Signed Two’s-Complement Range by Bit Width

Signed two’s-complement range

In an n-bit signed two’s-complement system, the bit pattern is used to represent both positive and negative integers. The key idea is that one bit pattern is reserved for the most negative value, and the remaining patterns are split between negative numbers, zero, and positive numbers.

Method

1. Identify the bit width: let it be n bits.
2. Find the total number of values: there are 2^n possible bit patterns.
3. Use the range rule:
   • Minimum value: -2^(n-1)
   • Maximum value: 2^(n-1) - 1
4. Write the interval from minimum to maximum.

Example structure

If the width is 8 bits, the range is:

• minimum: -2^7 = -128
• maximum: 2^7 - 1 = 127 So the range is -128 to 127.

Check

Make sure the count of integers in the interval matches 2^n. For n bits, the number of values from -2^(n-1) to 2^(n-1) - 1 is exactly 2^n.

Common mistake to avoid

Do not use symmetric endpoints like -2^(n-1) to 2^(n-1); the positive side stops one short because zero is included among the nonnegative values.