Informationcputnam/Comp516/Overheads/PDFChap03.pdf · 2009-03-19 · 1111 10110 11101 100100 10...

Chapter

Information Representation

3

(a)

(c)

A seven-bit cell.

Some possible values in a

seven-bit cell.

0 1 1 0 1 0 1

1 1 0 1 1 0 0

0 0 0 0 0 0 0

Some impossible values in a

seven-bit cell.

6 8 0 7 2 5 1

J A N U A R Y

1 1 0 1 0

(b)

Figure 3.1

(a)

(c)

A seven-bit cell.


seven-bit cell.

0 1 1 0 1 0 1

1 1 0 1 1 0 0

0 0 0 0 0 0 0


seven-bit cell.

6 8 0 7 2 5 1

J A N U A R Y

1 1 0 1 0

(b)

Figure 3.1(Continued)

(a)

(c)

A seven-bit cell.


seven-bit cell.

0 1 1 0 1 0 1

1 1 0 1 1 0 0

0 0 0 0 0 0 0


seven-bit cell.

6 8 0 7 2 5 1

J A N U A R Y

1 1 0 1 0

(b) Figure 3.1(Continued)

0 7 14 21 28 35 1 8 15 22 29 36 2 9 16 23 30 37 3 10 17 24 31 38 4 11 18 25 32 . 5 12 19 26 33 . 6 13 20 27 34 .

Counting in decimal

0 7 16 25 34 43 1 10 17 26 35 44 2 11 20 27 36 45 3 12 21 30 37 46 4 13 22 31 40 . 5 14 23 32 41 . 6 15 24 33 42 .

Counting in octal

0 21 112 210 1001 1022 1 22 120 211 1002 1100 2 100 121 212 1010 1101 10 101 122 220 1011 1102 11 102 200 221 1012 . 12 110 201 222 1020 . 20 111 202 1000 1021 .

Counting in base 3

0 111 1110 10101 11100 100011 1 1000 1111 10110 11101 100100 10 1001 10000 10111 11110 100101 11 1010 10001 11000 11111 100110 100 1011 10010 11001 100000 . 101 1100 10011 11010 100001 . 110 1101 10100 11011 100010 .

Counting in binary

(b)(a) The place values for

10110 (bin).

1’s place

2’s place

4’s place

8’s place

16’s place

1 0 1 1 0

0

2

4

0

16

0

1

1

0

1

1’s place

2’s place

4’s place

8’s place

16’s place

=

=

=

=

=

22 (dec)

Converting 10110 (bin) to

decimal.

Figure 3.2

1’s place

10’s place

100’s place

1,000’s place

10,000’s place

5 8 0 63

Figure 3.3

(a) The binary number 10110.

1 2 0 14

2 21

0 20

1 22

5 10 8 0410

3 102

3 101 6 100

(b) The decimal number 58,036.

Figure 3.4

Remainders

Dividends

22

11

5

2

1

0

0

1

1

0

1

Figure 3.5

0 0 1 0 1 1 0

Figure 3.6

Binary addition rules

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10

– 0 0 1 0 1

Figure 3.7

Magnitude

Sign bit

Figure 3.8

• The NEG operation

‣ Taking the two’s complement

• The NOT operation

‣ Change the 1’s to 0’s and the 0’s to 1’s

• The two’s complement of a number is 1 plus its one’s complement

• NEG x = 1 + NOT x

The two’s complement rule

Decimal

–7

–6

–5

–4

–3

–2

–1

Binary

1001

1010

1011

1100

1101

1110

1111

Figure 3.9

Figure 3.10

Decimal

–8

–7

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

7

Binary

1000

1001

1010

1011

1100

1101

1110

1111

0000

0001

0010

0011

0100

0101

0110

0111

Decimal

–8

–7

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

7

Binary

1000

1001

1010

1011

1100

1101

1110

1111

0000

0001

0010

0011

0100

0101

0110

0111

1 1 1 1 0 1 1 0 1 0

1’s place

4’s place

32’s place

Figure 3.11

0 1 2 3 4 5 6 7

111110101100011010001000

Figure 3.12

–4 –3 –2 –1 0 1 2 3

011010001000111110101100

(b) Shifting the right part to the left side.

