Computer Systems 1 Fundamentals of Computing

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Computer Systems 1Fundamentals of Computing

• Negative Real Number Binary

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Negative Real Binary

• Binary subtraction
• Real numbers in binary
• Ranges
• Fixed point
• Floating point notation

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Binary Subtraction

• Using twos complement we can perform binary
  subtraction
• Essentially addition
• e.g- 20 - 13 20 (-13)
• Take two binary numbers
• The first number is called the minuend
• Second number is called the subtrahend
• Second number negated using twos complement
• Binary addition is then applied
• Drop carry from result if required

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Binary Subtraction

• E.g.
• 20 - 13 20 (-13) 7
• 010100 (20)
• 110011 (t/c -13)(001101 1100101) 1000111
• Drop carried leading 1
• 000111 decimal 7

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Binary Subtraction

• What if the minuend is smaller than the
  subtrahend (negative number result)?
• E.g.
• 20 - 23 -3
• Or
• 20 (-23) (-3)
• 010100 (20)
• 101001 (t/c -23)(010111 1010001)
  111101
• 111101 -3

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Number Range

• Number of bits in a word determine the number
  range
• The sign bit from twos complement and sign
  magnitude representation methods reduce the range
  from that of an unsigned number
• Sign bit is ignored using Sign Magnitude
• RANGE (4 bit) 1111 to 0111 (-7 to 7)
• RANGE (8 bit) 11111111 to 01111111 (-127 to 127)
• Sign bit is used in the calculation of Twos
  Complement numbers but still reduces the range
• Because if sign indicator is used negate value
• RANGE (4 bit) 1000 to 0111 (-8 to 7)
• RANGE (8 bit) 10000000 to 01111111 (-128 to 127)

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Number Range - Overflow

• If the result of a calculation goes beyond the
  fixed bit size of a word then overflow has
  occurred
• Overflow is detected and highlighted by the ALU
• A common problem is that the sign bit is used as
  part of the magnitude of a number
• Especially when potentially working with negative
  numbers
• E.g. for 96 64 160
• 01100000
• 01000000
• 10100000 160 (is -96 in twos complement)
• Overcome by adding more bits to the word size

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Real Numbers

• So far we have looked at integer methods for
  number representation
• Real numbers are integers and fractions of a
  number system
• Essentially all values
• Real numbers provide greater accuracy than
  integers
• To be useful the computer must be able to work
  with real numbers
• Methods of representing real numbers using binary
• Fixed point notation
• Floating point notation

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Fixed Point Notation

• Splits numbers using a point .
• Integer section
• Fractional section
• Fraction is assumed to be to the right of the
  point
• Degree of precision is decided by position of
  point
• Point can be placed anywhere in the binary word
• More digits to the left means greater number of
  integers to be represented
• Essentially a greater numerical range
• More digits to the right means greater accuracy
  of real numbers
• Fractional information can be more precise
• Point position within a word is assumed in
  practice
• The same binary word can have many different
  values depending on the position of the point

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Fixed Point Notation

• E.g-
• Using 8 bit words
• 000010.11 2.75
• 1x2 (1x0.5) (1x0.25)
• 000101.1 5.5
• (1x4) (1x1) 1x0.5
• Fixed point numbers are prone to overflow and
  underflow when using fixed word sizes
• Overflow with larger numbers
• Underflow when fractions are too small

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Floating Point Notation

• The point is capable of moving
• Floating
• Converting fixed point to floating point is
  normalisation
• Representation is achieved by
• A mantissa
• An exponent
• And a radix
• m re
• m mantissa ( or -), r radix, e exponent (
  or -)
• In decimal Radix is 10
• In binary Radix is 2
• E.g.- in decimal 5.2 106 (5,200,000)

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Floating Point Notation

• E.g.- Normal Decimal numbers
• 2.5 103 2500
• 8.9 108 890,000,000
• 4.3 102 430
• E.g.- Decimal Numbers lt 1 and gt 0
• 5.24 10-5 0.0000524
• 2.531 10-7 0.0000002531
• 6.7 10-2 0.067

