Question

The maximum and minimum value represented in signed 16 bit 2's complement representations are

• A. -16384 and 16383
• B. 0 and 32767
• C. 0 and 65535
• D. -32768 and 32767

Hint:

The range for an n-bit number representation is crucial to remember: \[\begin{array}{rl} \bullet & \text{Unsigned: 0 to \( 2^n - 1 \).} \\ \bullet & \text{Signed (2's Complement): \( -2^{n-1} \) to \( 2^{n-1} - 1 \).} \\ \end{array}\] For 16 bits, \( 2^{15} \) is 32768, which is the key number to derive the range.

Answer 1

The Correct Option is D

For a signed n-bit 2's complement representation, the range of values that can be represented is from \( -2^{n-1} \) to \( 2^{n-1} - 1 \).
In this case, we have \( n = 16 \).
Minimum Value (most negative): Minimum value = \( -2^{16-1} = -2^{15} \). \( 2^{10} = 1024 \), so \( 2^{15} = 2^5 \times 2^{10} = 32 \times 1024 = 32768 \).
So, the minimum value is -32768.
Maximum Value (most positive):
Maximum value = \( 2^{16-1} - 1 = 2^{15} - 1 \). \( = 32768 - 1 = 32767 \).
The range is from -32768 to 32767.