Computer Architecture: Numeric Systems

Computer Architecture - Numeric Systems

• This is chapter 2 of the computer architecture series
• The information covers numerical systems

Agenda

• The agenda includes positional notation systems
• Methods for converting integers and fractional numbers
• Fundamental operations in binary, octal, and hexadecimal systems
• The concept of a number's complement is also covered

Positional Notation Systems

• In positional notation, each digit or symbol has a distinct weight based on its position
• N = dn * B^n + dn-1 * B^(n-1) + ... + d1 * B^1 + d0 * B^0, shows how to calculate a number's value

Decimal System

• The decimal system has ten symbols, making its base equal to 10
• Numbers can be converted by summing symbols multiplied by their weight, according to a formula

Binary System

• Composed of two symbols; its base is 2
• Traditionally referred to as bits
• Binary symbols can associate with "states" like OFF/ON, False/True, or No/Yes

Octal and Hexadecimal Systems

• The octal system uses eight symbols and has a base of 8
• The hexadecimal system uses sixteen symbols and has a base of 16

Equivalencies Between Number Systems

• A table shows the equivalents between decimal, binary, octal, and hexadecimal
• Decimal 0-15, Binary 0-1111, Octal 0-17, Hexadecimal 0-F

Max Representable Quantity in Decimal

• If n symbols are available, decimals between 0 and B^n - 1 can be represented
  • Binary: 2^n
  • Octal: 8^n
  • Hexadecimal: 16^n

Conversion of other Bases to Decimal

• To convert integers, the formula mentioned previously is use: N = dn * B^n + dn-1 * B^(n-1) + ... + d1 * B^1 + d0 * B^0

Conversion of Fractions

• Each symbol is multiplied by the base raised to the negative power of its position after the decimal point
• The previous formula can be expanded and generalized

Conversion of Decimal to other Bases

• Conversion from decimal to other bases can be done using successive divisions or an algorithm of successive subtractions

Conversion of Decimal Fractions to other Bases

• Multiply the fraction by the base
• Separate the result into integer and fractional parts
• Use the resulting fraction in a new multiplication
• The successive multiplications end when the fractional part equals zero or a periodicity is detected

Direct Conversion Between Bases 2 & 8 - 2 & 16

• Octal and hexadecimal systems emerged as shorthand for the binary system
• 3 bits (binary symbols) correspond to one octal symbol
• 4 bits correspond to one hexadecimal symbol

Binary/Octal/Hexadecimal Chart

• Binary 000-111, Octal 0 - 7, Binary 0000 - 1111, Hexadecimal 0 - F

Fundamental Operations in Binary

• For addition:
  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 0 = 1
  • 1 + 1 = 0 (with a carry of 1)
• For subtraction:
  • 0 - 0 = 0
  • 1 - 0 = 1
  • 0 - 1 = 1 (borrows from the next position)
  • 1 - 1 = 0

Fundamental Operations in Octal and Hexadecimal

• Octal Addition Examples:
  • 1 + 5 = 6
  • 2 + 3 = 5
  • 3 + 5 = ???
• Hexadecimal Addition Examples:
  • 1 + 5 = 6
  • 2 + 3 = 5
  • 3 + 5 = 8
  • 9 + 2 = ????
• A method exists for adding large numbers in these systems

Complement of a Number

• Digital computers base their arithmetic operations mainly on addition
• The complement of a number must satisfy the equation: X + X' = 10^n, where n is the number of digits being complemented
• Two methods exist for obtaining the complement of a number

Number Complements Method

• Complement to the base, or the radix complement: X + X' = B^n => X' = B^n - X
• Base-1 complement, or diminished radix complement: X + X' = B^n - 1 => X' = (B^n - 1) - X
• Complements facilitate subtraction through addition and representing signed numbers