The Multiplication Operation Inner View

1
The Multiplication Operation Inner View

• Two types of Multiplication
• Unsigned ( Conventional way) Treats sign
  magnitude separately.
• Signed ( Machine Implementation) Uses rs
  complement scheme ( the base of the number system
  being r).

2
The Conventional Way Unsigned Multiplication

• Task at hand
• Given Operand A m digit magnitude
• ( The Multiplicand ( MPD ) .
• Given Operand B m digit magnitude
• ( The Multiplier ( MPR ) .
• Produce the 2m digit Result C ? A X B.

3
Unsigned Multiplication Examples - 1

• Size of MPD (A) MPR (B) 3 digits.
• Hence size of Result (C ) 6 digits.
• Bases used
• 1. Decimal
• 2. Binary

4
Unsigned Multiplication in Decimal

• Multiplicand (MPD) A 125
• Multiplier (MPR) B 232
• Multiplication Process
• 
  1 2 5 1 2 5
• 
  X 2 3 9 X 2 3 9
• 
  --------------- ---------------
• 
  1 1 2 5 0 1 1 2 5
• 
  3 7 5 X 0 3 7 5 0
• 
  2 5 0 X 2 5 0 0 0
• 
  --------------- -----------------
• 6 Digit Result C 0 2 9 8 7 5
  0 2 9 8 7 5

5
Unsigned Multiplication in Binary

• Multiplicand (MPD) A 110
• Multiplier (MPR) B 101
• Multiplication Process
• 
  1 1 0 1 1 0
• 
  X 1 0 1 X 1 0 1
• 
  --------------- ---------------
• 
  1 1 0 0 0 0 1 1 0
• 
  0 0 0 X 0 0 0 0 0 0
• 
  1 1 0 X 0 1 1 0 0 0
• 
  --------------- -----------------
• 6 Digit Result C 0 1 1 1 1 0
  0 1 1 1 1 0

6
Unsigned MultiplicationCommon Observations

• Essentially a process of shifted addition.
• At each step, one Multiplier Digit is considered
  , starting from Least Significant Digit , moving
  towards left.
• Multiplication Table is needed.
• Requires 2m digit addition involving more than 2
  operands. Cannot be performed by machine .

7
Unsigned Multiplicationtowards Machine
Implementation

• Use m digit adder only.
• Add only 2 Operands at a time.
• Eliminate need for multiplication table by
  repeated addition.
• Ease out the task of considering each multiplier
  digit starting from Least Significant Digit.

8
Unsigned Binary Multiplication Example - 1

• Consider the Multiplicand (MPD) A 110
• The Multiplier (MPR) B 101 Lower Half of
  Result
• We are to Compute C A X B
• Step 1 Set up Upper Half of the Result C 0 0 0
  3 Digits
• Set Operation / Shift Counter ? 3
• Set Cy ? 0 . Copy/Preserve MPR B
  in E .
• Step 2 The Current Rightmost Multiplier Digit
  MPRd 1.
• Step 3 Addition Counter Add_Count 1 . One
  ADDITION
• Step 4 C ? C PLUS MPD A ?
• C ? 000 PLUS 110 110 , Carry 0

9
Unsigned Binary Multiplication Example - 2

• Cy Result C MPR B Add_Count
  MPD A
• 0 1 1 0 1 0 1
  1 1 1 0
• Step 5 Decrement Add_Counter 1 / 0.
• Step 6 Add_Count is ZERO hence Right Shift
  Logically Cy, C B.
• Cy Result C MPR B Add_Count
  MPD A
• 0 0 1 1 0 1 0
  0 1 1 0
• Step 7 Decrement Operation/ Shift Counter 3 /
  (2) NOT 0 hence Continue.

10
Unsigned Binary Multiplication Example - 3

• Step 8 The Current Rightmost Multiplier Digit
  MPRd 0.
• Step 9 Addition Counter Add_Count 0 . NO
  ADDITION
• Step 10 Right Shift Logically Cy, C B.
• Cy Result C MPR B Add_Count
  MPD A
• 0 0 0 1 1 0 1
  0 1 1 0
• Step 11 Decrement Operation / Shift Counter
  2 /( 1)
• NOT 0 Hence Continue.

