If x = 111101 and y = 000100 be two 8-bit 2s complement numbers. Their product in 2s complement is:a)110011b)001100c)110100d)110110Correct answer is option 'C'. Can you explain this answer?

Question

Answer

To find the product of two 8-bit 2s complement numbers, x = 111101 and y = 000100, we can follow these steps:

Step 1: Convert x and y to their decimal equivalent
- x = 111101 in 2s complement represents a negative number. To convert it to decimal, we take the 2s complement of the positive number it represents, which is 000011.
- y = 000100 in 2s complement represents a positive number. So, its decimal equivalent is 000100 itself.

Step 2: Multiply x and y using binary multiplication rules
- Multiply the binary representation of x (000011) and y (000100) using the standard binary multiplication rules.
- The result is 0000000100.

Step 3: Convert the result back to 2s complement representation
- Since the result of the multiplication is positive, we don't need to take the 2s complement.
- The result is 0000000100, which represents the decimal number 000100.

Step 4: Convert the decimal result back to 8-bit 2s complement representation
- The decimal result 000100 needs to be converted back to 8-bit 2s complement representation.
- Since the number is positive, the 2s complement representation is the same as the binary representation.
- The 8-bit 2s complement representation of the result is 000100.

Therefore, the product of x = 111101 and y = 000100 in 2s complement is 110100 (option C).

Answer

Given,

x = 111101

y = 000100

By converting 2's complement x = 111101 and y = 000100 into decimal:

x = 111101

The first bit is 1 so, the number is negative.

x = −3

The first bit is 0 so the number is positive.

y = 4

As we know,

Positive numbers have the same representation in simple binary and 2's complement.

x × y = −3 × 4 = −12

Now,

If decimal is negative:

STEP 1: Convert magnitude to binary.

STEP 2: Pad 0's to the desired bit-size.

STEP 3: Invert bits to achieve 1's-complement.

STEP 4: Add 1 to achieve 2's-complement.

By converting −12 to 2's complement which is 110100.

Hence, the correct option is (C).

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