complex.Rmd

# Complex Numbers {#complex}

## Motivation {-}

The following videos explain why imaginary numbers are _necessary_ in mathematics. Note that these videos 
use $i$ to denote the imaginary number $\sqrt{-1}$, whereas we use $j$ (which is common in electrical engineering, 
to avoid confusion with current).

<iframe width="560" height="315" src="https://www.youtube.com/embed/T647CGsuOVU" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

<iframe width="560" height="315" src="https://www.youtube.com/embed/2HrSG0fdxLY" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

<iframe width="560" height="315" src="https://www.youtube.com/embed/N9QOLrfcKNc" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

<iframe width="560" height="315" src="https://www.youtube.com/embed/DThAoT3q2V4" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

## Theory {-}

The following videos explain the core concepts: the complex plane and the geometric interpretations of 
complex addition and multiplication.

<iframe width="560" height="315" src="https://www.youtube.com/embed/65wYmy8Pf-Y" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

<iframe width="560" height="315" src="https://www.youtube.com/embed/z5IG_6_zPDo" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

Be sure to try the calculations suggested at the end of the previous video before moving on to the next video:

- $(4 + 3j) \cdot j$
- $(4 + 3j) \cdot 2j$
- $(4 + 3j) \cdot (4 + 3j)$
- $(2 + j) \cdot (1 + 2j)$

<iframe width="560" height="315" src="https://www.youtube.com/embed/YHvR8siIiD0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

```{definition magnitude, name="Magnitude of a Complex Number"}
The **magnitude** of a complex number $z = a + jb$ is 
\[ |z| \overset{\text{def}}{=} \sqrt{a^2 + b^2}. \]
This follows from the Pythagorean Theorem.
```

```{definition complex-conjugate, name="Complex Conjugate"}
The **complex conjugate** of a complex number $z = a + jb$ is 
\[ z^* \overset{\text{def}}{=} a - jb. \]
```

To conjugate a complex number, we simply flip the sign in front of any $j$s in the expression. For example:
\[ \left(\frac{1 + 2j}{2 - j} \right)^* = \frac{1 - 2j}{2 + j}.  \]

Because it is so easy to calculate the complex conjugate, conjugation is often the preferred way to 
calculate magnitudes.

```{theorem calculating-magnitude, name="Calculating Magnitude"}
The _squared_ magnitude of a complex number $z$ is the product of the number and its complex conjugate:
\[ |z|^2 = z z^*. \]
This means that the magnitude of a complex number can be calculated as 
\[ |z| = \sqrt{z z^*}. \]
```

```{proof}
If $z = a + jb$, then $z^* = a - jb$, and 
\[ z z^* = (a + jb)(a - jb) = a^2 - j^2 b^2 = a^2 + b^2 = |z|^2. \]
```

For example, the magnitude of the number above is:
\[ \left|\frac{1 + 2j}{2 - j} \right| = \sqrt{\frac{1 + 2j}{2 - j} \cdot \frac{1 - 2j}{2 + j}} = \sqrt{\frac{1 + 4}{4 + 1}} = 1. \]
It would have been much more tedious to calculate this magnitude by finding $a$ and $b$ such that 
$\frac{1 + 2j}{2 - j} = a + jb$ and using Definition \@ref(def:magnitude).

```{theorem euler, name="Euler's Identity"}
Euler's identity relates the complex exponential $e^{j\theta}$ to the trigonometric functions:
\[ e^{j\theta} = \cos \theta + j \sin\theta \]
Two immediate corollaries of Euler's identity are:
\begin{align*}
\cos\theta &= \frac{e^{j\theta} + e^{-j\theta}}{2} \\
\sin\theta &= \frac{e^{j\theta} - e^{-j\theta}}{2j}.
\end{align*}
```

## Essential Practice {-}

1. Express the complex number $\frac{1 - j}{1 + j}$ in $a + jb$ form, where $a$ and $b$ are real numbers.
2. Calculate $e^{j\pi/4} + e^{-j\pi/4}$. Your answer should be a real number.
3. Calculate $\left| \frac{1 - j}{1 + e^{j\pi/4}} \right|$.