11
Unsigned Binary Multiplication Example - 4

• Step 11 The Current Rightmost Multiplier Digit
  MPRd 1.
• Step 12 Addition Counter Add_Count 1 .
• C ? C PLUS MPD A ?
• C ? 001 PLUS 110 111 , Carry 0.
• Add _Count 0
• Step 13 Right Shift Logically Cy, C B.
• Cy Result C MPR B Add_Count
  MPD A
• 0 0 1 1 1 1 0
  0 1 1 0
• Step 14 Decrement Operation / Shift Counter
  1 /( 0)
• Equals 0 Hence STOP.
• Hence final Result 0 1 1 1 1 0

12
Unsigned Binary Multiplication Compact Process
Example

• Multiplicand A 0101 5 Decimal
• Multiplier B 0100 4 Decimal
• Result C 0 0 0 1 0 1 0 0 20 Decimal
• Step 1
• Partial Result 0 0 0 0 0 1 0 0
• 1st two Multiplier bits 0 , Hence after
  two Logical Right Shifts ( Step 1 Step 2) the
  Partial Result 0 0 0 0 0 0 0 1
• Next Step ( Step 3 ) , Current Multiplier Bit
  1
• Add Multiplicand to the MS Nibble of the
  Partial Result
• 0 0 0 0 0 0 0 1
• 0 1 0 1
• ----------
• 0 1 0 1 0 0 0 1 gtgt 0 0 1 0 1 0 0 0.
  Now last Multiplier bit 0
• After one more Right Logical Shift Result 0
  0 0 1 0 1 0 0

13
Unsigned Decimal Multiplication The Modified
Process - 1

• MPD A m Digits , MPR B m Digits . Hence
  Result will be 2m Digits long.
• Step 1 Initialize Upper Half of the Result C
  0 0 0
• 
  m Digits
• Copy MPR B into E. MPR B will hold Lower Half
  of Result
• Clear Carry Cy 0 . Preserve it.
• Set up Operation / Shift Counter m
• Step 2 Consider the current rightmost
  multiplier digit MPRd.
• Step 3 Addition Counter (ADD_Count) ? MPRd k
  .
• Step 4 C ? ( C ) PLUS MPD A .
• Step 5 For any Carry produced , Increment Cy .
• Step 6 Decrement ADD_Count.
• Step 7 IF (ADD_Count is NON ZERO) GO TO Step 4.

14
Unsigned Decimal Multiplication The Modified
Process - 2

• Step 8 ADD_Count is Zero. One Multiplier
  Digit Considered.
• a. Right Shift Logically the Cy, Upper Half of
  Result C MPR (B) .
• b. Cy Comes in at Left Most position of the
  Upper Half of Result C .
• c. Right most Digit of C comes to the left
  most place of MPR (B).
• d. Right most Digit of the MPR B goes out.
• e. 0 comes into Cy.
• Step 9 Decrement Operation Counter.
• Step 10 If Operation Counter is NON Zero GOTO
  Step 2 ELSE STOP.

15
Unsigned Decimal Multiplication Example - 1

• Consider the Multiplicand (MPD) A 125
• The Multiplier (MPR) B 239
• We are to Compute C A X B
• Step 1 Set up 3 digit Upper Half of Result C
  0 0 0
• Here Operation Counter 3. Copy
  B(239) to D
• Set Cy ? 0 . Preserve it
• Step 2 The Current Rightmost Multiplier Digit
  MPRd 9.
• Step 3 Addition Counter Add_Count 9
• Step 4 C ? C PLUS MPD A ?
• C ? 000 PLUS 125 125 , No Carry.

