(36) COMPLEMENT

COMPLEMENT

Ø  It is a method.

Ø It is a rules or producer.

Ø Complements are used in digital computer for simplifying the subtraction and for logical calculation.

Ø There are two types of complements for each base is r.

1. (r-1)’s  complement

2. r’s        complement

1.   (r-1)’s complement :-

                                        Given  a number n is base,digit  n (r-1)’s complement of n is defined as (rⁿ-1)-n.

2.   r’s complement:-

given a number n is base r a  nd digit                                                          n,r’s complements n defined as (rⁿ-n)=(rⁿ-1)-n+1 =(r-1)’s complement+1.

Ø In the case of  decimal number base is equal +10.

          There are complements

1.    9’s complement

2.    10’s complement

1.   9’s complement- 9’s complement defined as (10ⁿ-1)-N where n= digit

               N= number

  For example:- 1.  (2356)₁₀

                              9’s complement

                                   N= 2356

                                    n= 4

                                    r =10

        9’s complement = (rⁿ-1)-N

                                   = (10⁴_­-1)-2356

                                   = (10000 -1) – 2356

                                   = 9999-2356

                                   = 7643 Ans

     10’s complement =( rⁿ_­­-N)

                                  = (10⁴-2356)

                                  = (10000 – 2356)

                                  = 7644 Ans

_2.        (4678)₁₀

N= 4678

n= 4

r = 10

9’s complement = as complement + 1

                           = 5321+1

                           = 5322

Ø  In the case of binary number:-

1.   1’s complement

2.   2’s complement

1.   1’s complement :- 1’s complement as defined as (2ⁿ-1)-N.

                                        n= digit

                                        N= number

                                        r = base or radix.

2.  2,s complement :- 2,s complement as defined as (2ⁿ-1)-n

                                        n=digit

                                        N=number

                                   

        Ex:- 1. (1010)₂

                                          N= 1010

                                       n= 4

                                       r=2

1^, complement = (rⁿ-1)-n

                        =   (2⁴-1)-1010

                        =(16-1)-1010

                        = 15-1010

                        = 1111-1010

                        = 0101

2^,s complement (rⁿ-n)

                        =(2⁴-1010)

                        =(16-1010)

                        =(16-10)

                        =  6

                        = 0110

2. (11001)₂

                         = n= 11001

                         = n= 5

                         = r=2

     1^s complement – (rⁿ-1)-n

                              =(2⁵-1)-11001

                              =(32-1)-11001

                              = 31-25

                              =6

                              =00110

2^s complement = 1^,s  complement +1

                         = 00110+1

                         = 00111 ans