Booth's algorithm allows for the binary multiplication of a positive multiplicand and negative multiplier. It involves 3 steps: 1) Representing the multiplicand and multiplier in binary, with the negative multiplier represented in 2's complement form. 2) Recoding the bits of the multiplier according to Booth's recoding table to translate it into a signed-digit form. 3) Multiplying the multiplicand by the recoded multiplier through successive additions or subtractions of the multiplicand according to the recoded bits. The result is the binary product of the positive and negative values.

1. BOOTH’S ALGORITHM
E.g.: Binary Multiplication of Positive Multiplicand & Negative Multiplier (+13 X -
7)

2. EXPLANATION
Binary Multiplication of (+13 X -7)
STEP 1: Number Representation
Multiplicand +13
Multiplier -7
1101
1110
0
1
Binary Representation 2’s Compliment Representation
01101
11001

3. BINARY MULTIPLICATION OF (+13 X -7)
STEP 2: Recoding of the Multiplier
Multiplier -7 11001
1 1 0 0 1
Recoded Multiplier
Multiplier
Multiplicand selected
Bit i Bit i-1
0 0 0 X Multiplicand
0 1 +1 X Multiplicand
1 0 -1 X Multiplicand
1 1 0 X Multiplicand
Booth’s Recoding Table
i
0
i-1
-1+1
i i-1
0
i i-1
-1
i i-1
0
i i-1

4. BINARY MULTIPLICATION OF (+13 X -7)
STEP 3: Multiplication
Multiplicand 01101
Recoded Multiplier 0-10+1-1
Note:
1. Multiplication with 0 – 0 (00000)
2. Multiplication with +1 – Multiplicand (01101)
3. Multiplication with -1 – 2’s compliment of Multiplicand (10011)
0 1 1 0 1
0 -1 0 +1 -1
1 0 0 1 11 1 1 1 1
0 1 1 0 10 0 0 0
0 0 0 0 00 0 0
1 0 0 1 11 1
0 0 0 0 00
101001
1
0
1
1
1
1
1
11
110

5. BINARY MULTIPLICATION OF (+13 X -7)
1110100101
+13 -7
01101 0-10+1-1
+13 x -7
1 0 1 1 0 1 1
64 32 16 8 4 2 1
64+16+8+2+1 = 91
-91
STEP 4: Verification
1110100101
0001011011
2’s Compliment