Two's Complement Method

Recall that $1$'s complement still allows the existence of 2 forms for $0$: $0000...000$ and $1111...111$. Another method called two's complement (or $\boldsymbol 2$'s complement) solves this issue! $2$'s complement works in the same way as $1$'s complement except that you need to add $1$ to every negative number you work with:

• If your number is positive, its representation is the same as with the sign & magnitude method.
• Otherwise, if your number is negative, do the following:
  1. Find the sign & magnitude representation of its absolute value, and then
  2. Flip all the bits: change $0$s to $1$s, and $1$s to $0$s.
  3. Add $1$ to this number.

Example: The absolute value of $-5_{10}$ is $5$, so its sign & magnitude ("normal") representation in an 8-bit format is $00000101_2$, the $1$'s complement representation is $11111010_2$, and the $2$'s complement representation is: $11111010_2 + 00000001_2 = 11111011_2$. That's it!