1. CHAPTER 3: Number Systems The Architecture of Computer Hardware and Systems Software & Networking: An Information Technology Approach 4th Edition, Irv Englander John Wiley and Sons  2010 PowerPoint slides authored by Wilson Wong, Bentley University PowerPoint slides for the 3 rd edition were co-authored with Lynne Senne, Bentley University

2. Why Binary?  Early computer design was decimal  Mark I and ENIAC  John von Neumann proposed binary data processing (1945)  Simplified computer design  Used for both instructions and data  Natural relationship between on/off switches and calculation using Boolean logic OnOff TrueFalse YesNo 10 3-2 Copyright 2010 John Wiley & Sons, Inc.

3. 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, 0 through 9  Binary or base 2  Bit (binary digit): 2 digits, 0 and 1  Octal or base 8: 8 digits, 0 through 7  Hexadecimal or base 16: 16 digits, 0 through F  Examples: 10 10 = A 16 ; 11 10 = B 16 3-3 Copyright 2010 John Wiley & Sons, Inc.

4. Keeping Track of the Bits  Bits commonly stored and manipulated in groups  8 bits = 1 byte  4 bytes = 1 word (in many systems)  Number of bits used in calculations  Affects accuracy of results  Limits size of numbers manipulated by the computer 3-4 Copyright 2010 John Wiley & Sons, Inc.

5. Numbers: Physical Representation  Different numerals, same number of oranges  Cave dweller: IIIII  Roman: V  Arabic: 5  Different bases, same number of oranges  5 10  101 2  12 3 3-5 Copyright 2010 John Wiley & Sons, Inc.

6. 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 2  Octal system: system of positional notation based on powers of 8  Hexadecimal system: system of positional notation based powers of 16 3-6 Copyright 2010 John Wiley & Sons, Inc.

7. Positional Notation: Base 10 Place10 2 10 1 10 0 Value100101 Evaluate5 x 1002 x 107 x1 Sum500207 1’s place 10’s place 527 = 5 x 10 2 + 2 x 10 1 + 7 x 10 0 100’s place 3-7 Copyright 2010 John Wiley & Sons, Inc.

8. Positional Notation: Octal 624 8 = 404 10 Place8282 8181 8080 Value6481 Evaluate6 x 642 x 84 x 1 Sum for Base 10 384164 64’s place8’s place1’s place 3-8 Copyright 2010 John Wiley & Sons, Inc.

9. Positional Notation: Hexadecimal 6,704 16 = 26,372 10 Place16 3 16 2 16 1 16 0 Value4,096256161 Evaluate 6 x 4,096 7 x 2560 x 164 x 1 Sum for Base 10 24,5761,79204 4,096’s place256’s place1’s place16’s place 3-9 Copyright 2010 John Wiley & Sons, Inc.

10. Positional Notation: Binary Place2727 2626 2525 2424 23232 2121 2020 Value1286432168421 Evaluate1 x 1281 x 640 x 321 x160 x 81 x 41 x 20 x 1 Sum for Base 10 128640160420 1101 0110 2 = 214 10 3-10 Copyright 2010 John Wiley & Sons, Inc.

11. Range of Possible Numbers  R = B K where  R = range  B = base  K = number of digits  Example #1: Base 10, 2 digits  R = 10 2 = 100 different numbers (0…99)  Example #2: Base 2, 16 digits  R = 2 16 = 65,536 or 64K  16-bit PC can store 65,536 different number values 3-11 Copyright 2010 John Wiley & Sons, Inc.

12. Decimal Range for Bit Widths BitsDigitsRange 10+2 (0 and 1) 41+16 (0 to 15) 82+256 1031,024 (1K) 164+65,536 (64K) 2061,048,576 (1M) 329+4,294,967,296 (4G) 6419+Approx. 1.6 x 10 19 12838+Approx. 2.6 x 10 38 3-12 Copyright 2010 John Wiley & Sons, Inc.

