Consider the following statements about the range of numbers in a 9-bit 1's complement and 2's complement system.
I. In 9-bit 1's complement, the range is -255 to +255, and there exist two representations of zero. II. In 9-bit 2's complement, the range is -256 to +255, and both 1's complement and 2's complement can represent exactly 512 unique values. III. The maximum positive number representable is +255 in both 1's complement and 2's complement 9-bit systems. Identify the CORRECT option.Step-by-step Solution:
Let us analyze the properties of an \( n \)-bit system where \( n=9 \): - **I. True:** The formula for the range in 1's complement is \( -(2^{n-1}-1) \) to \( +(2^{n-1}-1) \). For \( n=9 \), this calculates to \( -(2^8-1) \) to \( +(2^8-1) \), which is \( -255 \) to \( +255 \). 1's complement inherently features two representations for zero (positive zero `000000000` and negative zero `111111111`). - **II. False:** While 2's complement indeed has a range of \( -256 \) to \( +255 \) and covers 512 unique values, 1's complement only represents 511 *unique* values because `+0` and `-0` evaluate to the exact same magnitude. - **III. True:** In both complement systems, the maximum positive integer is determined by utilizing all available bits excluding the sign bit as 1s. This is \( 2^{n-1}-1 \), which evaluates to \( +255 \) for both. Therefore, only statements I and III are correct.