0 1 2 3 4 5 6 7

111110101100011010001000

(a) Breaking the number line in the middle.

Figure 3.13

–4 –3 –2 –1 0 1 2 3

32107654

Figure 3.14

The status bits• N = 1 if the result is negative

N = 0 otherwise

• Z = 1 if the result is all zeros

Z = 0 otherwise

• V = 1 if a signed integer overflow occurred

V = 0 otherwise

• C = 1 if an unsigned integer overflow occurred

C = 0 otherwise

p AND q

0

0

0

1

(a)

0

0

1

1

p q

0

1

0

1

p OR q

0

1

1

1

(b)

0

0

1

1

p q

0

1

0

1

p XOR q

0

1

1

0

(c)

0

0

1

1

p q

0

1

0

1

Figure 3.15

p AND q

0

0

0

1

(a)

0

0

1

1

p q

0

1

0

1

p OR q

0

1

1

1

(b)

0

0

1

1

p q

0

1

0

1

p XOR q

0

1

1

0

(c)

0

0

1

1

p q

0

1

0

1

p AND q

true

false

false

false

true

true

false

false

p

(a)

q

true

false

true

false

p OR q

true

true

true

false

(b)

true

true

false

false

p q

true

false

true

false

p XOR q

false

true

true

false

(c)

true

true

false

false

p q

true

false

true

false

Figure 3.16

p AND q

true

false

false

false

true

true

false

false

p

(a)

q

true

false

true

false

p OR q

true

true

true

false

(b)

true

true

false

false

p q

true

false

true

false

p XOR q

false

true

true

false

(c)

true

true

false

false

p q

true

false

true

false

Operation RTL Symbol

< >{ }

;

,

AND

OR

XOR

NOT

Implies

Transfer

Bit index

Informal description

Sequential separator

Concurrent separator

Figure 3.17

RTL specification of OR operation

c! a"b ; N! c< 0 , Z! c= 0

C

0

Figure 3.18

C! r"0# , r"0..4# ! r"1..5# , r"5# ! 0 ;

N! r < 0 , Z! r = 0 , V! {overflow}

Figure 3.19

RTL specification is a problem for the student

C

0 7 E 15 1C 23 1 8 F 16 1D 24 2 9 10 17 1E 25 3 A 11 18 1F 26 4 B 12 19 20 . 5 C 13 1A 21 . 6 D 14 1B 22 .

Counting in hexadecimal

(b)(a)

1’s place

16’s place

256’s place

4096’s place

8 B E 7

7

14

11

8

35,815

The place values for 8BE7. Converting 8BE7 to decimal.

