Chapter I, Problem 42E

To determine

To write: The number $e^{2\pi i}$ in the form $a+bi$.

Answer to Problem 42E

The number $e^{2\pi i}$ expressed in the form of complex number as $1$.

Explanation of Solution

Formula used:

Euler’s formula:

$e^{iy}=\cos y+i\sin y$

Calculation:

Simplify the number $e^{2\pi i}$ by the use of Euler’s formula as follows.

$\begin{array}{c}{e}^{2\pi i}=\cos 2\pi +i\sin 2\pi \\ =1+i\left(0\right)\\ =1\end{array}$

Thus, the number $e^{2\pi i}$ expressed in the form of complex number as $1$.