# Complex Number

Three ways to represent a complex number.

### Standard Form

**Definition**: A *complex number* in *standard form* is an expression of the form $x+yj$ where $x,y∈R$ and $j$ satisfies $j_{2}=−1$. The set of all complex numbers is denoted by
$C={x+yj∣x,y∈R}$

```
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 $z=x+yj∈C$ with $x,y∈R$. $x=Re(z)$ $y=Im(z)$

### Inverse of Complex Number

For $z∈C$, $z=x+yj$, $z_{−1}$ exists if and only if $z=0$. Besides, when it exists, $z_{−1}=x_{2}+y_{2}x −ix_{2}+y_{2}y $

### Conjugate

The *complex conjugate* of $z=x+yj$ with $x,y∈R$ is $z=x−yj$

**Properties of Conjugates**. Let $z,w∈C$ with $z=x+yj$ where $x,y∈R$. Then
$zˉˉ=z$
$z∈R⟺zˉ=z$
$zis purely imaginary⟺zˉ=−z$
$z+w =z+w$
$zw=zw$
$z_{k}=z_{k}fork∈Z,k≥0,(k=0ifz=0)$
$(wz ) =wz $
$z+zˉ=2x=2Re(z)$
$z−zˉ=2yj=2jIm(z)$
$zzˉ=x_{2}+y_{2}$

### Modulus

The *modulus* of $z=x+yj$ with $x,y∈R$ is the nonnegative real number
$∣z∣=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 $z,w∈C$. Then
$∣z∣=0⟺z=0$
$∣zˉ∣=∣z∣$
$zzˉ=∣z∣_{2}$
$∣zw∣=∣z∣∣w∣$
$∣wz ∣=∣w∣∣z∣ providedw=0$
$∣z+w∣≤∣z∣+∣w∣which is known as the Triangle Inequality$

### Geometry of $C$ 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 $cosθ=rx sinθ=ry $

The *polar form* of a complex number $z=0$ is given by
$z=r(cosθ+jsinθ)=r∠θ$
where $r=∣z∣$ and $θ$ 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 $z=1+3 j$ in polar form.

$r=∣1+3 j∣=1_{2}+3 _{2} =2$ $z=2(21 +23 j)$

Since we have $cosθ=21 $ and $sinθ=23 $, we know that $θ=3π $
*Answer*: $z=2(cos3π +jsin3π )$

### Polar Multiplication and Division

**Multiplication**
$z_{1}z_{2}=r_{1}r_{2}∠(θ_{1}+θ_{2})$
**Division**
$z_{2}z_{1} =(r_{2}r_{1} )∠(θ_{1}−θ_{2})$

The MATH115 notation… $z_{1}z_{2}=r_{1}r_{2}(cos(θ_{1}+θ_{2})+jsin(θ_{1}+θ_{2}))$ $z_{2}z_{1} =r_{2}r_{1} (cos(θ_{1}−θ_{2})+jsin(θ_{1}−θ_{2}))$

### Powers of Complex Numbers

**de Moivre’s Theorem**. If $z=r(cos(nθ)+jsin(nθ))=0$, then
$z_{n}=r_{n}(cos(nθ)+jsin(nθ))$ for any $n∈Z$.

### Complex *n*th Roots

Let $z=r(cosθ=jsinθ)$ be nonzero, and let n be a positive integer. Then the n distinct nth roots of z are given by $w_{k}=r_{1/n}(cos(nθ+2kπ )+jsin(nθ+2kπ ))$ for $k=0,1,…,n−1$

### 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
$z=re_{jθ}$

### Complex Polynomials

Theorem 5.2. Let $p(x)=a_{n}x_{n}+a_{n−1}x_{n−1}+⋅⋅⋅+a_{1}x+a_{0}$ be a real polynomial. If $z∈C$ is a root of $p(x)$, then so too is $zˉ$.