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Floating Point Notation

• Converting FIXED to FLOAT
• Binary Floating Point
• If the number is greater than one
• Point floats to the left
• Position before the MSB
• If the number is a fraction less than ONE but
  greater than 0, and binary 1 doesnt follow the
  point (0.5)
• Point floats to the right
• Position before first non-zero bit
• Negative exponent (e)

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Floating Point Notation

• E.g.- Normal Binary numbers
• 101.01010 0.10101010 23
• 1010.0110 0.10100110 24
• 111011.01 0.11101101 26
• E.g.- Binary Numbers less than 1 and gt 0
• 0.0011001 0.11001 2-2
• 0.0000101 0.101 2-4
• 0.0000011 0.11 2-5

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Floating Point Notation

• Storing floating point numbers
• Two parts to be stored
• Mantissa
• More bits the greater the magnitude
• Exponent
• More bits the greater the precision
• Usually allocated a third to a half of the bits
• E.g- in a 16-bit word
• Mantissa is given 12 bits
• Exponent gets 4 bits
• Or- in a 16-bit word
• Mantissa 10 bits
• Exponent 6 bits

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Floating Point Notation

• Allocation of bits
• E.g.- 16 bit allocation
• Binary point appears to the right of the sign bit

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Representing floating point numbers

• E.g.- 16-bit system
• 1110.0000011 0.11100000011 24
• Mantissa 0111000000112
• Exponent 01002
• Can be written 0111000000112 01002
• Actually stored as
• 01110000001101002
• because number format for floating point has
  already been decided
• We know where the mantissa ends, etc.

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Negative floating point

• Negative floating point numbers can be stored
  using the Twos complement method
• A normalised positive mantissa must be between
  0.5 and less than 1 (0.100..n 0.111..n)
• A normalised negative mantissa must be between -1
  and less than -0.5 (1.000..n 1.011..n)
• Mantissa
• Sign bit before binary point is used
• 1 negative
• 0 positive
• MSB to right of binary point
• 1 positive
• 0 negative
• If sign bit and MSB are the same then
  normalisation is required

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Negative floating point

• E.g.-
• 0.1xxxxxxxxxx positive mantissa
• 1.0xxxxxxxxxx negative mantissa
• 0.0xxxxxxxxxx invalid positive
• 1.1xxxxxxxxxx invalid negative
• Exponent
• Usually, normal Twos complement rules apply
• Because exponent does not use a binary point
• E.g.- 11002
• (1 -8) (1 4)
• -410

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Floating point accuracy

• We can see that precision can be lost when using
  floating point numbers
• Especially when performing calculations
• Methods of improving precision become useful
• Ways to do this
• Mantissa length
• Rounding
• Double precision numbers arithmetic

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Floating point accuracy

• Mantissa length
• Increase the word size allocated to the mantissa
  storage
• Still never completely solves the problem
• Rounding
• Attempt to reduce the problem cause by losing a
  binary 1 from the number
• Occurs during shifting and normalisation
• In rounding if the last bit to be discarded is a
  1 then a 1 should be added to the LSB position
• E.g.(5-bit num) 1.0101 shifted 0.10101 (last
  1 is lost)
• 0.1010 when rounded with the 1 lost 0.1011
• Can also cause more problems than it solves
• Many different rounding algorithms

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Double precision numbers

• Uses two sequential memory words to store data
  when required
• Most significant half
• Least significant half
• Can accommodate numbers being used or hold the
  result of calculations
• Essentially expands the available size for the
  mantissa

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CS1 - Week 23

• Binary subtraction using twos complement
• Actually addition
• Normal subtraction
• Subtraction with negative result
• Ranges
• Overflow
• Underflow
• Real number representation
• Fixed point notation
• Binary
• Floating point notation
• Decimal
• Binary
• Storage