systems 1,639 / 256 =6 103 /16 =6 2-33 From Base 10 to Base 16 Base 10 5,735 Remainde r 16 ) 5,735 ( Least significant bit Quotient 16 ) 16 ) 16 ) 16 ) Base 16 358 ( 22 ( ( Most significant bit 1667 Chapter Number Systems 2-34 From Base 10 to Base 16 Base 10 8,039 Remainde r 16 ) 8,039 ( Least significant bit Quotient 16 ) 16 ) 16 ) 16 ) Base 16 502 ( 31 ( 15 ( Most significant bit 1F67 Chapter Number Systems 2-35 From Base to Base 10 72638 = 3,76310 Power Sum for Base 10 83 82 81 80 512 64 x7 x2 x6 x3 3,584 128 48 Chapter Number Systems 2-36 From Base to Base 10 72638 = 3,76310 x8 56 + = 58 x8 464 + = 470 x8 3760 + = 3,763 Chapter Number Systems 2-37 From Base 16 to Base  The nibble approach  Hex easier to read and write than binary Base 16 Base 0001  F 1111 0110 0111 Why hexadecimal?  Modern computer operating systems and networks present variety of troubleshooting data in hex format Chapter Number Systems 2-38 Fractions  Number point or radix point    Decimal point in base 10 Binary point in base No exact relationship between fractional numbers in different number bases  Exact conversion may be impossible Chapter Number Systems 2-39 Decimal Fractions  Move the number point one place to the right    Effect: multiplies the number by the base number Example: 139.010 139010 Move the number point one place to the left   Effect: divides the number by the base number Example: 139.010 Chapter Number Systems 13.910 2-40 Fractions: Base 10 and Base 258910 Place 10-1 10-2 10-3 10-4 Value 1/10 1/100 1/1000 1/10000 x 1/10 x 1/100 x 1/1000 x1/1000 05 008 0009 Evaluate Sum 1010112 = 0.67187510 Place 2-1 2-2 2-3 2-4 2-5 2-6 Value 1/2 1/4 1/8 1/16 1/32 1/64 x 1/2 x 1/4 1x 1/8 x 1/16 x 1/32 x 1/64 0.03125 0.015625 Evaluate Sum Chapter Number Systems 0.125 2-41 Fractions: Base 10 and Base  k k No general relationship between fractions of types 1/10 and 1/2    Therefore a number representable in base 10 may not be representable in base But: the converse is true: all fractions of the form 1/2 k can be represented in base 10 Fractional conversions from one base to another are stopped   If there is a rational solution or When the desired accuracy is attained Chapter Number Systems 2-42 Mixed Number Conversion   Integer and fraction parts must be converted separately Radix point: fixed reference for the conversion   Digit to the left is a unit digit in every base B0 is always regardless of the base Chapter Number Systems 2-43 ... stored and manipulated in groups    bits = byte bytes = word (in many systems) Number of bits used in calculations   Affects accuracy of results Limits size of numbers manipulated by the computer. .. Chapter Number Systems Range (0 and 1) 1,024 (1K) 65,536 (64K) 2-14 Base or Radix  Base:   The number of different symbols required to represent any given number The larger the base, the more numerals... Number Systems 1 1 1 1 0 0 1 2-24 Binary Arithmetic  Addition   Multiplication    Boolean using XOR and AND + AND Shift Division x Chapter Number Systems 1 10 0 2-25 Binary Arithmetic: Boolean