1

16

256

4096 32,768

2,816

224

7

Figure 3.20

4

20

36

52

68

84

100

116

132

148

164

180

196

212

228

244

4

5

21

37

53

69

85

101

117

133

149

165

181

197

213

229

245

5

6

22

38

54

70

86

102

118

134

150

166

182

198

214

230

246

6

7

23

39

55

71

87

103

119

135

151

167

183

199

215

231

247

7

8

24

40

56

72

88

104

120

136

152

168

184

200

216

232

248

8

9

25

41

57

73

89

105

121

137

153

169

185

201

217

233

249

9 A

10

26

42

58

74

90

106

122

138

154

170

186

202

218

234

250

B

11

27

43

59

75

91

107

123

139

155

171

187

203

219

235

251

0_

1_

2_

3_

4_

5_

6_

7_

8_

9_

A_

B_

C_

D_

E_

F_

C

12

28

44

60

76

92

108

124

140

156

172

188

204

220

236

252

D

13

29

45

61

77

93

109

125

141

157

173

189

205

221

237

253

E

14

30

46

62

78

94

110

126

142

158

174

190

206

222

238

254

F

15

31

47

63

79

95

111

127

143

159

175

191

207

223

239

255

1

17

33

49

65

81

97

113

129

145

161

177

193

209

225

241

1

2

18

34

50

66

82

98

114

130

146

162

178

194

210

226

242

2

3

19

35

51

67

83

99

115

131

147

163

179

195

211

227

243

3

0

16

32

48

64

80

96

112

128

144

160

176

192

208

224

240

0

Figure 3.21

Binary

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

Hexadecimal

Figure 3.22

Binary

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

Hexadecimal

Char

NUL

SOH

STX

ETX

EOT

ENQ

ACK

BEL

BS

HT

LF

VT

FF

CR

SO

SI

DLE

DC1

DC2

DC3

DC4

NAK

SYN

ETB

CAN

EM

SUB

ESC

FS

GS

RS

US

Bin

000 0000

000 0001

000 0010

000 0011

000 0100

000 0101

000 0110

000 0111

000 1000

000 1001

000 1010

000 1011

000 1100

000 1101

000 1110

000 1111

001 0000

001 0001

001 0010

001 0011

001 0100

001 0101

001 0110

001 0111

001 1000

001 1001

001 1010

001 1011

001 1100

001 1101

001 1110

001 1111

Hex

00

01

02

03

04

05

06

07

08

09

0A

0B

0C

0D

0E

0F

10

11

12

13

14

15

16

17

18

19

1A

1B

1C

1D

1E

1F

Char

SP

Bin

010 0000

010 0001

010 0010

010 0011

010 0100

010 0101

010 0110

010 0111

010 1000

010 1001

010 1010

010 1011

010 1100

010 1101

010 1110

010 1111

011 0000

011 0001

011 0010

011 0011

011 0100

011 0101

011 0110

011 0111

011 1000

011 1001

011 1010

011 1011

011 1100

011 1101

011 1110

011 1111

Hex

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

Char Bin

100 0000

100 0001

100 0010

100 0011

100 0100

100 0101

100 0110

100 0111

100 1000

100 1001

100 1010

100 1011

100 1100

100 1101

100 1110

100 1111

101 0000

101 0001

101 0010

101 0011

101 0100

101 0101

101 0110

101 0111

101 1000

101 1001

101 1010

101 1011

101 1100

101 1101

101 1110

101 1111

Hex

40

41

42

43

44

45

46

47

48

49

4A

4B

4C

4D

4E

4F

50

51

52

53

54

55

56

57

58

59

5A

5B

5C

5D

5E

5F

Char

DEL

Bin

110 0000

110 0001

110 0010

110 0011

110 0100

110 0101

110 0110

110 0111

110 1000

110 1001

110 1010

110 1011

110 1100

110 1101

110 1110

110 1111

111 0000

111 0001

111 0010

111 0011

111 0100

111 0101

111 0110

111 0111

111 1000

111 1001

111 1010

111 1011

111 1100

111 1101

111 1110

111 1111

Hex

60

61

62

63

64

65

66

67

68

69

6A

6B

6C

6D

6E

6F

70

71

72

73

74

75

76

77

78

79

7A

7B

7C

7D

7E

7F

Abbreviations for Control Characters

NUL

SOH

STX

ETX

EOT

ENQ

ACK

BEL

BS

HT

LF

VT

null, or all zeros

start of heading

start of text

end of text

end of transmission

enquiry

acknowledge

bell

backspace

horizontal tabulation

line feed

vertical tabulation

FF

CR

SO

SI

DLE

DC1

DC2

DC3

DC4

NAK

SYN

ETB

form feed

carriage return

shift out

shift in

data link escape

device control 1

device control 2

device control 3

device control 4

negative acknowledge

synchronous idle

end of transmission block

CAN

EM

SUB

ESC

FS

GS

RS

US

SP

DEL

cancel

end of medium

substitute

escape

file separator

group separator

record separator

unit separator

space

delete

Binary

1100 0001

1100 0010

1100 0011

·

·

·

1111 0000

1111 0001

1111 0010

·

·

·

A

B

C

·

·

·

0

1

2

·

·

·

EBCDIC

Figure 3.24

(b)(a) The place values for

101.011 (bin).

