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Sign magnitude, ones' complement, and two's complement

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In computers, all data is stored in binary form, with 0 representing positive numbers and 1 representing negative numbers. The use of original code (the binary representation of a number) in calculations can lead to incorrect results, so the concept of complement code is introduced. Complement code includes the use of both the original code and the inverse code, with the addition of 1 for negative numbers. By using complement code, accurate calculations can be performed in computers.

First of all, all data in a computer is stored in binary (either 0 or 1), including the representation of positive and negative numbers, which can only be expressed using 0 or 1. Therefore, if the highest bit of a binary number is 0, it represents a positive number, and if it is 1, it represents a negative number.

Using the Sign-Magnitude Calculation#

Let's assume we have the values 2 and -2 as shown below:

  • The sign-magnitude representation of +2 is 0000_0010.
  • The sign-magnitude representation of -2 is 1000_0010.

Now, let's perform the calculation 2 + (-2):

0000_0010 + 1000_0010 = 1000_0100 (in decimal, -4)

Clearly, the above calculation result is not what we expected.

Using the Ones' Complement Calculation#

To address the issue mentioned above, we introduce the concept of ones' complement.

Ones' complement: The ones' complement of a positive number is the same as its sign-magnitude representation, while the ones' complement of a negative number is obtained by inverting all the bits except the sign bit.

  • The ones' complement of +2 is 0000_0010.
  • The ones' complement of -2 is 1111_1101.

Now, let's perform the calculation 2 + (-2):

0000_0010 + 1111_1101 = 1111_1111
The above result 1111_1111 is in ones' complement form. Converting it to sign-magnitude representation gives 1000_0000, which is equivalent to -0 in decimal.

From the above, it can be observed that using ones' complement introduces the issue of +0 and -0.

Using the Two's Complement Calculation#

To address the issue mentioned above, we introduce the concept of two's complement.

Two's complement: The two's complement of a positive number is the same as its sign-magnitude representation, while the two's complement of a negative number is obtained by taking the ones' complement and adding 1 to the least significant bit.

  • The two's complement of +2 is 0000_0010.
  • The two's complement of -2 is 1111_1110.

Now, let's perform the calculation 2 + (-2):

0000_0010 + 1111_1110 = 0000_0000
The above result is exactly what we expected, so computers use the two's complement for calculations.

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