TWO’S COMPLEMENT

The two’s complement is a method used in computing to represent both positive and negative integers using binary numbers. It’s particularly efficient for arithmetic operations because it simplifies addition and subtraction by treating positive and negative numbers uniformly.

To find the two’s complement of a number:

  1. Invert (flip) all the bits, changing 1s to 0s and 0s to 1s;
  2. After inverting the bits, add 1 to the result;

Let’s find the two’s complement of the binary number 0000 0010:

> 0000 0010 (Represents 2)
> 1111 1101 (Inverted all bits) (One's Complement)
> 1111 1110 (Added 1) (Two's Complement)

The two’s complement ensures that the sum of a number and its negative counterpart equals zero:

    1111 1110 (Two's Complement)
+   0000 0010 (represents 2)
-------------
(1) 0000 0000 (result is 0)
  • The overflow, represented by (1), is handled by a special register called the carry register, which is not directly visible in the result;
  • With this method, I satisfy the law of arithmetic A + (−A) = 0;

The two’s complement representation has several advantages:

  • It allows for the representation of both positive and negative numbers using the same binary system;
  • It simplifies arithmetic operations like addition and subtraction because negative numbers are represented in a way that aligns well with addition;
  • It eliminates the need for a separate sign bit, making the representation more compact and efficient for computation in computer hardware;