18’s place

14’s place

12’s place

1’s place

2’s place

0 1 0 1 1

0.125

0.25

0.0

1.0

0.0

1

1

0

1

0

18’s place

14’s place

12’s place

1’s place

2’s place

=

=

=

=

=

5.375 (dec)

Converting 101.011 (bin) to

decimal.

4’s place

1 .

4.01 4’s place =

Figure 3.25

(a) The binary number 101.011.

1 2 0 02

21

21

1 20

5 10 0 6210

1 100

7 101

2 102

1 103

(b) The decimal number 506.721.

1 22

1 23

Figure 3.26

.5859375

1 .171875

0 .34375

0 .6875

1 .375

0 .75

1 .5

1 .0

(b) Convert the fractional part

6.5859375

6 (dec) = 110 (bin)

(a) Convert the whole partFigure 3.27

.5859375

1 .171875

0 .34375

0 .6875

1 .375

0 .75

1 .5

1 .0

(b) Convert the fractional part

6.5859375

6 (dec) = 110 (bin)

(a) Convert the whole part

.2

0 .4

0 .8

1 .6

1 .2

0 .4

0 .8

1 .6

Figure 3.28

Significand

Exponent

Sign

Figure 3.29

Two’s

Decimal Excess 3 Complement

4 100

3 000 101

2 001 110

1 010 111

0 011 000

1 100 001

2 101 010

3 110 011

4 111

Figure 3.30

Special value• Zero

‣ Exponent field all 0’s

‣ Significand all 0’s

‣ There is a +0 and a –0

31.0 0.1328125 0.13281250.0 31.0

Neg

ativ

e over

flow

Neg

ativ

e norm

aliz

ed

Neg

ativ

e under

flow

Zer

o

Posi

tive

under

flow

Posi

tive

norm

aliz

ed

Posi

tive

over

flow

Figure 3.31

Special value• Infinity


‣ Significand all 0’s

‣ There is a +∞ and a –∞‣ Produced by operation that gives result in

overflow region

Special value• Not a Number (NaN)


‣ Significand nonzero

‣ Produced by illegal math operations

Special value• Denormalized number


‣ Significand nonzero

‣ Hidden bit is assumed to be 0 instead of 1

‣ If the exponent is stored in excess n for normalized numbers, it is stored in excess n – 1 for denormalized numbers

3.5 Floating Point Representation 123

Not a Number 1 111 nonzero

Negative infinity 1 111 0000 !"

Negative 1 110 1111 !1.1111 # 23 !15.5normalized 1 110 1110 !1.1110 # 23 !15.0

... ... ...1 011 0001 !1.0001 # 20 !1.06251 011 0000 !1.0000 # 20 !1.01 010 1111 !1.1111 # 2!1 !0.96875... ... ...1 001 0001 !1.0001 # 2!2 !0.2656251 001 0000 !1.0000 # 2!2 !0.25

Negative 1 000 1111 !0.1111 # 2!2 !0.234375denormalized 1 000 1110 !0.1110 # 2!2 !0.21875

... ... ...1 000 0010 !0.0010 # 2!2 !0.031251 000 0001 !0.0001 # 2!2 !0.015625

Negative zero 1 000 0000 !0.0

Positive zero 0 000 0000 +0.0

Positive 0 000 0001 0.0001 # 2!2 0.015625denormalized 0 000 0010 0.0010 # 2!2 0.03125

... ... ...0 000 1110 0.1110 # 2!2 0.218750 000 1111 0.1111 # 2!2 0.234375

Positive 0 001 0000 1.0000 # 2!2 0.25normalized 0 001 0001 1.0001 # 2!2 0.265625

... ... ...0 010 1111 1.1111 # 2!1 0.968750 011 0000 1.0000 # 20 1.00 011 0001 1.0001 # 20 1.0625... ... ...0 110 1110 1.1110 # 23 15.00 110 1111 1.1111 # 23 15.5

Positive infinity 0 111 0000 +"


ScientificBinary notation Decimal

Figure 3.32Floating point values for a three-bitoperand and four-bit significand.

