Question

How do you find $\left| 1+i \right|$ ?

Answer 1

Hint: Assume the given complex number as Z. For the modulus part find the square root of the addition of the square of both real part and imaginary part of the complex number. Do the necessary calculations for finding the value of modulus.

Complete step by step answer:
As we know, a complex number of the form $Z=a+ib$
Where ‘a’ is the real part of Z which is denoted as $\operatorname{Re}\left( Z \right)$ and ‘b’ is the imaginary part of Z which is denoted as $\operatorname{Im}\left( Z \right)$.
The modulus of Z$=\left| Z \right|=\sqrt{\operatorname{Re}{{\left( Z \right)}^{2}}+\operatorname{Im}{{\left( Z \right)}^{2}}}$
So, the modulus of $\left| a+ib \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$
Now for our question
Let $Z=1+i$
So, the modulus of Z
$\begin{align}
  & \left| Z \right|=\left| 1+i \right| \\
 & \Rightarrow \left| Z \right|=\sqrt{{{1}^{2}}+{{1}^{2}}} \\
\end{align}$
$\begin{align}
  & \Rightarrow \left| Z \right|=\sqrt{1+1} \\
 & \Rightarrow \left| Z \right|=\sqrt{2} \\
\end{align}$
This is the required solution of the given question.

Note: A complex number has both real and imaginary parts. If a complex number has only a real part, then we can assume it’s imaginary part as 0. Similarly, If a complex number has only an imaginary part, then we can assume it’s real part as 0. Now for the modulus part, first we have to find squares of both real and imaginary parts and add them. Then taking the square root we can obtain the value of the modulus. Since we are taking the square of both the real and the imaginary part hence the modulus can never be a negative value. It is always a positive quantity.