1. Two’s Complement Shortcut
To take the two’s complement of a number:
copy bits from right to left up to, and including, the first “1”
flip remaining bits to the left
(1’s comp)
+ 1
(flip)
(copy)

2. Why does it work?
Adding 1 to any binary number will invert digits until the first zero is encountered
When the bit is zero, adding 1 yields 1, with no carry
remaining bits are unaffected
If the bit is 1, adding 1 yields 0, with a 1 carry
Process will then repeat at the next bit
Since 1 is being added to the already inverted bits of the one’s complement, inverted 1 bits get converted back to their original zero state
Until the first 0 is encountered, then the inversion stops
“original” 1 is preserved
Remaining bits stay “flipped” relative to original number

3. 2’s Complement “Sign” bit
Restrict positive numbers to have a leading “0”
Restricted to the N-1 lower bits of an N-bit 2’s complement number
When, the one’s complement is taken…
Highest order bit becomes a one
Adding 1 to make it 2’s complement cannot change that bit (based on previous discussion)
So…
Positive 2’s complement numbers have a leading zero
Negative 2’s complement numbers have a leading one
Highest order bit acts as a sign bit!

4. Two’s Complement Signed Integers
MS bit is sign bit – it has weight –2n-1.
Range of an n-bit number: -2n-1 through 2n-1 – 1.
The most negative number (-2n-1) has no positive counterpart.
-23 22 21 20 1 2 3 4 5 6 7
-23 22 21 20 1 -8 -7 -6 -5 -4 -3 -2 -1