Figure 3.32

3.5 Floating Point Representation 123


Negative infinity 1 111 0000 !"

Negative 1 110 1111 !1.1111 # 23 !15.5normalized 1 110 1110 !1.1110 # 23 !15.0

... ... ...1 011 0001 !1.0001 # 20 !1.06251 011 0000 !1.0000 # 20 !1.01 010 1111 !1.1111 # 2!1 !0.96875... ... ...1 001 0001 !1.0001 # 2!2 !0.2656251 001 0000 !1.0000 # 2!2 !0.25

Negative 1 000 1111 !0.1111 # 2!2 !0.234375denormalized 1 000 1110 !0.1110 # 2!2 !0.21875

... ... ...1 000 0010 !0.0010 # 2!2 !0.031251 000 0001 !0.0001 # 2!2 !0.015625

Negative zero 1 000 0000 !0.0

Positive zero 0 000 0000 +0.0

Positive 0 000 0001 0.0001 # 2!2 0.015625denormalized 0 000 0010 0.0010 # 2!2 0.03125

... ... ...0 000 1110 0.1110 # 2!2 0.218750 000 1111 0.1111 # 2!2 0.234375

Positive 0 001 0000 1.0000 # 2!2 0.25normalized 0 001 0001 1.0001 # 2!2 0.265625

... ... ...0 010 1111 1.1111 # 2!1 0.968750 011 0000 1.0000 # 20 1.00 011 0001 1.0001 # 20 1.0625... ... ...0 110 1110 1.1110 # 23 15.00 110 1111 1.1111 # 23 15.5

Positive infinity 0 111 0000 +"


ScientificBinary notation Decimal

Figure 3.32Floating point values for a three-bitoperand and four-bit significand.


Bits 1 8 23

Bits 1 11 52

(a) Single precision

(b) Double precision

Figure 3.33

NevadaCalifornia

Las Vegas

Santa Barbara

San Francisco

Sacramento

Reno

40

60

40

65

55

70

Palm Springs

San Diego

Los Angeles

70

60

35

45

40

80

50

30

Figure 3.34

4 San Francisco

5 Sacramento

6 Reno

7 Las Vegas

0

1

2

3

4

5

6

7

1e30

35

45

40

55

65

80

70

0

City Numbers

0 Los Angeles

1 San Diego

2 Palm Springs

3 Santa Barbara

35

1e30

30

1e30

1e30

1e30

1e30

1e30

1

45

30

1e30

1e30

1e30

1e30

1e30

60

2

40

1e30

1e30

1e30

70

1e30

1e30

1e30

3

55

1e30

1e30

70

1e30

40

60

1e30

4

65

1e30

1e30

1e30

40

1e30

40

1e30

5

80

1e30

1e30

1e30

60

40

1e30

50

6

70

1e30

60

1e30

1e30

1e30

50

1e30

7

Figure 3.35

Route

number

0

1

2

3

4

5

6

7

8

9

10

11

12

13

Cost

35

45

40

55

65

80

70

30

60

70

40

60

40

50

From

0

0

0

0

0

0

0

1

2

3

4

4

5

6

To

1

2

3

4

5

6

7

2

7

4

5

6

6

7

Figure 3.36

Informationcputnam/Comp516/Overheads/PDFChap03.pdf · 2009-03-19 · 1111 10110 11101 100100 10...

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