DIGITAL LOGIC - COMBINATION CIRCUIT(2'S COMPLEMENT NUMBER OPERATION )

2'S COMPLEMENT NUMBER OPERATION 

Definition of one’s complement

• The ones' complement of a binary number is defined as the value obtained by inverting all the bits in the binary representation of the number (swapping 0's for 1's and vice-versa).

• In this form, a negative number is the 1’s complement of the corresponding positive number.

• The 1’s complement of a binary number is found by changing all 1s to 0s and all 0s to 1s as shown below:

+28₁₀ = 00011100₂

-28₁₀ = 11100011₂ (1’s complement of +25)

Definition of two’s complement

• Two's complement is a mathematical operation on binary numbers, as well as a representation of signed binary numbers based on this operation. The two's complement of an N-bit number is defined as the complement with respect to 2^N, in other words the result of subtracting the number from 2^N

•  The 2’s complement of a binary number can be obtained by adding 1 to one’s complement:

2’s complement = (1’s complement) + 1

Example:

• The negative (-) decimal number conversion is done in next steps .Lets we convert -118.

1. Separate the sign and magnitude number of -1 .If the sign bits is 1, its represent      as  negative sign in the 2s complement conversion.

2. Convert the decimal number to its 7-bits binary equivalent.

Decimal	8-bits binary number			Note
	sign	Magnitude
118	0	1110110	Convert to 7-bits binary
		0001001	1stcomplement	Each 0 is changed to 1 and each 1 to 0.
		0001010	2nd

Another Example:

Express the decimal number -55 as an 8-bit in the sign-magnitude, 1’s complement, and 2’s complement forms.

SOLUTION:

8-bit number for + 55₁₀ = 00110111₂

Sign-magnitude form for -55₁₀ = 10110111₂

•  Change the sign bit to a 1 and leave the magnitude bits as they are 1’s complement form for -55₁₀ = 11001000₂

• Take the 1’s complement of +55 by changing all 1’s to 0s and 0s to 1s

2’s complement form for -55_{10 =}

          11001000    1's complement

+                       1                        

           11001001    2's complement

2’s Complement Operations

• Two Positive Numbers

    1410                              000011102

+  2510          -------->     +    000110012

    3910                               001001112  

Positive Number and Smaller Negative Number

    

                2910  

           -    1510

SOLUTION:

1. Find the binary numbers for 2910
            2910        =     000111012

2. Find the binary form for -1510 using 2's complement
            -1510      =     111100012
3. Add the binary numbers

                                             000111012
                                       +   111 100012
                                            1000011102
                                                          |           
                                    |
                                    |
                            Discard carry over

Positive Number and Larger Negative Number

               1210

        -      2510

 

Tasks:

1. Find the binary form for 12₁₀

2. Find the binary form for −25₁₀ using 2’s complement

3. Add the binary numbers

Solution:

1 :+122   -------->  000011002

2 : 

             +2510    ------>   000110012

             -2510     ------>  111001102

                                  +             1  

                                    111001112

3 :

           1210                              000011002

     +( -2510)         -------->   +  111001102

                                                                            111100112= -1310

Check the answer using  2's complement

       111100112     -------->        000011002

                                               +               1  

                                                  000011012 = +1310