Question

What is the complex conjugate of $i$?

Answer 1

Hint: First understand the meaning of conjugate of a complex number. Now, considering $i$ as the imaginary number $\sqrt{-1}$, write it in the form $a+ib$ where ‘a’ is the real part and ‘b’ is the imaginary part. Now, change the sign of the imaginary part from (+) to (–) or from (–) to (+), whichever required, to get the answer.

Complete step-by-step solution:
Here we have been provided with the imaginary number $i$ and we are asked to write its complex conjugate. First we need to understand the meaning of conjugate of a complex number.
Now, first of all $i$ is an imaginary number given as $\sqrt{-1}$ and it is the solution of the quadratic equation ${{x}^{2}}+1=0$. A complex number contains both real and imaginary parts. The general representation of a complex number is given as $a+ib$ where ‘a’ is the real part and ‘b’ is the imaginary part.
The conjugate of a complex number is the reflection of the given complex number about the real axis on the complex plane (argand plane). To find the conjugate of a complex number we reverse the sign of the imaginary part present in the given expression, so the conjugate of the complex number $a+ib$ becomes $a-ib$. If the complex number is of the form $a-ib$ then its conjugate will be $a+ib$.
Let us come to the question, we have $i$. Writing it in the form $a+ib$ we get,
$\Rightarrow i=0+i$
Here, real part is 0 and imaginary part is 1, so changing the sign of the imaginary part we get,
$\Rightarrow $ Complex conjugate = $0-i$
$\therefore $ Complex conjugate = $-i$
Hence, the complex conjugate of $i$ is $-i$.

Note: Remember the basic terms of complex numbers like conjugate, argument, modulus etc. Note that if the complex number $a+ib$ is inclined at an angle $\theta ={{\tan }^{-1}}\left( \dfrac{b}{a} \right)$ then the conjugate $a-ib$ is inclined at an angle $-\theta $ to the real axis. The angle $\theta $ is known as the argument of the complex number represented as $\arg \left( z \right)={{\tan }^{-1}}\left( \dfrac{\operatorname{Re}\left( z \right)}{\operatorname{Im}\left( z \right)} \right)$ where z denotes the complex number, Re (z) is the real part and Im (z) is the imaginary part.