13. Base or Radix  Base:  The number of different symbols required to represent any given number  The larger the base, the more numerals are required  Base 10 : 0,1, 2,3,4,5,6,7,8,9  Base 2 : 0,1  Base 8 : 0,1,2, 3,4,5,6,7  Base 16 : 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F 3-13 Copyright 2010 John Wiley & Sons, Inc.

14. Number of Symbols vs. Number of Digits  For a given number, the larger the base  the more symbols required  but the fewer digits needed  Example #1:  65 16 101 10 145 8 110 0101 2  Example #2:  11C 16 284 10 434 8 1 0001 1100 2 3-14 Copyright 2010 John Wiley & Sons, Inc.

15. Counting in Base 2 Binary Number Equivalent Decimal Number 8’s (2 3 )4’s (2 2 )2’s (2 1 )1’s (2 0 ) 00 x 2 0 0 11 x 2 0 1 101 x 2 1 0 x 2 0 2 111 x 2 1 1 x 2 0 3 1001 x 2 2 4 1011 x 2 2 1 x 2 0 5 1101 x 2 2 1 x 2 1 6 1111 x 2 2 1 x 2 1 1 x 2 0 7 10001 x 2 3 8 10011 x 2 3 1 x 2 0 9 10101 x 2 3 1 x 2 1 10 3-15 Copyright 2010 John Wiley & Sons, Inc.

16. Base 10 Addition Table +0123456789 00123456789 112345678910 223456789 11 33456789101112 445678 etc 910111213 3 10 + 6 10 = 9 10 3-16 Copyright 2010 John Wiley & Sons, Inc.

17. Base 8 Addition Table +01234567 001234567 1123456710 2234567 11 334567101112 4456710111213 55671011121314 667101112131415 7710111213141516 3 8 + 6 8 = 11 8 (no 8 or 9, of course) 3-17 Copyright 2010 John Wiley & Sons, Inc.

18. Base 10 Multiplication Table x0123456789 00 1123456789 224681012141618 3369121518212427 404812162024283236 551015202530354045 661218243036424854 771421283542495663 etc. 3 10 x 6 10 = 18 10 3-18 Copyright 2010 John Wiley & Sons, Inc.

19. Base 8 Multiplication Table x01234567 00 11234567 224610121416 30361114172225 44101420243034 55121724313643 66142230364452 77162534435261 3 8 x 6 8 = 22 8 3-19 Copyright 2010 John Wiley & Sons, Inc.

20. Addition BaseProblemLargest Single Digit Decimal 6 +3 9 Octal 6 +1 7 Hexadecimal 6 +9 F Binary 1 +0 1 3-20 Copyright 2010 John Wiley & Sons, Inc.

21. Addition BaseProblemCarryAnswer Decimal 6 +4 Carry the 1010 Octal 6 +2 Carry the 810 Hexadecimal 6 +A Carry the 1610 Binary 1 +1 Carry the 210 3-21 Copyright 2010 John Wiley & Sons, Inc.

22. Binary Arithmetic 11111 1101101 +10110 10000011 3-22 Copyright 2010 John Wiley & Sons, Inc.

23. Binary Arithmetic  Addition  Boolean using XOR and AND  Multiplication  AND  Shift  Division + 01 001 1110 x 01 000 101 3-23 Copyright 2010 John Wiley & Sons, Inc.

24. Binary Arithmetic: Boolean Logic  Boolean logic without performing arithmetic  EXCLUSIVE-OR  Output is “1” only if either input, but not both inputs, is a “1”  AND (carry bit)  Output is “1” if and only both inputs are a “1” 11111 1101101 +10110 10000011 3-24 Copyright 2010 John Wiley & Sons, Inc.

25. Binary Multiplication  Boolean logic without performing arithmetic  AND (carry bit)  Output is “1” if and only both inputs are a “1”  Shift  Shifting a number in any base left one digit multiplies its value by the base  Shifting a number in any base right one digit divides its value by the base  Examples:  10 10 shift left = 100 10  10 10 shift right = 1 10  10 2 shift left = 100 2  10 2 shift right = 1 2 3-25 Copyright 2010 John Wiley & Sons, Inc.

