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) \( (11111010)_2 \)
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;
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- 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- 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- 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- 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- 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- 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 !