The result of binary addition of – 8 and – 2 in two’s complement form

The result of binary addition of – 8 and – 2 in two’s complement form is :

10110
0110
1110
0111
This question was previously asked in
UPSC CISF-AC-EXE – 2023
To perform binary addition of -8 and -2 in two’s complement, we first need to represent these numbers in two’s complement form using a sufficient number of bits. Since the result -10 is needed, and the options suggest 4 or 5 bits, let’s use 5 bits as 4 bits is insufficient to represent -10 (-8 to +7 range for 4 bits).
– Convert positive 8 and 2 to binary (5 bits):
– 8 = 01000
– 2 = 00010
– Find two’s complement of 8 (for -8):
– Invert bits: 10111
– Add 1: 10111 + 1 = 11000 (This is -8 in 5-bit two’s complement)
– Find two’s complement of 2 (for -2):
– Invert bits: 11101
– Add 1: 11101 + 1 = 11110 (This is -2 in 5-bit two’s complement)
– Perform binary addition:
11000 (-8)
+ 11110 (-2)
——-
110110
– In 5-bit two’s complement addition, the carry-out from the most significant bit is discarded if the result is within the representable range. The result within 5 bits is 10110.
– To verify 10110, it is a negative number (starts with 1). Take two’s complement: Invert (01001), add 1 (01010). 01010 is 10 in decimal. Since it was negative, 10110 represents -10.
The sum -8 + (-2) = -10. The 5-bit two’s complement representation of -10 is 10110, which matches option A. If we had used 4 bits, -8 is 1000, -2 is 1110. 1000 + 1110 = 10110. Truncating to 4 bits gives 0110, which is +6, an incorrect result due to overflow/insufficient bits for the sum. Therefore, 5 bits is necessary and 10110 is the correct result in that format.
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