16
Unsigned Decimal Multiplication Example - 2

• Step 5 Cy Result C MPR B
  Add_Count MPD A
• 0 1 2 5 2 3
  9 9 1 2 5
• Step 6 Decrement Addition Counter Add_Count
  9 / 8.
• Step 7 Add_Count is NON ZERO hence GO TO STEP
  4.
• After repeating this 6 more i.e. Adding MPD A (
  125 ) to the Upper Half of thel Result C , a
  total of 7 times the status becomes
• Cy Result C MPR B
  Add_Count MPD A
• 0 8 7 5 2 3 9
  2 1 2 5
• Step 4 C ? C PLUS MPD A ?
• C ? 875 PLUS 125 000 , Carry 1

17
Unsigned Decimal Multiplication Example - 3

• Step 5 Cy Result C MPR B
  Add_Count MPD A
• 1 0 0 0 2 3 9
  2 1 2 5
• Step 6 Decrement Addition Counter Add_Count
  2 / 1.
• Step 7 Add_Count is NON ZERO hence GO TO STEP
  4.
• After Looping one more time i.e.a total of 9
  times the status becomes
• Cy Result C MPR B
  Add_Count MPD A
• 1 1 2 5 2 3
  9 0 1 2 5
• Step 8 Right Shift Logically Cy, C B.

18
Unsigned Decimal Multiplication Example - 4

• Step 8 Cy Result C MPR B
  Add_Count MPD A
• 0 1 1 2 5 2
  3 3 1 2 5
• Add_Count 3. Operation Counter 2 .
  
• Add_Count is NON ZERO hence GO TO STEP 4.
• Step 4 C ? C PLUS A ? C ? 1 12 PLUS 125 2
  3 7 , Cy 0
• Decrement Add_Count.
• Cy Result C MPR B
  Add_Count MPD A
• 0 2 3 7 5 2
  3 2 1 2 5
• Step 5 C ? C PLUS A ? C ? 237 PLUS 125 3 6
  2 , Cy 0
• Decrement Add_Count.
• Cy Result C MPR B
  Add_Count MPD A
• 0 3 6 2 5 2
  3 1 1 2 5

19
Unsigned Decimal Multiplication Example - 5

• Step 5 Cy Result C MPR B
  Add_Count MPD A
• 0 3 6 2 5 2
  3 1 1 2 5
• Step 6 C ? C PLUS A ? C ? 3 6 2 PLUS 125 4
  8 7 , Cy 0
• Decrement Add_Count.
• Cy Result C MPR B
  Add_Count MPD A
• 0 4 8 7 5 2
  3 0 1 2 5
• Step 7 RSHIFT Logical Cy C B Decrement
  Operation Count.
• Cy Result C MPR B
  Add_Count MPD A
• 0 0 4 8 7 5
  2 2 1 2 5

20
Unsigned Decimal Multiplication Example - 6

• Step 7 Cy Result C MPR B
  Add_Count MPD A
• 0 0 4 8 7
  5 2 2 1 2 5
• Step 8 C ? C PLUS A ? C ? 0 4 8 PLUS 125 1
  7 3 , Cy 0
• Decrement Add_Count.
• Cy Result C MPR B
  Add_Count MPD A
• 0 1 7 3 7 5
  2 1 1 2 5
• Step 9 C ? C PLUS A ? C ? 1 7 3 PLUS 125 2
  9 8 , Cy 0
• Decrement Add_Count.
• Cy Result C MPR B
  Add_Count MPD A
• 0 2 9 8 7 5
  2 0 1 2 5

21
Unsigned Decimal Multiplication Example - 7

• Step 10 Cy Result C MPR B
  Add_Count MPD A
• 0 2 9 8 7
  5 2 0 1 2 5
• RSHIFT LOGICAL Cy C B .
• Decrement Operation Counter 1 / 0. Multiplication
  Over
• Cy Result C MPR B
  Add_Count MPD A
• 0 0 2 9 8 7 5
  N.A. 1 2 5
• Final Result 125 X 239 0 2 9 8 7 5
• Unsigned HEX Unsigned Octal Multiplication Left
  as an Exercise.