Representations

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Representations

• Example Numbers
• 145
• CVL
• 10010001
• 91
• 
  

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Meaning of Number Representation

• Examples
• 145 1102 4101 5100 Decimal
• CVL 100 5 50 Roman
• 10010001127 124 120 Binary
• 91 9161 1160
  Hexadecimal
• 100 101010101111
  1
• 
  Egypt
  
• 260 10105
  Babylon
• 720 5
  Maya

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Meaning of Numbers Convention/Agreement

• Any number consists of symbols
• The value of a number is defined by a set of
  rules of how to interpret these symbols
• Most systems have a base number
• 10 Decimal
• 2 Binary
• 8 Octal
• 16 Hexadecimal

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What makes a good representations?

• Meet certain constraints on the symbols
• Intuitive interpretation
• Can express everything you need
• !! Support for frequent operations
• Efficiency
• Space

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What do we want to represent Data Types

• Set of objects of the same kind
• Defined by
• a way of representing each object
• a group of operations to perform on such objects
• Basic data types of computer
• integers (unsigned and signed)
• plain text characters
• bit vectors
• (floating point numbers)

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Computer Representation

• Computer representation
• Symbols 0,1
• Words sequence of k symbols (bits)
• 8 bit 1 byte
• notation for an unknown k-bit word
• ak-1a k-2 a1a0
• ak-1is called the most significant bit
• a0 is called the least significant bit
• k is always a power of 2 16 or 32

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Unsigned Integer Representation

• 1451102 4101 5100 Decimal
• 10010001ui Binary
• 127 026025 124 023 022 021 120
• 112806403211608040210
• 145
• Multiply and Add Algorithm

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How good is unsigned integer?

• Positive
• Uses only 0 and 1
• Easy addition and conversion to decimal
• Negative
• Limited size (2k) for k-bit word
• No negative
• Limited subtraction

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Signed Integers

• need to represent both non-negative and negative
  integers
• need to be able to perform the following
  operations
• addition (using the same rules as before)
• negation
• subtraction (trivial)
• three different representations will be
  considered
• in all three representations words whose most
  significant bit
• is 0 represent the same non-negative integer

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Signed Magnitude

• Most significant bit determines whether the
  number is positive (ak-10, as before ) or
  negative (ak-11)
• 1 1 1 0 0 k4
• 1 1 1 0 (-6)
• 0 1 1 1 (7)
• 0 1 0 1 (5)
• We now have negative numbers
• Easy negation, only change first bit
• -
• Addition does not work anymore
• Does not work!

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Ones Compliment

• Positive number as before
• Negation is performed by inverting all bits
• Example -6 Inverse (6)
• Inverse (0110) 1001
• test addition by adding 1001 with 0111
• 1 1 1 1 0
• 1 0 0 1 (-6)
• 0 1 1 1 (7)
• 0 0 0 0 (0)
• Does not work!

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Twos Compliment

• Positive as before
• Negation is performed by inverting all of the
  bits, and then adding 1 (binary)
• -6 Inverse(6)0001 Inverse(0110)0001
• 011000010111
• test addition by adding 1010 with 0111
• 1 1 1 0 0
• 1 0 1 0 (-6)
• 0 1 1 1 (7)
• 0 0 0 1 (1)
• Twos compliment is useful for representing
  signed integers

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Things you should be able to do

• Convert decimal number to binary and vice versa
  to all 4 forms of binary representation
• Addition in unsigned integer
• Addition and subtraction for twos complement
• Negation in twos complement
• Recognize the different subscripts ui, sm, 1c,2c
• Understand why 2c is better than sm
• Know which representation (of the 4) is used for
  integer