What range of signed decimal values can be represented in 12 bits (including the sign bit)?
The correct answer and explanation is :
In a 12-bit signed binary system, one bit is reserved for the sign (positive or negative), and the remaining 11 bits are used to represent the magnitude of the number. This system typically uses a two’s complement representation, which is a method for encoding signed integers in binary form. The two’s complement system is efficient because it allows for easy arithmetic operations and simplifies the representation of negative numbers.
Range of Signed Decimal Values:
In a 12-bit system, the first bit (leftmost) is the sign bit. A 0 in the sign bit represents a positive number, and a 1 represents a negative number. The remaining 11 bits are used to represent the magnitude of the number.
- Positive Range: The largest positive value occurs when the sign bit is
0and all other bits are set to1. In binary, this is011111111111, which equals to: [
2^{11} – 1 = 2047
] - Negative Range: The smallest negative value occurs when the sign bit is
1and all other bits are set to0(in two’s complement, the number with a leading1represents a negative number). This value is: [
-2^{11} = -2048
]
Therefore, the range of signed decimal values that can be represented in a 12-bit signed binary system is from -2048 to 2047.
Two’s Complement Representation:
In two’s complement, to represent a negative number, you subtract the magnitude from the maximum possible value (which is ( 2^{n-1} ), where ( n ) is the total number of bits). For example:
- To represent
-1in 12-bit, the binary value is111111111111(all bits are set to1). - To represent
-2048in 12-bit, the binary value is100000000000.
The two’s complement method provides an elegant solution for representing both positive and negative numbers in a fixed bit width, as it eliminates the need for separate representations for positive and negative numbers, streamlining arithmetic operations.