Module 2:

1

• Module 2
• Arithmetic of Computers Arithmetic Operations
• (Book Computer Organization and Design,3ed,
  David L. Patterson and John L. Hannessy, Morgan
  Kaufmann Publishers)

2
Introduction

• Numbers are represented by binary bits
• How are negative numbers represented?
• What is the largest number that can be
  represented in a computer world?
• What happens if an operation creates a number
  bigger than can be represented?
• What about fractions and real numbers?
• A mystery How does hardware really multiply or
  divide numbers?

3
Binary Numbers
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Twos Complement (32-bit)

• 0111 ... 1111 1111 1111 1111two
  2,147,483,647ten
• 0111 ... 1111 1111 1111 1110two
  2,147,483,646ten
• 0111 ... 1111 1111 1111 1101two
  2,147,483,645ten
• . . .
• 0000 ... 0000 0000 0000 0010two 2ten
• 0000 ... 0000 0000 0000 0001two 1ten
• 0000 ... 0000 0000 0000 0000two 0ten
• 1111 ... 1111 1111 1111 1111two -1ten
• 1111 ... 1111 1111 1111 1110two -2ten
• 1111 ... 1111 1111 1111 1101two -3ten
• . . .
• 1000 ... 0000 0000 0000 0001two
  -2,147,483,647ten
• 1000 ... 0000 0000 0000 0000two
  -2,147,483,648ten

• To represent positive and negative numbers
• MSB - 0 means positive
• MSB - 1 means negative

Indicates sign of the integer
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Twos Complement Shortcut NegationConvert
Decimal to Twos ComplementConvert Twos
Complement to Decimal
Therefore -2 since the MSB is 1
6
Binary Addition and Subtraction
6
7
Detecting Overflow in Two Complement Numbers

• Overflow occurs when adding two positive numbers
  and the sum is negative, or vice versa
• A carry out occurred into the sign bit

Overflow conditions for addition and subtraction
8
Binary Mutiplication
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1st Version of Multiplication Hardware
Flows of 1st Version Multiplication

32-bit multiplicand starts at right half of
multiplicand register
Product register is initialized at 0
Multiplicand register, product register, ALU
are 64-bit wide multiplier register is 32-bit
wide
Algorithm
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Example of Multiplication 4 bits
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Signed Multiplication

• So far we have dealt with positive numbers. But
  how to deal with signed numbers?
• To first convert the multiplier and multiplicand
  to positive numbers and then remember the
  original signs
• Leave the sign out of the calculation
• To negate the product only if the original signs
  disagree

12
Binary Division
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1st Version of Division Hardware
Flows of 1st Version Division

32-bit divisor starts at left half of divisor
register
Quotient register is initialized to be 0
Remainder register is initialized with the
dividend at right
Divisor register, remainder register, ALU
are 64-bit wide quotient register is 32-bit wide
Algorithm
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Example of Division
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Signed Division

• Signed divide
• - make both divisor and dividend positive and
  perform division
• - negate the quotient if divisor and dividend
  were of opposite signs
• - make the sign of the remainder match that of
  the dividend
• - this ensures always
• dividend (quotient divisor) remainder
• quotient (x/y) quotient (x/y) (e.g. 7 32
  1 7 32 1)