Question

Find the modulus and argument of the following complex number and hence express each of them in the polar form:
$\sqrt{3}+i$

Answer 1

Hint: First, let us suppose the complex number be z as $z=\sqrt{3}+i$. Then, the modulus of the complex number is defined as the square root of the sum of the square of the real and imaginary part of the complex number. Then, let $\theta $ be the argument of complex number z and the argument is given by the formula as $\tan \theta =\dfrac{\left| \operatorname{Im}\left( z \right) \right|}{\left| \operatorname{Re}\left( z \right) \right|}$.

Complete step-by-step answer:
In this question, we are supposed to find the modulus and argument of the complex number $\sqrt{3}+i$.
So, let us suppose the complex number be z.
Then, we get it as:
$z=\sqrt{3}+i$
Now, the modulus of the complex number is defined as the square root of the sum of the square of the real and imaginary part of the complex number.
Then, by using the above definition we get the modulus of the complex number as:
$\left| z \right|=\sqrt{{{\left( \sqrt{3} \right)}^{2}}+{{1}^{2}}}$
Now, solve the above expression to get the modulus of the complex number as:
$\begin{align}
  & \left| z \right|=\sqrt{3+1} \\
 & \Rightarrow \left| z \right|=\sqrt{4} \\
 & \Rightarrow \left| z \right|=2 \\
\end{align}$
So, the modulus of the complex number $\sqrt{3}+i$ is 2.
Now, let $\theta $ be the argument of complex number z and the argument id given by the formula as:
$\tan \theta =\dfrac{\left| \operatorname{Im}\left( z \right) \right|}{\left| \operatorname{Re}\left( z \right) \right|}$
So, by using the above formula, we get the argument as:
$\tan \theta =\dfrac{\left| 1 \right|}{\left| \sqrt{3} \right|}$
Now, solve the above expression to get the argument of the complex number as:
$\begin{align}
  & \tan \theta =\dfrac{1}{\sqrt{3}} \\
 & \Rightarrow \theta ={{\tan }^{-1}}\left( \dfrac{1}{\sqrt{3}} \right) \\
 & \Rightarrow \theta =\dfrac{\pi }{6} \\
\end{align}$
So, the argument of the complex number$\sqrt{3}+i$ is $\dfrac{\pi }{6}$.
Hence, the modulus and argument of the complex number $\sqrt{3}+i$ is 2 and $\dfrac{\pi }{6}$ respectively.

Note:Now, to solve these types of the questions we need to know some of the two basic formulas of the complex number that is argument and modulus of the complex number. Then the above two basic formulas as follows:
Modulus of complex number $a+bi$ is $\sqrt{{{a}^{2}}+{{b}^{2}}}$.
Argument of complex number is $\tan \theta =\dfrac{\left| \operatorname{Im}\left( z \right) \right|}{\left| \operatorname{Re}\left( z \right) \right|}$.