Question

What is the complex conjugate of \[\sqrt { - 7} \]?

Answer 1

Hint: We know that \[i = \sqrt { - 1} \] . In order to solve this question, first of all we will write the given number in the form of \[i\] and then in the standard form of the complex numbers, i.e., \[z = a + ib\] where \[a\] is the real part and \[b\] is the imaginary part. After that we know that the complex conjugate of \[z\] is \[\overline z \] and \[\overline z = a - ib\].So, we will substitute the real and imaginary part and get the complex conjugate of the given complex number.

Complete step by step answer:
We know that,
A complex number is a number which can be expressed in the form of \[a + ib\] where \[a\] is the real part and \[b\] is the imaginary part.
We have given, the complex number as \[\sqrt { - 7} \]
Now we know that
\[i = \sqrt { - 1} \]
Therefore, \[\sqrt { - 7} = \sqrt { - 1 \times 7} = \sqrt 7 i\]
Now we know that
the standard form of the complex numbers, i.e., \[z = a + ib\] where \[a\] is the real part and \[b\] is the imaginary part
Therefore, on transforming \[\sqrt 7 i\] in the standard form, we have
\[z = 0 + \sqrt 7 i\]
So, on comparing we get
\[a = 0,{\text{ }}b = \sqrt 7 \]
which means that the real part of the given complex number is \[0\] and the imaginary part of the given complex number is \[\sqrt 7 \]
Now we know that
Complex conjugate of \[z\] is \[\overline z \]
where \[\overline z = a - ib\]
which means that the complex conjugate of the complex number is found by just changing the sign of its imaginary part.
So, by changing the sign of the imaginary part of the given number, we get
\[\overline z = 0 - \sqrt 7 i\]
Therefore, the complex conjugate of \[0 + \sqrt 7 i\] is \[0 - \sqrt 7 i\]
Hence, the complex conjugate of \[\sqrt { - 7} \] is \[ - \sqrt 7 i\].

Note: Always remember the complex conjugate of the complex number is found by just changing the sign of its imaginary part by its opposite sign. Students make mistakes when the negative signs are given instead of positive in the imaginary part. When a negative sign is given, you have to conjugate it to get a positive value in the imaginary part. Also note that if the imaginary part is zero, then it is equal to its complex conjugate.