Which of the following is the most efficient to perform arithmetic operations…

Answer: 2’s complement is the most efficient representation for arithmetic operations.

Key reason: two’s complement allows addition and subtraction to use the same binary adder with no special sign handling and has a single representation of zero, which simplifies hardware and improves efficiency.

• Sign-magnitude — stores a separate sign bit and the magnitude. Arithmetic requires extra logic to handle signs, magnitude comparison for subtraction, and produces two zeros (+0 and -0). Not efficient for arithmetic hardware.

• 1’s complement — negative numbers are the bitwise inverse of the positive value. Addition requires an end-around carry correction and there are two zeros, so hardware needs extra steps, reducing efficiency.

• 2’s complement — negative numbers are formed by inverting bits and adding one. There is a single zero representation, and addition/subtraction use the same adder without special rules or end-around carry, which makes arithmetic simpler and faster in hardware.

• 9’s complement — a decimal (base-10) complement used for decimal/BCD arithmetic. It is not a binary representation and is not appropriate or efficient for binary arithmetic operations.

Short example (8-bit): +5 = 00000101. -3 in two’s complement = 11111101. Adding: 00000101 + 11111101 = 00000010 (discard carry) = 2, which is correct and requires no extra correction step.

Conclusion: use 2’s complement for efficient binary arithmetic because it minimizes special-case logic and simplifies hardware implementation.