Question

How do you find the absolute value of the complex number $ z=2+5i $ .

Answer 1

Hint: In this problem, we need to calculate the absolute value of the complex number. We know that the complex number consists of the real part and the imaginary part. Now the absolute value of a complex number is the square root of the sum of the squares of the real part and the imaginary part. So, we will write the real part and imaginary part of the given complex number. From this value, we will calculate the squares of the real part and imaginary part. After that, we will sum up the above-calculated values and find the square root to get the absolute value of the given complex number.

Complete step by step answer:
Given a complex number is $ z=2+5i $ .
The real part of the above complex number is $ 2 $.
The imaginary part of the above complex number is $ 5 $.
Square of the real part of the given complex number is $ {{2}^{2}}=4 $.
Square of imaginary part of the given complex number is $ {{5}^{2}}=25 $ .
Sum of the squares of the real part and the imaginary part is $ 4+25=29 $.
Now the square root of the sum of squares of a real part and the imaginary part is $ \sqrt{29} $.
Hence the absolute value of the complex number $ z=2+5i $ is $ \sqrt{29} $ .

Note:
We can also directly calculate the absolute value of the complex number $ z=2+5i $ which is denoted by $ \left| z \right| $ as mentioned below
 $ \left| z \right|=\left| 2+5i \right| $
We know that the $ \left| a+bi \right|=\sqrt{{{a}^{2}}+{{b}^{2}}} $ , then we will get
 $ \Rightarrow \left| z \right|=\sqrt{{{2}^{2}}+{{5}^{2}}} $
Substituting the know values $ {{2}^{2}}=4 $ , $ {{5}^{2}}=25 $ , then we will get
 $ \begin{align}
  & \Rightarrow \left| z \right|=\sqrt{4+25} \\
 & \Rightarrow \left| z \right|=\sqrt{29} \\
\end{align} $
From both the methods we got the same result.