Questions: Plot the complex number on the complex plane and write it in polar form and in exponential form. 16+16 i

Solution Steps

The complex number is given as \( z = 16 + 16i \).

To convert this to polar form, we need to find the magnitude \( r \) and the angle \( \theta \).

The magnitude \( r \) is given by: \[ r = \sqrt{16^2 + 16^2} = \sqrt{512} = 16\sqrt{2} \]

The angle \( \theta \) is given by: \[ \theta = \tan^{-1}\left(\frac{16}{16}\right) = \tan^{-1}(1) = \frac{\pi}{4} \]

Thus, the polar form is: \[ z = 16\sqrt{2} \left( \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} \right) \]

The exponential form of a complex number is given by: \[ z = re^{i\theta} \]

Substituting the values of \( r \) and \( \theta \) we found: \[ z = 16\sqrt{2} e^{i\frac{\pi}{4}} \]

Final Answer