Booth's Algorithm Tutorial - CS Course Webpages

Booth’s Algorithm Tutorial (Tim Berger)
Signed multiplication is a careful process. With unsigned multiplication there is no need
to take the sign of the number into consideration. However in signed multiplication the
same process cannot be applied because the signed number is in a 2’s compliment form
which would yield an incorrect result if multiplied in a similar fashion to unsigned
multiplication. That’s where Booth’s algorithm comes in. Booth’s algorithm preserves
the sign of the result.
Methods Used
There are 2 methods that you should know before attempting Booth’s algorithm. Rightshift
circulant and right-shift arithmetic.
Right-shift circulant, or RSC for short, is simply shifting the bit, in a binary string, to
the right 1 bit position and take the last bit in the string and append it to the beginning of
the string.
Example:
10110
after right-shift circulant now equals – 01011
Right-shift arithmetic, or RSA for short, is where you add 2 binary number together
and shift the result to the right 1 bit position
Example:
0100
+0110
result = 1010
Now shift all bits right and put the first bit of the result at the beginning of the new
string:
result 1010
shift 11010
The Process
Step 1: Convert the two operands into binary. Then determine which operand has the
least transitions between bits and set X equal to that operand and Y equal to the other
operand.
Example:
13 = 1101
10 = 1010
1101 has less bit transitions so set X = 1101 and set Y=1010
Note: It’s a good idea to calculate –Y as well because you will most likely be
subtracting Y from the value in U which is the same as add –Y to the value in U
-Y = 0110

Step 2: Set up 4 columns as follows:
1 st column is U. This column holds the results from each step in the algorithm
2 nd column is V. This column hold the overflow from U when right-shifting
3 rd column is X. This holds X operand. This will show each RSC step.
4 th column is X-1. This holds the least significant bit from X before RSC.
Initially set this to 0.
Example Setup:
X=1101
Y=1010
U V X X-1
0000 0000 1101 0
Step 3: Analyze the least significant bit of X and the bit in X-1. From that string take
the following action:
If the string = “01” Add Y to the value in U and right-shift the result
If the string = “10” Subtract Y from the value in U and right-shift the result
If the string = “00” Take no action
If the string = “11”Right-shift the value in U 1 bit position
Example:
String = “10” so subtract Y(or add –Y) from the value in U
U V X X-1
0000 0000 1101 0
0110 0000
0011 0000
Step 4: RSC X . Go to step 3 and repeat the process until the X has been RSC to its
original position
Example:
String = “10” so subtract Y(or add –Y) from the value in U
U V X X-1
0000 0000 1101 0
0110 0000
0011 0000 1110 1
An Example:
X=0100 = 4
Y=0110 = 6
-Y=1010 = -6

U V X X-1
0000 0000 0100 0
0000 0000 0010 0
0000 0000 0001 0
+1010 0000
1010 0000
1101 0000 1000 1
+0110 0000
0011 0000
0001 1000 0100 0
Take U and V together and you get 00011000 which is 24
4 x 6 = 24 so the answer is correct.
Another Example:
This time with a signed number.
X=0100 = 4
Y=1010 = -6
-Y=0110 = 6
U V X X-1
0000 0000 0100 0
0000 0000 0010 0
0000 0000 0001 0
+0110 0000
0110 0000
0011 0000 1000 1
+1010 0000
1101 0000
1110 1000 0100 0
Take U and V together and you get 11101000 which is -24
4 x -6 = -24 so the answer is correct.