The following two signed 2's complement numbers (multiplicand M and multiplier Q) are being multiplied using Booth's algorithm:
M : 1100 1101 1110 1101 and Q: 1010 0100 1010 1010
The total number of addition and subtraction operations to be performed is ______.

The correct answer is 13.

Step-by-Step Explanation:

In Booth's multiplication algorithm, the total number of addition and subtraction operations is determined by scanning the multiplier, Q, from right to left (from the least significant bit to the most significant bit) and analyzing adjacent bit pairs. We also append an implicit 0 to the right of the least significant bit, which we denote as:

$q_{-1}=0$

For each step $i$ from 0 to $n-1$ (where $n$ is the bit-width of the multiplier, which is 16 in this case), we examine the bit pair $(q_i,q_{i-1})$ and decide on the operation according to the following rules:
1. If the pair is 01, we perform an addition operation.
2. If the pair is 10, we perform a subtraction operation.
3. If the pair is 00 or 11, no arithmetic operation is performed (only shifting occurs).

Given the multiplier Q = 1010 0100 1010 1010, the bits are index-mapped as follows:

$q_{15}=1$

$q_{14}=0$

$q_{13}=1$

$q_{12}=0$

$q_{11}=0$

$q_{10}=1$

$q_9=0$

$q_8=0$

$q_7=1$

$q_6=0$

$q_5=1$

$q_4=0$

$q_3=1$

$q_2=0$

$q_1=1$

$q_0=0$

$q_{-1}=0$

Now, let us examine the pairs sequentially and track the count of arithmetic operations:

- For $i=0$: pair $(q_0,q_{-1})=(0,0)$ ⇒ No operation.
- For $i=1$: pair $(q_1,q_0)=(1,0)$ ⇒ Subtraction (Operation 1)
- For $i=2$: pair $(q_2,q_1)=(0,1)$ ⇒ Addition (Operation 2)
- For $i=3$: pair $(q_3,q_2)=(1,0)$ ⇒ Subtraction (Operation 3)
- For $i=4$: pair $(q_4,q_3)=(0,1)$ ⇒ Addition (Operation 4)
- For $i=5$: pair $(q_5,q_4)=(1,0)$ ⇒ Subtraction (Operation 5)
- For $i=6$: pair $(q_6,q_5)=(0,1)$ ⇒ Addition (Operation 6)
- For $i=7$: pair $(q_7,q_6)=(1,0)$ ⇒ Subtraction (Operation 7)
- For $i=8$: pair $(q_8,q_7)=(0,1)$ ⇒ Addition (Operation 8)
- For $i=9$: pair $(q_9,q_8)=(0,0)$ ⇒ No operation.
- For $i=10$: pair $(q_{10},q_9)=(1,0)$ ⇒ Subtraction (Operation 9)
- For $i=11$: pair $(q_{11},q_{10})=(0,1)$ ⇒ Addition (Operation 10)
- For $i=12$: pair $(q_{12},q_{11})=(0,0)$ ⇒ No operation.
- For $i=13$: pair $(q_{13},q_{12})=(1,0)$ ⇒ Subtraction (Operation 11)
- For $i=14$: pair $(q_{14},q_{13})=(0,1)$ ⇒ Addition (Operation 12)
- For $i=15$: pair $(q_{15},q_{14})=(1,0)$ ⇒ Subtraction (Operation 13)

Summing up all the arithmetic operations, we perform 6 addition operations and 7 subtraction operations, yielding a total of 13 operations.