2. numbers

3. SA BY

11. 1
bit
(one of two states)

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

14. unary
(the tally system)

15. Aiwt$e#&!)9?

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

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

18. Arabic numerals

19. 39 40

20. 9 09 …

21. 09 10

22. 86999
87000

23. 78 seventy-eight seven eight

24. Understand that a positional notation system can use any number of symbols as long as it uses at least 2. So you could have a base-3 system, or a base-17 system, or a base-1065 system, or whatever. The decision to use base-10 was really just an arbitrary historical decision, likely made because we have 10 fingers and 10 toes, so it seemed natural to people to use base-10. And in fact, some civilizations used other bases, like the ancient Babylonians used base-12, probably because it fit best with their calendar.

25. (positional notation using 8 symbols)
octal base-8
(positional notation using 8 symbols)
So let's look at a positional notation system which uses something other than base-10. The base-8 system goes by the name octal.

26.
Rather than contriving new symbols, we use the same Arabic numerals as we do in decimal; we just use 8 of them instead of all 10.

27. 07 10
When you have the quantity seven and you wish to express one higher, the '7' cycles back to '0', and the digit to its left cycles to '1', so the quantity we call 'eight' in decimal is written as '1' '0' in octal.
While it's very, very tempting to read '1' '0' as the quantity 'ten', it is rather, in octal, the same quantity which we write as '8' in decimal. (After a lifetime of habit, it's really hard to stop yourself from reflexively reading numerals as decimal numbers.)

28. octal base-8 decimal base-10 1 2 3 4 5 6 7 10 11 12 13 14 1 2 3 4 5 69 10 11 12
octal base-8
decimal base-10

29. octal base-8 decimal base-10 15 16 17 20 21 22 23 24 25 26 27 30 13 1418 19 20 21 22 23 24
octal base-8
decimal base-10

30. 317777
320000

31. 61 (decimal)
061 (octal)

32. 05673

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

34.
8 9 A B C D E F

35. 0F 10

36. hex base-16 decimal base-10 1 2 3 4 5 6 7 8 9 A B C 1 2 3 4 5 6 7 8 911 12
hex base-16
decimal base-10

37. hex base-16 decimal base-10 D E F 10 11 12 13 14 15 16 17 18 13 14 1519 20 21 22 23 24
hex base-16
decimal base-10

38. hex base-16 decimal base-10 19 1A 1B 1C 1D 1E 1F 20 21 22 23 24 25 2627 28 29 30 31 32 33 34 35 36
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. (positional notation using 2 symbols)
binary base-2
(positional notation using 2 symbols)

43. 0 1

44. 01 10

45. binary base-2 decimal base-10 1 10 11 100 101 110 111 1000 1001 1010
1011
1100 1 2 3 4 5 6 7 8 9 10 11 12
binary base-2
decimal base-10

46. binary base-2 decimal base-10 1101 1110 1111 10000 10001 10010 10011
10100
10101
10110
10111
11000 13 14 15 16 17 18 19 20 21 22 23 24
binary base-2
decimal base-10

47. 101b (binary)
101 (decimal)

48. b b

50. base conversions

51. 36259

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

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

54. (3 * 104)
+ (6 * 103)
+ (2 * 102)
+ (5 * 101)
+ (9 * 100)

55. 36259
104
103
102
101
100
10000
1000
100 10 1

56. octal to decimal

57. 03675

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

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

60. (03 * 0103)
+ (06 * 0102)
+ (07 * 0101)
+ (05 * 0100)

61. (3 * 83)
+ (6 * 82)
+ (7 * 81)
+ (5 * 80)

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 * 104)
+ (6 * 103)
+ (2 * 102)
+ (5 * 101)
+ (6 * 100)

68. (03 * 0124)
+ (06 * 0123)
+ (02 * 0122)
+ (05 * 0121)
+ (06 * 0120)

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 * 164) + (0x6 * 163) + (0xE * 162) + (0x5 * 161) + (0x9 * 160)

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

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

77. 0x36E59 =
224857

78. binary to decimal

79. 10101b

80. (1 * 24)
+ (0 * 23)
+ (1 * 22)
+ (0 * 21)
+ (1 * 20)

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

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

83. 10101b = 21

84. b

85. b
128 32 8 4 1

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

87. b =
173

89. decimal to binary

90. find biggest fitting power of two
subtract it out
repeat until left with 0

91. 35872

92. 35872
32768 (215)

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

94. =
3104

95. 3104
2048 (211)

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

97. 3104 – 2048 =
1056

98. 1056
1024 (210)

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

100. 1056 – 1024 = 32

101. 32
32 (25)

102. ?????b

103. 32 – 32 =

104. b

105. why use hex and octal?

106. octal to binary
(and vice versa)

107. octal base-8 binary base-2 1 2 3 4 5 6 7 000 001 010 011 100 101 1101 2 3 4 5 6 7
000
001
010
011
100
101
110
111
octal base-8
binary base-2

108. 03673

109. octal base-8 binary base-2 1 2 3 4 5 6 7 000 001 010 011 100 101 1101 2 3 4 5 6 7
000
001
010
011
100
101
110
111
octal base-8
binary base-2

110. 03673 b

111. b

112. b
01526

113. hex to binary
(and vice versa)

114. hex base-16 binary base-2 1 2 3 4 5 6 7 8 9 A B C D E F 0000 0001 00101 2 3 4 5 6 7 8 9 A B C D E F
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
hex base-16
binary base-2

115. 0x7E9 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.

119. meaning relies upon agreement

120. integer (whole number) 5878 0 -87 234

121. signed integer unsigned integer

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

123. n bits = 2n values 1 bit = 2 values 2 bits = 4 values
etc…

124. unsigned value range 1 bit = 0..1 2 bits = 0..3 3 bits = 0..7
etc…

125. sign bit
(positive three)
(negative three)

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

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

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

129. 00000101 (negative three in excess-8)
excess-n
(positive three in excess-8)
(negative three in excess-8)

130. 00100111 (negative three in excess-42)
excess-n
(positive three in excess-42)
(negative three in excess-42)

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

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
138/20
0.75
0.125
-7.0
6.9

134. 36.259

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

136. (3 * 101) + (6 * 100) + (2 * 10-1) + (5 * 10-2) + (9 * 10-3)

137. 36.259
101
100
10-1
10-2
10-3 10 1
1/10
1/100
1/1000

138. 81 80
8-1
8-2
8-3 8 1
1/8
1/64
1/512

139. 10.111b 21 20
2-1
2-2
2-3 2 1
1/2
1/4
1/8

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
738/61
0.75
-7.0
6.9
0.33 …

141. a ratio with denominator 2a5b is finite in decimal a ratio with denominator 2a is finite in octal a ratio with denominator 2a is finite in hex a ratio with denominator 2a 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:
-7/13
Numerator:
Denominator:

143. (the computing equivalent of radix-point notation)
fixed-point
(the computing equivalent
of radix-point notation)
57/8 b
Integer:
Fraction:

144. scientific notation engineering notation
= * 102
= * 10-3
= * 105

145. (the computing equivalent of scientific notation)
floating-point
(the computing equivalent
of scientific notation)
21/2
1010.1b
1.0101b * 23
Significand:
Exponent:

146. 6.75

147. 6.75
110b + 0.?b

148. 0.75
3/4

149. 3/4
1/2 + 1/4

150. 6.75 =
110.11b

151. 110.11b =
1.1011b * 22

152. 1.1011b * 22 Significand: 11011000 Exponent: 00000010

153. (an international standard)
IEEE floating-point
(an international standard)