Question

What is \[{\left( {1001} \right)_2}\] equal to?
\[
  {\text{A}}{\text{. }}{\left( 5 \right)_{10}} \\
  {\text{B}}{\text{. }}{\left( 9 \right)_{10}} \\
  {\text{C}}{\text{. }}{\left( {17} \right)_{10}} \\
  {\text{D}}{\text{. }}{\left( {11} \right)_{10}} \\
 \]

Answer 1

Hint: In the above question they gave a binary number of base 2 so to find the decimal form we need to write the summation of multiplication of each bit with increasing power of 2 from the right to left of the binary number \[1001\].

Complete step by step answer:
Before going into solving, we need to know the concept of Binary to Decimal conversion which is given as follows.
\[ \Rightarrow \]Let we have a binary number as \[{({x_0}{x_1}{x_2}..............{x_n})_2}\] then its decimal number is given as Decimal \[ = {x_0} \times {2^0} + {x_1} \times {2^1} + {x_2} \times {2^2} + ............ + {x_n} \times {2^n}\] ……….. \[(1)\]
Here in the question they gave a four digit binary number \[{\left( {1001} \right)_2}\]
So we write using the above concept \[{x_0} = 1\] , \[{x_1} = 0\], \[{x_2} = 0\] , \[{x_3} = 1\] .
Now we substitute these values in the above equation \[(1)\]
Decimal \[ = 1 \times {2^0} + 0 \times {2^1} + 0 \times {2^2} + 1 \times {2^3}\]
\[ = 1 + 0 + 0 + 8\]
\[ = 9\]
As we know decimal numbers have the base as \[10\] we can also represent any decimal number \[x\] as \[{x_{10}}\] .
So now we write \[9\] as \[{\left( 9 \right)_{10}}\] .
The correct answer is option (A).

Note:
> Always remember while writing the summation of multiplication of each bit of binary number with increasing power of 2, go from right to left. Using the same method as above you can convert any n-digit binary number to its decimal form.
> Binary is the simplest kind of number system that uses only two digits of 0 and 1 (i.e. value of base \[2\] ). A “BIT” is the abbreviated term derived from BInary digiT. Whereas Decimal number is the most familiar number system to the general public. It is a base \[10\] which has only \[10\] symbols: \[0,1,2,3,4,5,6,7,8{\text{ and }}9\]. When we convert numbers from binary to decimal, or decimal to binary, subscripts are used to avoid errors.