Question

Value of \[{\left( {10101101} \right)_2} = {\left( { \ldots \ldots \ldots } \right)_{10}}\] is
(a) 137
(b) 173
(c) 170
(d) None of these

Answer 1

Hint: Here, we need to find the missing number. The number with base 2 is written in a binary number system. The missing number with base 10 is in the decimal number system. We will convert the binary number to decimal form.

Complete step-by-step answer:
The given number 10101101 is written with the base 2.
This means that it is written in the binary number system.
We will use binary to decimal conversion to find the missing number in the decimal number system, which is equal to 10101101 in the binary number system.
When a binary number is converted from binary to decimal number system, we follow the following system:
The first digit from the right is equivalent to \[{2^0} = 1\].
The second digit from the right is equivalent to \[{2^1} = 2\].
The third digit from the right is equivalent to \[{2^2} = 4\].
The fourth digit from the right is equivalent to \[{2^3} = 8\].
The fifth digit from the right is equivalent to \[{2^4} = 16\].
The six digit from the right is equivalent to \[{2^5} = 32\].
The seventh digit from the right is equivalent to \[{2^6} = 64\].
The eighth digit from the right is equivalent to \[{2^7} = 128\].
The values of the digits that have 1 in the binary number are added to convert the number in the decimal number system. If the digit in the binary number is 0, then it is not added to find the decimal number.
The given number is 10101101.
We can observe that the binary number has a 1 as the first digit, third digit, fourth digit, sixth digit, and eighth digit from the right.
Therefore, we will add the values 1, 4, 8, 32, 128 to get the required number.
Thus, we get
 \[ \Rightarrow {\left( {10101101} \right)_2} = {\left( {1 + 4 + 8 + 32 + 128} \right)_{10}}\]
Adding the terms, we get
\[ \Rightarrow {\left( {10101101} \right)_2} = {\left( {173} \right)_{10}}\]
Therefore, we get the missing number as 173.
Thus, the correct option is option (b).

Note: In a number system, a base represents each number. If the base is 10, then it is a decimal number system. If the base is 8 it is called octal number. We used the term ‘binary number system’ in the solution. Binary number system is the number system used by computers to compute the various operations. Each number in the binary number system is written using the digits 0 and 1. All binary numbers are a combination of 8 digits, having either 0 or 1 at each place.