Question

Solve: ${{\left( 100 \right)}_{2}}-{{\left( 10 \right)}_{2}}$
$\left( A \right)\text{ }{{\left( 11 \right)}_{2}}$
$\left( B \right)\text{ }{{\left( 01 \right)}_{2}}$
$\left( C \right)\text{ }{{\left( 10 \right)}_{2}}$
$\left( D \right)\text{ }{{\left( 101 \right)}_{2}}$

Answer 1

Hint: In this question we have been given two numbers with their base as $2$ for which we have to do subtraction. To solve this question, we will use the rules of binary subtraction. We know the rules of binary subtraction that $0-0=0$, $1-0=1$, $1-1=0$ and $0-1$ with a borrow of $1$. We will use these rules to subtract and get the required solution.

Complete step-by-step solution:
We have the expression given to us as:
$\Rightarrow {{\left( 100 \right)}_{2}}-{{\left( 10 \right)}_{2}}$
We can see in the expression that we have a three-digit binary number and a two-digit binary number.
To simplify the subtraction, we will convert the two-digit binary number into a three-digit number by adding a $0$ as its prefix since adding a $0$ does not change its value.
Therefore, we can write the expression as:
 $\Rightarrow {{\left( 100 \right)}_{2}}-{{\left( 010 \right)}_{2}}$
On writing the numbers in subtraction and using the rules of binary subtraction, we get:
\[\Rightarrow \dfrac{\begin{align}
  & \text{ 100} \\
 & -\text{010} \\
\end{align}}{\text{ }010}\], since $0-0=0$, $0-1$ gives us a borrow of $1$ and with that borrow, we have $1-1=0$.
Therefore, the required answer is
 $\Rightarrow {{\left( 100 \right)}_{2}}-{{\left( 010 \right)}_{2}}={{\left( 010 \right)}_{2}}$
Now since removing the zeros from the left-hand side of a number does not change its value, we can write:
$\Rightarrow {{\left( 100 \right)}_{2}}-{{\left( 10 \right)}_{2}}={{\left( 10 \right)}_{2}}$, which is the required solution therefore, the correct option is $\left( C \right)$.

Note: In these types of questions the rules regarding the addition and subtraction of the $1$ and $0$ should be remembered in all the cases. Binary numbers have their base as $2$. There also exist various other forms of numbers such as octal which has base $8$, hexadecimal which has $16$ as the base. But the most commonly used number system is the decimal number system which has the base as $10$.