Binary, Octal, and Decimal Conversion in Digital Logic Design

Binary Values

• Binary values are represented abstractly by digits 0 and 1, words (symbols) False (F) and True (T), words (symbols) Low (L) and High (H), and words On and Off
• Binary values are represented by values or ranges of values of physical quantities, such as V(t) Logic 1 undefine and Logic 0 t

Number Systems

• Decimal Number System:
  • Base (radix) = 10
  • 10 digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
  • Digit Position: Integer & fraction
  • Digit Weight: Weight = (Base) Position Magnitude
  • Sum of “Digit x Weight” Formal Notation: 500 d2B2+d1B1+d0B0+d-1B-1+d-2*B-2
• 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: 8 1 1/8 1/64
• Binary Number System:
  • Base = 2
  • 2 digits: {0, 1}, called binary digits or “bits”
  • Weights: Weight = (Base) Position Magnitude
  • Sum of “Bit x Weight” Formal Notation: 2 1 1/2 1/4
• 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 “Digit x Weight” Formal Notation: 256 16 1 1/16 1/256

The Power of 2

• 2n n 2n
• Examples:
  • 2^0 = 1
  • 2^1 = 2
  • 2^2 = 4
  • ...
  • 2^20 = 1 Mega (M)
  • 2^30 = 1 Giga (G)
  • 2^40 = 1 Tera (T)

Addition and Subtraction

• Decimal Addition:
  • 1 + 1 = 1 Carry
  • Example: 5 + 5 = 10
• Binary Addition:
  • Column Addition
  • Example: 1 + 1 = 10
• Binary Subtraction:
  • Borrow a “Base” when needed
  • Example: 0 - 1 = 1 (borrowing 1)

Binary Multiplication

• Bit by bit
• Example: 1 x 1 = 1

Number Base Conversions

• Evaluate Magnitude:
  • Octal (Base 8)
  • Decimal (Base 10)
  • Binary (Base 2)
  • Hexadecimal (Base 16)
• Conversion Examples:
  • Decimal (Integer) to Binary Conversion
  • Decimal (Fraction) to Binary Conversion
  • Binary to Octal Conversion
  • Binary to Hexadecimal Conversion
  • Octal to Hexadecimal Conversion

ASCII Codes

• American Standard Code for Information Interchange (ASCII)
• Character Code
• 7-bit code to represent:
  • 94 Graphic printing characters
  • 34 Non-printing characters
• ASCII Properties:
  • Digits 0 to 9 span Hexadecimal values 30 to 39
  • Upper case A-Z span 41 to 5A
  • Lower case a-z span 61 to 7A
  • Lower to upper case translation (and vice versa) occurs by flipping bit 6

Binary Codes

• Error-Detecting Code:
  • Adding an eighth bit to the ASCII character to indicate its parity
  • Parity bit is an extra bit included with a message to make the total number of 1's either even or odd

Complements

• Two types of complements for each base-r system:
  • Radix complement
  • Diminished radix complement