The document describes Booth's algorithm for multiplying two binary numbers in two's complement notation. It was invented by Andrew Booth in 1950 to increase the speed of calculations on desk calculators that were faster at shifting than adding. The algorithm handles both positive and negative multipliers uniformly. It then provides an example of multiplying two positive binary numbers (+13 x +7) using Booth's algorithm in four steps: 1) representing the numbers in binary, 2) recoding the multiplier, 3) performing the multiplication, and 4) verifying the result.

1. BOOTH’S ALGORITHM
E.g.: Binary Multiplication of two Positive Numbers (+13 X +7)

2. BOOTH’S ALGORITHM
Booth's multiplication algorithm is used to multiply two signed binary numbers
in two's complement notation.
The algorithm was invented by Andrew Donald Booth in 1950.
Booth used desk calculators that were faster at shifting than adding and created
the algorithm to increase their speed.
It handles both positive and negative multiplier uniformly.
Basic Understandings required to learn the topic are:
1. Binary Number Representation
2. Binary Multiplication

3. EXPLANATION WITH AN EXAMPLE
Binary Multiplication of two Positive Numbers (+13 X +7)
STEP 1: Number Representation
Multiplicand +13
Multiplier +7
1101
1110
0
0
Binary Representation 2’s Compliment Representation
01101
00111

4. BINARY MULTIPLICATION OF TWO POSITIVE NUMBERS (+13 X +7)
STEP 2: Recoding of the Multiplier
Multiplier +7 00111
0 0 1 1 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
-10
i i-1
0
i i-1
+1
i i-1
0
i i-1

5. BINARY MULTIPLICATION OF TWO POSITIVE NUMBERS (+13 X +7)
STEP 3: Multiplication
Multiplicand 01101
Recoded Multiplier 0+100-1
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 0 -1
1 0 0 1 11 1 1 1 1
0 0 0 0 00 0 0 0
0 0 0 0 00 0 0
0 1 1 0 10 0
0 0 0 0 00
110110
1
1
1
0
1
0
1
01

6. BINARY MULTIPLICATION OF TWO POSITIVE NUMBERS (+13 X +7)
0001011011
+13 +7
01101 0+100-1
+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