Computer Arithmetic:

1
Computer Arithmetic

• Unsigned Notation
• By Sherwin Chiu

2
Unsigned Notation

• Introduction
• Definition of Unsigned Notation
• Addition
• Subtraction
• Multiplication
• Division

3
Introduction

• Most frequently preformed operation is copying
  data from one place to another
• From one register to another
• Between a register and a memory location
• Value is not modified as it is copied

4
Definition of Unsigned Notation

• Unsigned Notation The notation does not have a
  separate bit to represent the sign of the number.

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Unsigned Notation

• There are two types of unsigned notation
• Non-negative notation
• 2s complement notation

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Non-negative versus 2s complement
Binary Representation Non-negative 2s Complement
0000 0 0
0001 1 1

0111 7 7 (2 - 1)
1000 8 -8 (2 )
1001 9 -7

1111 15 (2ª - 1) -1
n-1
n-1
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Addition

• Can be easily done when values fit the
  limitations for non-negative or 2s complement
• A little more complicated when it does not.

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Straightforward Addition

• Adding numbers that fit the limitations can
  perform a straightforward addition of binary
  numbers.
• When adding two numbers of different signs (/-),
  a valid result will always occur.

9
Implementation of ADD X?XY
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Overflow

• When two numbers being added exceed the number of
  bits available in the register, an overflow
  occurs.
• In non-negative, an overflow flag is notified by
  the carry out bit.
• In 2s complement, an overflow flag is notified
  by both the carry in and carry out bit.

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Subtraction

• Essentially treated the same as addition, but
  negating one of the values. X (-Y)
• Overflow is still caused by exceeding the number
  of bits available in the register.

12
Implementation ofSUB X? X Y
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Multiplication

• We could multiply by adding n copies of the
  number together, but it would be inefficient.
• It is also not how people would do
  multiplication.

14
People to Shift-Add Multiplication
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Binary Multiplication

• Using binary notation makes multiplication even
  easier by having only two possible values, 0 (X
  0 0) or 1 (1 X 1 1)

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Shift-Add Multiplication in RTL Form

• X,U,V n bit register U
  high order half
• C 1 bit register for carry V
  low order half
• Start initiates
  Finish terminates
• 1 U?0, i ?n
• Vo 2 CU ?U X
• 2 i ?i 1
• 3 shr(CUV)
• Z3 GOTO 2
• Z 3 FINISH?1

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1 U?0, i ?n Vo 2 CU?U X 2 i ?i
13 shr(CUV) Z3 GOTO 2 Z
3 FINISH?1
Conditions Micro-operations i C U V Z FINISH
START x x xxxx 1011 0
1 U?0, i ? 4 4 0000 0
Vo 2,2 CU?UX, i?i1 3 0 1101 0
3, Z3 shr(CUV), GOTO 2 0 0110 1101
Vo 2,2 CU?UX, i?i 1 2 1 0011 0
3, Z3 shr(CUV), GOTO 2 0 1001 1110
2 i?i 1 1 0
3, Z3 shr(CUV), GOTO 2 0 0100 1111
Vo 2,2 CU?UX, i?i 1 0 1 0001 1
3, Z 3 shr(CUV), FINISH?1 0 1000 1111 1
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Implementation of Shift-Add Multiplication
Algorithm
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Division

• Can be done with the same idea of repeated
  additions like multiplication, but instead for
  division, it is repeated subtractions.
• Shifting is also applicable in the same way as
  multiplication, just shifting left and
  subtracting instead.

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Shift-Sub Division in RTL Form

• X,Y,U,V n bit register
  U high order half
• C 1 bit register for carry
  V low order half
• Start initiates
  Finish terminates
• 1 CU?UX1
• 1 U ?UX
• C1 FINISH ?1,OVERFLOW ?1
• 2 Y ?0,OVERLOW ?0,i ?n
• 3 shl(CUV),shl(Y),i ?i-1
• C4 U ?UX1
• C4 CU ?UX1
• C4 Y ?1
• C4 U ?UX
• Z 4 FINISH ?1
• Z4 GOTO 3

1
2
2
1
1
0
2
2
2
2
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X,Y,U,V n bit register
U high order halfC 1 bit register for
carry V low order
halfStart initiates
Finish terminates
Conditions Micro-operations C U V Y Z FINISH
START x x 1001 0011 xxxx 0
1 1 CU?UX1 0 1100
1 2 U ?UX 1001
2 Y ?0,OVERFLOW ?0,i ?4 4 0000 0
3 shl(CUV),shl(Y),i ?i-1 3 1 0010 0110 0000 0
C41 U ?UX1 0101
C42,Z42 Y0 ?1,GOTO 3 0001
3 shl(CUV),shl(Y),i ?i-1 2 0 1010 1100 0010 0
C41 CU ?UX1 0 1101
C42,Z42 U ?UX,GOTO 3 1010
3 shl(CUV),shl(Y),i ?i-1 1 1 0101 1000 0100 0
C41 U ?UX1 1000
C42,Z42 Y0 ?1,GOTO 3 0101
3 shl(CUV),shl(Y),i ?i-1 0 1 0001 0000 1010 1
C41 U ?UX1 0100
C42,Z42 Y0 ?1,FINISH?1 1011 1
i
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Implementation of shift-sub division algorithm
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Conclusion

• Basic arithmetic functions are a lot more
  complicated than what it seems.
• These implementations are only for unsigned
  notation.
• There still exist signed notation and
  binary-coded decimal.
• Any Questions?