Question

The following two signed 2’s complement numbers (multiplicand \( M \) and multiplier \( Q \)) are being multiplied using Booth’s algorithm: 

Multiplicand (\( M \))	Multiplier (\( Q \))
1100 1101 1110 1101	1010 0100 1010 1010

The total number of addition and subtraction operations to be performed is __________. (Answer in integer)

Hint:

Booth's multiplication algorithm reduces the number of addition/subtraction operations by encoding runs of 1s efficiently. The number of operations depends on the bit pattern of the multiplier.

Answer 1

Correct Answer: 13

The number of arithmetic operations performed during the multiplication depends only on how many times the multiplier changes between consecutive bit values when observed from least significant to most significant position.

Each change between two adjacent bits corresponds to one arithmetic action, while uninterrupted sequences of identical bits require only shifting. Thus, counting the total number of such transitions in the multiplier directly gives the total number of additions and subtractions performed.

For the given multiplier \(Q = 1010\;0100\;1010\;1010\), the number of transitions between adjacent bits is found to be 13.

Hence, the total number of arithmetic operations carried out during the multiplication is:

\(\boxed{13}\)