Why we need to add 1 while doing 2's complement

Question

The 2\'s complement of a number which is represented by N bits is 2^N-number.
For example: if number is 7 (0111) and i\'m representing it using 4 bits then, 2\'s comple

Answer 1

The +1 is added so that the carry over in the technique is taken care of.

Take the 7 and -7 example.

If you represent 7 as 00000111

In order to find -7:

Invert all bits and add one

11111000 -> 11111001

Now you can add following standard math rules:

  00000111
+ 11111001
 -----------
  00000000

For the computer this operation is relatively easy, as it involves basically comparing bit by bit and carrying one.

If instead you represented -7 as 10000111, this won't make sense:

   00000111
+  10000111
  -----------
   10001110 (-14)

To add them, you will involve more complex rules like analyzing the first bit, and transforming the values.

A more detailed explanation can be found here.