Question

What is the decimal equivalent of the binary number \[101010\]?

Answer 1

Hint:We know that the base number for binary is \[2\] so we will write the provided number from the right side (rather than the left) and calculate the sum when the successive numbers are multiplied by increasing powers of \[2\]. For representation, we shall use base numbers, such as \[2\] for binary and \[10\] for decimal.

Complete step by step solution:
The binary number is \[101010\]. We need to find the decimal value of the given binary number. Now let us assume that base number 2 is used for binary and 10 for decimal
\[{\left( {101010} \right)_2} = {\left( x \right)_{10}}\]
Now we multiply each digit of the binary number by 2, and we get
\[0 \times {2^0} = 0 \\
\Rightarrow 1 \times {2^1} = 2 \\
\Rightarrow 0 \times {2^2} = 0 \times 4 = 0 \]

And
\[1 \times {2^3} = 1 \times 8 = 8 \\
\Rightarrow 0 \times {2^4} = 0 \times 16 = 0 \\
\Rightarrow 1 \times {2^5} = 1 \times 32 = 32 \]

Now to get the value of x we need to add all the above values:
\[x = 0 + 2 + 0 + 8 + 0 + 32 \\
\Rightarrow x = 10 + 32 \\
\therefore x = 42 \]
Therefore, \[42\] is the decimal equivalent of the binary number \[101010\].

Hence, option (B) is correct option

Additional information: A binary number is a number that can be used in the binary system or the base \[2\] numeral system. It only has two numerical values: \[1\] and \[0\]. The binary system is an internal application used by practically every modern computer and computer-based device.

Note: Students must know that Binary numerals have only two values, 1 and 2, whereas decimal numerals can have any digit from 0 to 9. Binary uses two digits whose base number is two, while decimal uses ten digits whose base number is ten.