The Sign of the Four Bits

Be sure that you understand This: it is by happy coincidence that the high order bit of a two's complement integer is 0 for positive and 1 for negative. But, to create the two's complement representation of the negative of a number you must "reflect, add one". Changing the sign bit by itself will not work.

To convert N bits of two's complement representation into decimal representation:

1. If the integer is negative, complement it to get a positive integer.
2. Convert (positive) integer to decimal (as usual).
3. If the integer was originally negative, put "-" in front of the decimal representation.

The number represented by 1001 is negative (since the "sign bit" is one), so complement:

 1001 →  0110 + 1 →  0111

Convert the result to decimal:  0111 = 7₁₀.  Add a negative sign:   -7₁₀.  So (in 4-bit two's complement representation) 1001 represents -7₁₀.