College Algebra, Chapter 1, 1.4, Section 1.4, Problem 54

Evaluate $\displaystyle \frac{1 - \sqrt{-1}}{1 + \sqrt{-1}}$ and express the result in the form $a + bi$.


$
\begin{equation}
\begin{aligned}

=& \frac{1 - \sqrt{-1}}{1 + \sqrt{-1}}
&& \text{Given}
\\
\\
=& \frac{1 - \sqrt{1 i^2}}{1 + \sqrt{1 i^2}}
&& \text{Recall that } i^2 = -1
\\
\\
=& \frac{1 - i}{1 + i}
&& \text{Multiply the complex conjugate of the denominator}
\\
\\
=& \left( \frac{1 - i}{1 + i}\right) \left( \frac{1 - i}{1 - i} \right)
&& \text{Use FOIL method to simplify}
\\
\\
=& \frac{1 - 2i + i^2}{1 - i^2}
&& \text{Recall that } i^2 = -1
\\
\\
=& \frac{1 - 2i + (-1)}{1 - (-1)}
&& \text{Simplify}
\\
\\
=& \frac{-2i}{2}
&&
\\
\\
=& -i
&&

\end{aligned}
\end{equation}
$