Understanding the Basics of Signed and Complement Representation in Number Systems

The signed numbers have a sign bit so that it can differentiate positive and negative integer numbers. The signed binary number technique has both the sign bit and the magnitude of the number. For representing the negative decimal number, the corresponding symbol in front of the binary number will be added. 

The signed numbers are represented in three ways. The signed bit makes two possible representations of zero (positive (0) and negative (1)), which is an ambiguous representation. The third representation is 2's complement representation in which no double representation of zero is possible, which makes it an unambiguous representation. There are the following types of representation of signed binary numbers: 

• Sign-Magnitude:  In this form, a binary number has a bit for a sign symbol. If this bit is set to 1, the number will be negative else the number will be positive if it is set to 0. Apart from this sign-bit, the n-1 bits represent the magnitude of the number. 

            E.g. 0 100 = + 4  

                   1 100 = - 4  

Note: - where 0 and 1 represent - and + respectively.

• 1's Complement: by inverting each bit of a number, we can obtain the 1's complement of a number. The negative numbers can be represented in the form of 1's complement. In this form, the binary number also has an extra bit for sign representation as a sign-magnitude form. 

• 2's Complement: By inverting each bit of a number and adding plus 1 to its least significant bit, we can obtain the 2's complement of a number. The negative numbers can also be represented in the form of 2's complement. In this form, the binary number also has an extra bit for sign representation as a sign-magnitude form. 

• N's Complement: N's complement, also known as one's complement, refers to a number obtained by flipping all the bits of a given number N. Here's how it works: 

      For each bit in N, if it's 0, change it to 1, and vice versa. 

      For example: 

   a)  If N = 5 (binary: 101), its one's complement is 100 (binary: 4). 

    b)  If N = 12 (binary: 1100), its one's complement is 0011 (binary: 3). 

• N –1’s Complement: N-1's complement is derived from N's complement. It is calculated by adding 1 to N's complement. Here's the process: 

a) Find the one's complement of N (as explained above). 

b) Add 1 to the obtained one's complement

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