26. Binary Multiplication 1101 101 1101 1’s place 0 2’s place 1101 4’s place (bits shifted to line up with 4’s place of multiplier) 1000001 Result (AND) 3-26 Copyright 2010 John Wiley & Sons, Inc.

27. Converting from Base 10 Power Base 876543210 2 2561286432168421 8 32,7684,0965126481 16 65,5364,096256161  Powers Table 3-27 Copyright 2010 John Wiley & Sons, Inc.

28. From Base 10 to Base 2 42 10 = 101010 2 Power Base 6543210 26432168421 101010 Integer42/32 = 1 Remainder 10/16 = 0 10 10/8 = 1 2 2/4 = 0 2 2/2 = 1 0 0/1 = 0 0 10 3-28 Copyright 2010 John Wiley & Sons, Inc.

29. From Base 10 to Base 2 Base 1042 2 )42( 0 Least significant bit 2 ) 21( 1 2 )10( 0 2 )5( 1 2 )2( 0 2 )1 Most significant bit Base 2101010 Remainder Quotient 3-29 Copyright 2010 John Wiley & Sons, Inc.

30. From Base 10 to Base 16 5,735 10 = 1667 16 Power Base 43210 1665,5364,096256161 1667 Integer 5,735 /4,096 = 1 1,639 / 256 = 6 103 /16 = 6 7 Remainder 5,735 - 4,096 = 1,639 1,639 –1,536 = 103 103 – 96 = 7 3-30 Copyright 2010 John Wiley & Sons, Inc.

31. From Base 10 to Base 16 Base 108,039 16 )8,039( 7 Least significant bit 16 )502( 6 16 )31( 15 16 )1( 1 Most significant bit 16 )0 Base 161F67 Quotient Remainder 3-31 Copyright 2010 John Wiley & Sons, Inc.

32. From Base 8 to Base 10 7263 8 = 3,763 10 Power8383 8282 8181 8080 5126481 x 7 x 2x 6x 3 Sum for Base 10 3,584128483 3-32 Copyright 2010 John Wiley & Sons, Inc.

33. From Base 8 to Base 10 7263 8 = 3,763 10 7 x 8 56+ 2 =58 x 8 464+ 6 =470 x 8 3760+ 3 =3,763 3-33 Copyright 2010 John Wiley & Sons, Inc.

34. From Base 16 to Base 2  The nibble approach  Hex easier to read and write than binary  Why hexadecimal?  Modern computer operating systems and networks present variety of troubleshooting data in hex format Base 161F67 Base 20001111101100111 3-34 Copyright 2010 John Wiley & Sons, Inc.

35. Fractions  Number point or radix point  Decimal point in base 10  Binary point in base 2  No exact relationship between fractional numbers in different number bases  Exact conversion may be impossible 3-35 Copyright 2010 John Wiley & Sons, Inc.

36. Decimal Fractions  Move the number point one place to the right  Effect: multiplies the number by the base number  Example: 139.0 10 1390 10  Move the number point one place to the left  Effect: divides the number by the base number  Example: 139.0 10 13.9 10 3-36 Copyright 2010 John Wiley & Sons, Inc.

37. Fractions: Base 10 and Base 2 Place10 -1 10 -2 10 -3 10 -4 Value1/101/1001/10001/10000 Evaluate2 x 1/105 x 1/1008 x 1/10009 x1/1000 Sum.2.05.008.0009.101011 2 = 0.671875 10 Place2 -1 2 -2 2 -3 2 -4 2 -5 2 -6 Value1/21/41/81/161/321/64 Evaluate1 x 1/20 x 1/41x 1/80 x 1/161 x 1/321 x 1/64 Sum.50.1250.031250.015625.2589 10 3-37 Copyright 2010 John Wiley & Sons, Inc.

38. Fractions: Base 10 and Base 2  No general relationship between fractions of types 1/10 k and 1/2 k  Therefore a number representable in base 10 may not be representable in base 2  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 3-38 Copyright 2010 John Wiley & Sons, Inc.

39. 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  B 0 is always 1 regardless of the base 3-39 Copyright 2010 John Wiley & Sons, Inc.

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