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numbers BY SA.

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Presentation on theme: "numbers BY SA."— Presentation transcript:

1

2 numbers

3 BY SA

4

5

6

7

8

9

10

11 0 1 bit (one of two states)

12

13 number system (a scheme for symbolically and verbally representing quantity)

14 unary (the tally system)

15  

16 positional notation (represents arbitrarily large quantities using a finite set of symbols)

17 decimal base-10 (positional notation using 10 symbols)

18 Arabic numerals

19 39 40

20

21

22

23 78 seventy-eight seven eight

24

25 octal base-8 (positional notation using 8 symbols)

26

27

28 octal base-8 decimal base-10

29 octal base-8 decimal base-10

30

31 61 (decimal) 061 (octal)

32 05673

33 hexadecimal hex base-16 (positional notation using 16 symbols)

34 A B C D E F

35 0F0F 10

36 ABC ABC hex base-16 decimal base-10

37 D E F hex base-16 decimal base-10

38 19 1A 1B 1C 1D 1E 1F hex base-16 decimal base-10

39 ADD BAD BEAD BEE BEEF CAFE DEAD DEAF DEED FAD FEED

40 0xADD 0xBAD 0xBEAD 0xBEE 0xBEEF 0xCAFE 0xDEAD 0xDEAF 0xDEED 0xFAD 0xFEED

41 0xA3BFFFFF 0xA3C00000

42 binary base-2 (positional notation using 2 symbols)

43 0 1

44

45 binary base-2 decimal base-10

46 binary base-2 decimal base

47 101b (binary) 101 (decimal)

48 b b

49

50 base conversions

51 36259

52 (30000) + (6000) + (200) + (50) + (9)

53 (3 * 10000) + (6 * 1000) + (2 * 100) + (5 * 10) + (9 * 1)

54 (3 * 10 4 ) + (6 * 10 3 ) + (2 * 10 2 ) + (5 * 10 1 ) + (9 * 10 0 )

55

56 octal to decimal

57 03675

58 (03000) + (0600) + (070) + (05)

59 (03 * 01000) + (06 * 0100) + (07 * 010) + (05 * 01)

60 (03 * ) + (06 * ) + (07 * ) + (05 * )

61 (3 * 8 3 ) + (6 * 8 2 ) + (7 * 8 1 ) + (5 * 8 0 )

62 (3 * 512) + (6 * 64) + (7 * 8) + (5 * 1)

63 (1636) + (384) + (56) + (5)

64 03675 = 2081

65 decimal to octal

66 36256

67 (3 * 10 4 ) + (6 * 10 3 ) + (2 * 10 2 ) + (5 * 10 1 ) + (6 * 10 0 )

68 (03 * ) + (06 * ) + (02 * ) + (05 * ) + (06 * )

69 (03 * ) + (06 * 01750) + (02 * 0144) + (05 * 012) + (06 * 01)

70 (072460) + (013560) + (0310) + (062) + (06)

71 36256 =

72 hex to decimal

73 0x36E59

74 (0x3 * 16 4 ) + (0x6 * 16 3 ) + (0xE * 16 2 ) + (0x5 * 16 1 ) + (0x9 * 16 0 )

75 (3 * 65536) + (6 * 4096) + (14 * 256) + (5 * 16) + (9 * 1)

76 (196608) + (24576) + (3584) + (80) + (9)

77 0x36E59 =

78 binary to decimal

79 10101b

80 (1 * 2 4 ) + (0 * 2 3 ) + (1 * 2 2 ) + (0 * 2 1 ) + (1 * 2 0 )

81 (1 * 16) + (0 * 8) + (1 * 4) + (0 * 2) + (1 * 1)

82 (16) + (0) + (4) + (0) + (1)

83 10101b = 21

84 b

85

86 (128) + (32) + (8) + (4) + (1)

87 b = 173

88

89 decimal to binary

90 1)find biggest fitting power of two 2)subtract it out 3)repeat until left with 0

91 35872

92 32768 (2 15 )

93 1???????????????b

94 = 3104

95 2048 (2 11 )

96 10001???????????b

97 3104 – 2048 = 1056

98 1024 (2 10 )

99 100011??????????b

100 1056 – 1024 = 32

101 32 (2 5 )

102 ?????b

103 32 – 32 = 0

104 b

105 why use hex and octal?

106 octal to binary (and vice versa)

107 octal base-8 binary base-2

108 03673

109 octal base-8 binary base-2

110 b

111 b

112 b 01526

113 hex to binary (and vice versa)

114 ABCDEF ABCDEF hex base-16 binary base-2

115 0x7E b

116 b 0x32D

117 recap Each digit is a coefficient of the number base raised to an increasing power. One technique works for all number base conversions. Shortcut conversions: decimal to binary (sum powers of 2) binary to decimal (subtract out powers of 2) octal and binary (1 octal digit = 3 binary digits) hex and binary (1 hex digit = 4 binary digits) Effectively, hex and octal serve as a compacter way of writing binary.

118

119 meaning relies upon agreement

120 integer (whole number)

121 signed integer unsigned integer

122 0, 1, 2, 3, 4, 5… 0b, 1b, 10b, 11b, 100b, 101b…

123 n bits = 2 n values 1 bit = 2 values 2 bits = 4 values 3 bits = 8 values 4 bits = 16 values 5 bits = 32 values 6 bits = 64 values 7 bits = 128 values 8 bits = 256 values etc…

124 unsigned value range 1 bit = bits = bits = bits = bits = bits = bits = bits = etc…

125 (negative three) (positive three) sign bit

126 (negative three) (positive three) one’s complement

127 (negative three) (positive three) two’s complement

128 (positive three) (negative three) two’s complement

129 (negative three in excess-8) (positive three in excess-8) excess-n

130 (negative three in excess-42) (positive three in excess-42) excess-n

131 sign bit: -127 to +127 one’s complement: -127 to +127 two’s complement: -128 to bit range

132 rational numbers 2/5 1/98 7/1 -61/1738

133 radix-point notation (ratio written as an integer component and a fractional component, separated by a radix point) 3/4 1/8 -7/1 138/

134 36.259

135 (30) + (6) + (0.2) + (0.05) + (0.009)

136 (3 * 10 1 ) + (6 * 10 0 ) + (2 * ) + (5 * ) + (9 * )

137 /101/1001/1000

138 /81/641/512

139 10.111b /21/41/8

140 finite rational (rational with a fractional component which can be expressed with a finite number of digits) 3/4 -7/1 138/20 1/3 738/ …

141 a ratio with denominator 2 a 5 b is finite in decimal a ratio with denominator 2 a is finite in octal a ratio with denominator 2 a is finite in hex a ratio with denominator 2 a is finite in binary all ratios which are finite in binary are also finite in decimal some ratios which are finite in decimal are also finite in binary

142 rational as two integers 3/4 Numerator: Denominator: /13 Numerator: Denominator:

143 fixed-point 57/ b Integer: Fraction: (the computing equivalent of radix-point notation)

144 scientific notation engineering notation = * = * = * 10 5

145 floating-point 21/ b b * 2 3 Significand: Exponent: (the computing equivalent of scientific notation)

146 6.75

147 110b + 0.?b

148 0.75 3/4

149 1/2 + 1/

150 6.75 = b

151 = b * 2 2

152 Significand: Exponent:

153 IEEE floating-point (an international standard)

154


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