Digital Design: Number Systems

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Digital Design Number Systems
Credits Slides adapted from J.F. Wakerly,
Digital Design, 4/e, Prentice Hall, 2006 C.H.
Roth, Fundamentals of Logic Design, 5/e, Thomson,
2004
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Positional Number Systems

• A number is represented by a string of digits,
  where each digit position has an associated
  weight and it has the following form
• dp-1dp-2 ??? d1d0 . d-1d-2 ??? d-n
• The value of the number is given by

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Binary Numbers

• The general form of a binary number of pn binary
  digits (bits) is
• bp-1bp-2 ??? b1b0 . b-1b-2 ??? b-n
• and its value is

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Octal and Hexadecimal Numbers

• The octal number system uses radix 8, while the
  hexadecimal number system uses radix 16
• The octal and hex number systems are useful for
  representing multibit numbers

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Conversion from Binary to Decimal

• Method summation

Example 101110110012 1 ? 210 0 ? 29 1 ? 28
1 ? 27 1 ? 26 0 ? 25
1 ? 24 1 ? 23 0 ? 22 0 ? 21 1 ? 20
149710
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Conversion from Decimal to Binary

• Method successive divisions
• Example

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EXAMPLE convert 5310 to binary
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EXAMPLE convert .625ten to binary
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EXAMPLE convert 0.710 to binary.
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EXAMPLE convert 231.34 to base 7.
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Addition of Binary Numbers
EXAMPLE Add 1310 and 1110 in binary.
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Subtraction of Binary Numbers
EXAMPLES
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Representation of Negative Numbers

• Signed-Magnitude Representation
• 10ten -10ten
• 001010two
  101010two
• The number zero has two representations (0 and
  -0)
• An n-bit signed-magnitude number lies within the
  range -(2n-1 - 1) through (2n-1 - 1)
• To add signed-magnitude numbers we must examine
  the signs of the addends to determine what to do
  
• Radix Complement Representation
• Diminished Radix Complement Representation

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Representing Numbers ????

• Key observation Numbers are just strings of
  symbols. The meaning (value) we assign to each
  string instance (pattern) is up to us. If the
  string is n symbols (digits) long and each symbol
  can take up to different r instances (radix) then
  we can form rn different patterns.
• Common sense characteristics of a system
  number
• Assign a different value to each different
  pattern
• Split the patterns equally between positive
  numbers and negative numbers
• The mechanic of doing arithmetic operations
  should be as simple as possible

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Complement Number Systems

• While the signed-magnitude system negate a number
  by changing its sign, a complement number system
  negates a number by taking its complement.
• Radix-complement RepresentationThe complement of
  an n-digit number D is obtained by subtracting it
  from rn
• rn D ((rn-1)-D) 1
• Diminished Radix-complement RepresentationIn a
  diminished radix-complement system the complement
  of an n-digit number D is obtained by subtracting
  it from rn-1

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Complement Number Systems
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Complement Number Systems
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Complement Number Systems

• Once we know how to compute the diminished-radix
  complement of a number, computing the
  radix-complement is very simple
• radix complement diminished-radix complement
  1

0
1
9
8
2
3
7
4
6
5
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C2 Number System

• For binary numbers, the radix complement is
  called twos complement (C2).
• The MSB of a number in this system serves as the
  sign bit.
• Negative numbers have MSB equal to 1
• Positive numbers have MSB equal to 0
• The range of representable numbers is (2n-1)
  through (2n-1-1)
• Zero has only one representation

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Twos Complement Number System
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C1 Number System

• For binary numbers, the diminished-radix
  complement is called ones complement (C1).
• The MSB of a number in this system serves as the
  sign bit.
• Negative numbers have MSB equal to 1
• Positive numbers have MSB equal to 0
• The range of representable numbers is (2n-1-1)
  through (2n-1-1)
• Zero has two representations positive zero (00
  ??? 00) and negative zero (11 ??? 11)

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Summary of Signed Number Systems
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C1 Number System

• In the C1 number system to negate an n-bit number
  all we have do is to flip (invert) all the bits

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C2 Number System

• In the C2 number system to negate an n-bit number
  requires two steps
• ? invert all bits of the number (i.e. take
  the C1 of the number) and then
• ? add 1

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Playing with the C2 notation

• The sum of a number and its inverted
  representation must be 111.111two, which in C2
  represent 1

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C2 EXAMPLES
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C2 sign extension

• As far as m gt n, it is possible to convert n-bit
  numbers into m-bit numbers, but some care is
  needed
• copy the most significant bit (the sign bit) into
  the other bits 0010 ? 0000 0010 1010 ? 1111
  1010
• This procedure is referred as "sign extension"

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C2 Addition and Subtraction
1. Addition of 2 positive numbers, sum lt 2n 1.
2. Addition of 2 positive numbers, sum 2n 1
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C2 Addition and Subtraction
3. Addition of positive and negative numbers
(negative number has greater
magnitude).
4. Addition of positive and negative numbers
(positive number has
greater magnitude).
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C2 Addition and Subtraction
5. Addition of two negative numbers, sum 2n
1.
6. Addition of two negative numbers, sum gt 2n
1.
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Detecting overflow

• Overflow occurs when the value affects the sign
  bit
• adding two positives yields a negative
• adding two negatives gives a positive
• subtract a negative from a positive and get a
  negative
• subtract a positive from a negative and get a
  positive
• No overflow when adding a positive and a negative
  number
• No overflow when subtracting two numbers of same
  sign
• Consider the operations A B, and A B
• Can overflow occur if B is 0 ?
• Can overflow occur if A is 0 ?

cannot occur !
can occur ! (for A-B if B-2n-1)
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Binary Codes for Decimal Numbers
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Gray Code
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Character Codes
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N-cubes and Hamming distance
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Traversing a 3-cube in Gray code order