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Question 11 (Addition of numbers in binary system )

Question 11 : Solve ​\( (10110111)_2 + (100011)_2 \)

a) ​\( (11011011)_2 \)​                            b) ​\( (11011010)_2 \)​          c) ​\( (11010010)_2 \)​       d) ​\( (11111010)_2 \)

Solution:

Before we start explanation, kindly keep in mind that this is base 2 number system problem, and hence, every time we count a sum as 2 or take an  even count we assume it to be equal to 10.

To find this sum of numbers in base 2, see the step wise explanation here;

Fig 1

1- In above figure , we started from right side column and took sum of entries in the extreme right column i-e 1+1 =2, — but as said in the note above we will assume it to be 10 instead i-e 1+1 = 10. We wrote 0 under line and carried 1 in circle over second column.
2

2- In second column (second fig above) now there are three entries i-e 1 carried over from first column ,and  —- 1 and 1 already present there . We took sum of these three entries as 1+1+1=2+1=10+1=11 (because 2 is equal to 10 as assumed ).

Of this 11, we wrote 1 under line in second column and carried  1 over third column.

3

3- In third column  (third fig above) now there are three entries i-e 1 – carried over from second column, and — 1 and 0 already present there.  We took sum of these three entries as 1+1+0=2,   or as per assumption 1+1+0 = 2= 10. We write 0 under line in third column and took 1 as carried over in circle over fourth column. 

4

4- In fourth column (fourth fig above) there are two entries i-e 1 and 0 and 0, we took sum as 1+0+0=1 and wrote it under line in fourth column.

5

5- In fifth column (fifth fig above) again there are two entries i-e 1 and 0, we took sum as 1+0=1, and wrote this 1 under line in fifth column.

6

6- In sixth column (sixth fig above) there are two entries i-e 1 and 1 of which sum is 1+1=2 which we assume as 10 i-e 1+1=10 instead , as per assumption.Of this 10 we wrote 0 under line in fifth column and carried 1 in circle  over the seventh column.

7

7- In the seventh column (seventh fig above) now there becomes two entries i-e 1 in circle —  carried over  from sixth column and already present 0. We took sum of these 1+0 = 1, which we wrote under seventh column.

8- In the last column there is the only entry 1 which we wrote under line in eighth column as it is i-e 1

In this way we completed the summation process and finally put small bracket around our answer and insert base 2 as below

\[ (11011010)_2 \]

 

So option b is correct !

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