Why use two's complement for signed integers? Select the primary advantage of the 2's complement number system for representing signed binary integers in digital hardware.
-
Acan perform subtraction by performing addition
-
Bcan perform addition by performing subtraction
-
Ccan perform division through repeated subtraction
-
Dnone of the above
Answer
Correct Answer: can perform subtraction by performing addition
Explanation
Introduction / Context:Two’s complement is the dominant signed-integer representation in general-purpose processors. Its popularity stems from the way it simplifies hardware design, especially for subtraction, overflow detection, and representation of zero and negative values without dual “+0/−0.”
Given Data / Assumptions:
- We are comparing signed encodings (signed magnitude, one’s complement, two’s complement).
- Goal: identify a key computational advantage in hardware.
- Standard binary adders are available.
Concept / Approach:In two’s complement, subtraction A − B is implemented as A + (two’s complement of B). The same adder circuit performs both addition and subtraction by conditionally inverting B and adding 1 via the carry-in. This unifies the hardware path and eliminates the need for a separate subtractor. Additional benefits include a unique zero and straightforward overflow rules.
Step-by-Step Solution:
Form −B by inverting B then adding 1.Compute A − B = A + (−B) using the existing adder.Use carry-in = 1 to add the +1 needed for two’s complement.Hardware re-use reduces complexity and speeds arithmetic units.Verification / Alternative check:Examine an ALU control table: addition and subtraction share the adder core, differing only by operand inversion and carry-in setting.
Why Other Options Are Wrong:
Addition by performing subtraction: not a typical or advantageous approach.Division through repeated subtraction: possible in principle but not a special benefit of the encoding.None of the above: incorrect because the core advantage is well known.Common Pitfalls:Confusing one’s complement with two’s complement (one’s complement has +0 and −0 representations).
Final Answer:can perform subtraction by performing addition