Two's complement representation establishes a unique mapping for signed integers where negative values correspond to $2^n - |X|$, enabling arithmetic operations on signed and unsigned numbers using identical hardware logic while distinguishing overflow behavior based on the numerical range of operands (signed vs. unsigned). The theoretical framework integrates Boolean algebraic functions, specifically Exclusive-OR and majority gates, to construct ripple-carry adders that support simultaneous addition, subtraction via operand inversion, ANDing, and ORing within a unified Arithmetic Logic Unit (ALU) design. This domain belongs to Computer Architecture and Digital Circuit Design, serving as the foundational hardware mechanism for instruction set architecture execution in processors like MIPS by abstracting micro-architectural realization of arithmetic logic operations into deterministic gate-level structures.
Two's complement representation establishes a unique mapping for signed integers where negative values correspond to $2^n - |X|$, enabling arithmetic operations on signed and unsigned numbers using i…