3000. Add the following numbers in binary:

          a.       78₁₀ + 78₁₀ using one’s complement

          b.       -AB₁₆ + 70₈  using one’s complement

          c.       439₅ + 64₇ using two’s complement

3.a.    78₁₀ + 78₁₀ using one’s complement:

          78 DIV 2 = 39        78 MOD 2 = 0                     78₁₀    = 1001110₂

          39 DIV 2 = 19        39 MOD 2 = 1

          19 DIV 2 = 9          19 MOD 2 = 1                     Chk:   = 2⁶   + 2³ + 2² + 2¹

           9   DIV 2 = 4           9   MOD 2 = 1                               = 64  + 8 + 4 + 2

           4   DIV 2 = 2           4   MOD 2 = 0                               = 78

           2   DIV 2 = 1           2   MOD 2 = 0

           1   DIV 1 = 0           1   MOD 2 = 1

Carry-over:                     1   111                           Chk:     = 2⁷    + 2⁴   + 2³ + 2²

                             0,000000001001110                                   = 128 + 16 + 8 + 4

+ 0,000000001001110                                   = 156 = 78 + 78

                             0,000000010011100

3.b.    -AB₁₆ + 70₈ using one’s complement

          AB₁₆ =    A              B

                       1010         1011       =                               0,000000010101011

                                                           Complementing:    1,111111101010100

           70₈      =   7               0

                        111         000        =                               0,000000000111000

                                                      Carry-Over:                              111

           Adding the two we get:                                         1,111111101010100     

+ 0,000000000111000

                                                                                       1,111111110001100

           But Since the result is negative:

           111111110001100  Complemented =  000000001110011₂ = 115₁₀

           Which we know to be true since AB₁₆ = 171₁₀ and 70₈ = 56₁₀ and

           -171₁₀ + 56₁₀ = 115₁₀

3.c.              439₅ + 64₇ using two’s complement:

                   439₅  =  4 * 5² + 3 * 5¹ + 9 * 5⁰              64₇ =  6 * 7¹  + 4 * 7⁰

                             =  4 * 25 + 3 * 5 + 9 * 1                      =  6 * 7   + 4 * 1

=  100     + 15      + 9                            =  42       + 4

                              =  124₁₀                                                 =  46₁₀

                                                               1111100                                              101110

                          124   DIV 2 = 62  124  MOD 2 = 0                       46  DIV 2 = 23                46  MOD 2 = 0

                             62  DIV 2 = 31  62      MOD 2 = 0                     23  DIV 2 = 11               23  MOD 2 = 1

                            31  DIV 2 = 15  31      MOD 2 = 1                     11  DIV 2 = 5                 11  MOD 2 = 1

                            15  DIV 2 = 7              15  MOD 2 = 1                 5  DIV 2 = 2                    5  MOD 2 = 1

                              7  DIV 2 = 3                7  MOD 2 = 1                 2  DIV 2 = 1                    2  MOD 2 = 0                    

                              3 DIV 2 = 1                3  MOD 2 = 1                 1 DIV 2 = 0                    1  MOD 2 = 1

                              1  DIV 2 = 0                1  MOD 2 = 1                

                   Adding:          1111100

+   0101110

                                       10101010₂ = 170₁₀

Since we are not dealing with negative numbers, we do not need to complement