Complex Number §

Three ways to represent a complex number.

• Standard Form
• Polar Form
• Complex Exponential Form

Standard Form §

Definition: A complex number in standard form is an expression of the form $x+yj$ where $x,y\in \mathbb{R}$ and $j$ satisfies $j^2=-1$. The set of all complex numbers is denoted by $\mathbb{C}=\{x+yj\mid x,y\in \mathbb{R}\}$

title:
Note that I learned to use $j$ rather than $i$ since  that letter is used in the modelling of electric networks.

$i$ is NOT equal to $\sqrt{-1}$. If it were, then
$$-1 = i^2 = i \cdot i = \sqrt{-1} \cdot\sqrt{-1} = \sqrt{-1 \cdot -1 } = \sqrt{1} = 1$$

Real and Imaginary Parts §

Let $z=x+yj\in \mathbb{C}$ with $x,y\in \mathbb{R}$. $x=\text{Re}(z)$ $y=\text{Im}(z)$

Inverse of Complex Number §

For $z\in \mathbb{C}$, $z=x+yj$, $z^{-1}$ exists if and only if $z\ne 0$. Besides, when it exists, $z^{-1}=\frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}$

Conjugate §

The complex conjugate of $z=x+yj$ with $x,y\in \mathbb{R}$ is $\overline{z}=x-yj$

Properties of Conjugates. Let $z,w\in \mathbb{C}$ with $z=x+yj$ where $x,y\in \mathbb{R}$. Then $\overline{\stackrel{ˉ}{z}}=z$ $z\in \mathbb{R}⟺\bar{z}=z$ $z\text{is purely imaginary}⟺\bar{z}=-z$ $\overline{z+w}=\overline{z}+\overline{w}$ $\overline{zw}=\overline{z}\overline{w}$ $\overline{z^k}={\overline{z}}^k\text{for}k\in \mathbb{Z},k\ge 0,(k\ne 0\text{if}z=0)$ $\overline{(\frac{z}{w})}=\frac{\overline{z}}{\overline{w}}$ $z+\bar{z}=2x=2\text{Re}(z)$ $z-\bar{z}=2yj=2j\text{Im}(z)$ $z\bar{z}=x^2+y^2$

Modulus §

The modulus of $z=x+yj$ with $x,y\in \mathbb{R}$ is the nonnegative real number $|z|=\sqrt{x^2+y^2}$

The modulus is basically the absolute value if we are dealing with real numbers, and this is actually where the modulus comes from. 

Properties of Modulus. Let $z,w\in \mathbb{C}$. Then $|z|=0⟺z=0$ $|\bar{z}|=|z|$ $z\bar{z}=|z{|}^2$ $|zw|=|z||w|$ $|\frac{z}{w}|=\frac{|z|}{|w|}\text{provided}w\ne 0$ $|z+w|\le |z|+|w|\text{which is known as the Triangle Inequality}$

Geometry of $\mathbb{C}$ Graphically §

Polar Form §

There is another way we can represent imaginary numbers, which will be super useful for complex multiplication and division.

We invent this plot, where the x-axis are the Real numbers and y-axis are the imaginary numbers. We see from the above image that $\cos \theta =\frac{x}{r}\sin \theta =\frac{y}{r}$

The polar form of a complex number $z\ne 0$ is given by $z=r(\cos \theta +j\sin \theta )=r\angle \theta$ where $r=|z|$ and $\theta$ is an argument of $z$.

title: Polar Form: Use $z=r\angle\theta$
When I first learned polar form in [[Linear Algebra|MATH115]], we always wrote it as $z = r(\cos\theta+j\sin\theta)$, but this is super long. Instead, just use 
$$z = r\angle\theta$$
which provides the same amount of information in a very concise way.

Exercise: Write $z=1+\sqrt{3}j$ in polar form. §

$r=|1+\sqrt{3}j|=\sqrt{1^2+{\sqrt{3}}^2}=2$ $z=2(\frac{1}{2}+\frac{\sqrt{3}}{2}j)$

Since we have $\cos \theta =\frac{1}{2}$ and $\sin \theta =\frac{\sqrt{3}}{2}$, we know that $\theta =\frac{\pi }{3}$ Answer: $z=2(\cos \frac{\pi }{3}+j\sin \frac{\pi }{3})$

Polar Multiplication and Division §

Multiplication $z_1{z}_2=r_1{r}_2\angle ({\theta }_1+{\theta }_2)$ Division $\frac{z_1}{z_2}=(\frac{r_1}{r_2})\angle ({\theta }_1-{\theta }_2)$

The MATH115 notation… $z_1{z}_2=r_1{r}_2(\cos ({\theta }_1+{\theta }_2)+j\sin ({\theta }_1+{\theta }_2))$ $\frac{z_1}{z_2}=\frac{r_1}{r_2}(\cos ({\theta }_1-{\theta }_2)+j\sin ({\theta }_1-{\theta }_2))$

Powers of Complex Numbers §

de Moivre’s Theorem. If $z=r(\cos (n\theta )+j\sin (n\theta ))\ne 0$, then $z^n=r^n(\cos (n\theta )+j\sin (n\theta ))$ for any $n\in \mathbb{Z}$.

Complex nth Roots §

Let $z=r(\cos \theta =j\sin \theta )$ be nonzero, and let n be a positive integer. Then the n distinct nth roots of z are given by $w_k=r^{1/n}(\cos (\frac{\theta +2k\pi }{n})+j\sin (\frac{\theta +2k\pi }{n}))$ for $k=0,1,\dots ,n-1$

Complex Exponential Form §

Exponential form is really cool, because we can actually start relating exponentials with sins and angles!

Derived from Euler Formula, the complex exponential form of $z$ is given by $z=r{e}^{j\theta }$

Complex Polynomials §

Theorem 5.2. Let $p(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdot \cdot \cdot +a_{1}x+a_0$ be a real polynomial. If $z\in \mathbb{C}$ is a root of $p(x)$, then so too is $\bar{z}$.