Computer Organization

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Computer Organization

• Integers and Arithmetic Operations
• Terrence Mak

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Counting to 9, oh no, 1 please

• Binary numbers (0, 1) are commonly used in
  computers as they are easily represented as
  on/off electrical signals
• Other related radix systems are Octal
  Hexadecimal
• Different number (representation) systems are
  used in computers
• Let's assume n bits (binary digits) are used in
  the following discussions

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Only 0 and 1 in a Sequence

• Numbers are represented as binary vectors
• B bn-1 bn-2 b1 b0
• MSB Most Significant Bit bn-1 i.e. leftmost
  digit in a binary vector
• LSB Least Significant Bit b0
• i.e. rightmost digit in a binary vector

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Representing Non-negative Numbers

• Unsigned (non-negative) numbers are in range 0 to
  2n 1
• Represented by Value(B) bn-1?2n-1 b1?21
  b0?20
• if B is a binary vector representing an unsigned
  integer

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Representing Signed Numbers

• In written decimal system, a signed number is
  "usually" represented by a "dash" or "plus" sign
  and followed by the magnitude
• e.g. 73, 215, 349
• In binary system, we have several choices
• Sign-and-magnitude
• 1s complement
• 2s complement

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Representing Signed Numbers

• Sign-and-magnitude
• MSB determines sign, remaining unsigned bits
  represent magnitude
• MSB 0 means "", 1 means ""
• 1s complement
• MSB determines sign
• To change sign from unsigned to negative, invert
  all the bits
• 2s complement (commonly used)
• MSB determines sign
• To change sign from unsigned to negative, invert
  all the bits and then add 1
• This is equivalent to subtracting the positive
  number from 2n

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Signed Number Systems
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2s Complement

• 2's complement numbers actually make sense since
  they follow normal modulo arithmetic except when
  they overflow
• Range is 2(n1) to 2(n1) 1 e.g. 8 to 7
  for n 4

Overflow cut-off
for n 4, N 2n 24 16
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1-bit Addition
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n-bit Addition/ Subtraction

• X Y
• Use 1-bit addition propagating carry to the most
  significant bit
• X Y
• ? X ( Y)
• Add X to the 2s complement of Y

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4-bit Addition/ Subtractionof numbers
represented in 2s Complement
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4-bit Addition/ Subtractionof numbers
represented in 2s Complement
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Sign Extension

• Given a 4-bit 2s complement number
• Make it into an 8-bit 2's complement number
• Positive number
• add 0s to LHS
• e.g. 0111 ? 00000111
• Negative number
• add 1s to LHS
• e.g. 1010 ? 11111010
• c.f. circle representation, i.e. extend the MSB

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Overflow

• In 2s complement arithmetic
• Addition of opposite sign numbers never overflow
• If the numbers are the same sign and the result
  is the opposite sign, overflow has occurred
• e.g. 0111 0100 1011 5!
• positive positive negative?!
• In unsigned arithmetic
• Carry out signals an overflow

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Exercise

• Using 4-bit 2s complement number system,
• What is the binary for -2?
• Calculate 23
• Calculate -2-3
• Calculate 55
• Using 5-bit number system
• What is the largest 2s complement number?
• What is the smallest 2s complement number?
• What is the largest unsigned number?
• Convert 56 to unsigned binary
• What is the decimal value of 10110101 in 2s
  complement?
• What is the decimal value of unsigned number
  10110101?

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Characters - 8-bit ASCII representation