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Question: Solve: \({{\left( 100 \right)}_{2}}-{{\left( 10 \right)}_{2}}\) \(\left( A \right)\text{ }{{\left(...

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

Explanation

Solution

In this question we have been given two numbers with their base as 22 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=00-0=0, 1−0=11-0=1, 1−1=01-1=0 and 0−10-1 with a borrow of 11. 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:
⇒(100)2−(10)2\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 00 as its prefix since adding a 00 does not change its value.
Therefore, we can write the expression as:
⇒(100)2−(010)2\Rightarrow {{\left( 100 \right)}_{2}}-{{\left( 010 \right)}_{2}}
On writing the numbers in subtraction and using the rules of binary subtraction, we get:

& \text{ 100} \\\ & -\text{010} \\\ \end{aligned}}{\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 $10$.