Question

The value of \[\sqrt i + \sqrt { - i} \] is
A). \[0\]
B). \[\sqrt 2 \]
C). \[i\]
D). \[ - i\]

Answer 1

Hint: In the given question, we have been given an expression involving the use of complex numbers. We have to find the value of the expression. It involves the sum of square root of the complex number ”iota” and square root of negative of the complex number ”iota”. To solve it, we are going to find the square root of the complex number ”iota” and the square root of negative of the complex number ”iota”. Then we will add them up to find the answer.

Complete step by step solution:
Let \[\sqrt i = a + bi\]
Squaring both sides,
\[i = {a^2} - {b^2} + 2abi\]
Comparing the imaginary parts,
\[1 = 2ab \Rightarrow ab = \dfrac{1}{2}\]
Comparing the real parts,
\[{a^2} - {b^2} = 0 \Rightarrow {a^2} = {b^2}\]
\[ \Rightarrow a = b = \pm \dfrac{1}{{\sqrt 2 }}\]
Hence, \[\sqrt i = \pm \dfrac{1}{{\sqrt 2 }}\left( {1 + i} \right)\]
Similarly, \[\sqrt { - i} = \mp \dfrac{1}{{\sqrt 2 }}\left( {1 + i} \right)\]
Hence, \[\sqrt i + \sqrt { - i} = \pm \dfrac{1}{{\sqrt 2 }}\left( {1 + i} \right) \mp \dfrac{1}{{\sqrt 2 }}\left( {1 + i} \right) = 0\]
Thus, the correct option is A.
Additional Information:
The “\[i\]” symbol multiplied with the constant is called the complex number. It has a value of \[\sqrt { - 1} \]. It is the imaginary part of the equation, as we know a negative number cannot be square rooted. There are a few properties of the number:
\[{i^2} = - 1\]
\[{i^3} = - i\]
\[{i^4} = 1\]

Note: In the given question, we had to find the sum of square root of the complex number ”iota” and square root of negative of the complex number ”iota”. To solve it, we first found the values of the two expressions by assuming them as the standard complex representation. Then we just added them and found our answer.