digital system and binary numbers, Study notes of Computer science

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I N T R O D U C T I O N T O L O G I C D E S I G N

Chapter 1

D i g i t a l S y s t e m s a n d B i n a r y N u m b e r s

g ü r t a ç y e m i ş ç i o ğ l u

O U T L I N E O F C H A P T E R 1

6 O c t o b e r , 2 0 1 6 I N T R O D U C T I O N T O L O G I C D E S I G N

Digital Systems

Binary Numbers

Number-base Conversions

Octal & Hexadecimal Numbers

Complements Signed Binary Numbers

Binary Codes

Binary Storage & Registers

Binary Logic

Binary Arithmetic

D I G I TA L S Y S T E M S

• Digital age and information age
• Digital computers
  • General purposes
  • Many scientific, industrial and commercial applications
• Digital systems
  • Telephone switching exchanges
  • Digital camera
  • Electronic calculators, PDA's
  • Digital TV
• Discrete information-processing systems
  • Manipulate discrete elements of information
  • For example, {1, 2, 3, …} and {A, B, C, …}… 6 O c t o b e r , 2 0 1 6 I N T R O D U C T I O N T O L O G I C D E S I G N

D I G I TA L S I G N A L

• The physical quantities or signals

can assume only discrete values.

• Greater accuracy

A N A L O G S I G N A L

• The physical quantities or signals may vary continuously over a specified range.

6 O c t o b e r , 2 0 1 6 D I G I T A L I N T E G R A T E D C I R C U I T D E S I G N

D I G I TA L S Y S T E M S

t

X(t)

t

X(t)

D I G I TA L S Y S T E M S

• Binary values are represented by values or ranges of values of

physical quantities.

6 O c t o b e r , 2 0 1 6 I N T R O D U C T I O N T O L O G I C D E S I G N

t

V(t)

Logic 1

Logic 0

undefine

1.2 BINARY NUMBERS

Octal Number System

• Base = 8
  • 8 digits
  • { 0, 1, 2, 3, 4, 5, 6, 7}
• Weights
  • Weight = ( Base) Position
• Magnitude
  • Sum of “ Digit x Weight ”
• Formal Notation

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B I N A R Y N U M B E R S

82 81 80 8 -1^8 -

5x64 1x8 2x1 7x1/8 4x1/

o 2 x B^2 + o 1 x B^1 + o 0 x B^0 + o-1 x B-1+o-2 x B-

Hexadecimal Number System

• Base = 16
  • 16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
• Weights
  • Weight = ( Base) Position
• Magnitude
  • Sum of “ bit x Weight ”
• Formal Notation

6 O c t o b e r , 2 0 1 6 I N T R O D U C T I O N T O L O G I C D E S I G N

B I N A R Y N U M B E R S

1 E 5 7 A

162 161 160 16 -1^16 -

1x256 14x16 5x1 7x1/16 10x1/

H 2 x B^2 + H 1 x B^1 + H 0 x B^0 +H-1 x B-1+H-2x B- (458.4765625) 10 (1E5.7A) 16

n 2 n 0 20 = 1 21 = 2 22 = 3 23 = 4 24 = 5 25 = 6 26 = 7 27 =

n 2 n 8 28 = 9 29 = 10 210 = 11 211 = 12 212 = 20 220 =1M 30 230 =1G 40 240 =1T

6 O c t o b e r , 2 0 1 6 D I G I T A L I N T E G R A T E D C I R C U I T D E S I G N

B I N A R Y N U M B E R S

The power of 2

1.3 BINARY

ARITHMETIC

B I N A R Y A R I T H M E T I C

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ADDITION

Binary Addition - Column Addition

B I N A R Y A R I T H M E T I C

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SUBTRACTION

Binary Subtraction - Borrow a “Base” when needed

1.4 NUMBER-BASE

CONVERSION

N U M B E R BA S E CO N V E R S I O N

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Decimal (Base 10)

Octal (Base 8)

Binary (Base 2)

Hexadecimal (Base 16)

Evaluate Magnitude