### <Success>Complex number </br>lecture 5</Success> <div style='float: left; width:50%;'> </div>

#### <Success>Polar representation of complex number</Success> <div style='float: left; width:50%;font-size:25px;'> <p>$ z=r(\cos (\theta) + i \sin (\theta))$</p> <p>$ \text{where}$ $r ≥ 0$ & $ 0 ≤ \theta ≤ 2 \pi $</p> <p>$ z=r(\cos (\theta +2k\pi) +i\sin (\theta+2k\pi))$</p> <p>$ k \in \Z$</p> <p>$ 1-i=\sqrt2\biggl(\cos (\frac {7\pi} {4})+i\sin \frac{7\pi} {4}\biggl) $</p> <p>$ 1-i=\sqrt2\biggl(\cos \biggl(\frac {-\pi} {4}\biggl)+i\sin\biggl(\frac{-\pi} {4}\biggl)\biggl) $</p> <p>$ \cos\frac {\pi} {4} = \frac {1} {\sqrt2}\sin\biggl(\frac {\pi} {4}\biggl) =\frac {1} {\sqrt2}$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/10a28b37-c7ca-4fec-fdf7-7111e00afa00/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/6c40a4bd-14c1-4238-7627-2e9447c4a200/public"></p> </div>

### <Success>De moivre's formula</Success> <div style='float: left; width:100%;font-size:30px;'> <p>$ z=r(\cos \theta + i \sin \theta)$</p> <p>$ \text{where}$ $r ≥ 0$ & $ 0 ≤ \theta ≤ 2 \pi $</p> <p>$ z=r(\cos (\theta +2k\pi) +i\sin (\theta+2k\pi))$</p> <p>$ k \in \Z$</p> <p>$ \text{De moivre’s formula} $</p> <p>$(\cos \theta + i \sin \theta)^n = \cos n\theta +i\ \sin n\theta $</p> </div>

### <Success>De moivre's formula</Success> <div style='float: left; width:90%;font-size:30px;'> <p>Division</p> <p>$ z_1 = r_1(\cos \theta_1 +i \sin \theta_1) $</p> <p>$ z_2 = r_2(\cos \theta_2 +i\sin \theta_2) $</p> <p>$ \frac {z_1} {z_2}=\frac{r_1( \cos \theta_1+ i \sin \theta_1)} {r_2(\cos \theta_2+ i \sin \theta_2)}= \frac{r_1( cos \theta_1+ i \sin \theta_1) (\cos \theta_2- i \sin \theta_2)} {r_2(\ cos^2\theta_2+ i \sin^2 \theta_2)}$</p> <p>$ \frac {r_1} {r_2} = [\cos (\theta_1 -\theta_2)+ i \sin (\theta_1 -\theta_2)] $</p> </div>

### <Success>De moivre's formula</Success> <div style='float: left; width:50%;font-size:30px;'> <p>$ z_1 =1+i \quad z_2=i $</p> <p>$ \frac {z_1} {z_2} =\frac {\sqrt2} {1}[\cos(\frac {-\pi} {4})+i \sin(\frac {-\pi} {4})] $</p> <p>$ \frac {z_1} {z_2} =(1-i) $</p> <p>$ {z_1} = {\sqrt2} \biggl(\cos \frac {\pi} {4}+i \sin \frac {\pi} {4}\biggl) $</p> <p>$ {z_2} = 1 \biggl(\cos \frac {\pi} {2}+i \sin \frac {\pi} {2}\biggl) $</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/d862ee86-8ed2-4710-a052-52b13e496d00/public"></p>

### <Success>Remark</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p><strong>(i)</strong> $\left|\frac{z_1}{z_2}\right|=\frac{r_1}{r_2}=\frac{\left|z_1\right|}{\left|z_2\right|} \quad\left(z_2 \neq 0\right)$</p> <p><strong>(ii)</strong> $z^n=(\cos \theta+i \sin \theta)^n$</p> <p>$n=-1,-2, \ldots$</p> <p>$n=-1 \quad z^{-1}=\frac{1}{z}=\frac{1}{\cos \theta+ i \sin \theta} $</p> <p>$= \frac{1}{1}[\cos (0-\theta)+i \sin (0-\theta)] $</p> <p>$= \cos (-\theta)+i \sin (-\theta)$</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/c664113e-d8bc-4960-65fe-4678a5fa6300/public"></p> </div>

### <Success>Remark</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$n=-2; $</p> <p>$z^{-2}=\frac {1} {z_2}=\frac{1}{z}\times\frac{1}{z}=[\cos(-\theta)+i\sin(-\theta)]^2 $</p> <p>$z^{-2}=\cos(-2\theta)+i\sin(-2\theta) $</p> <p>$z^{n}=\cos\ n\theta+i\sin\ n\theta \quad n\epsilon \Z.$</p> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>$z=(1+i\sqrt3)^n+(1-i\sqrt3)^n $</p> <p>$ z={2\biggl[\cos\frac{\pi}{3}+i \sin\frac{\pi}{3}\biggl]}^n+{2\biggl(\cos\biggl(\frac{-\pi}{3}\biggl)+i \sin\biggl(\frac{-\pi}{3}\biggl)\biggl)}^n$</p> </div> <div style='float: right; width:50%;font-size:30px;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/0fcd83de-7b62-4214-ba59-7012f9eaf400/public"> $\tan^{-1}(\frac{\sqrt3} {1})=\frac {\pi}{3}$</p> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:99%;font-size:30px;'> <p>$ z=2^n\biggl(\cos\frac{n\pi}{3}+i \sin\frac{n\pi}{3}\biggl)+2^n[\biggl(\cos\biggl(\frac{-n\pi}{3}\biggl)+i \sin\biggl(\frac{-n\pi}{3}\biggl)\biggl)]$</p> <p>$z=2^{n+1} \cos \frac {n\pi}{3}$</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/cfdcdc3d-cbb8-4ad9-6d19-ff014b508f00/public"></p> </div>

### <Success>$n^{th} \text{roots of unity}$</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>Find the $n^{th}$ roots of unity</p> <p>$z^n =1 $</p> <p>observation :suppose there is a $z_0\epsilon$ C</p> <p>such that $\quad z_0^n =1. \quad then |z_0|=1.$</p> <p>$|z_0^n|=|1|$</p> <p>$|z_0|^n=1$</p> <p>$\Leftrightarrow|z_0|=1$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/c40301dc-0d18-48ac-910e-eac42299e100/public"></p> <!----> </div>

### <Success>$n^{th} \text{roots of unity}$</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>Fix $n=1 ; \quad z^1=1 \quad \Rightarrow z=1 $</p> <p>$n=2 ; \quad z^2=1 $</p> <p>$\Rightarrow z_1=1, \quad z_2=-1$</p> <p>$n=3 ; \quad z^3=1 $</p> <p>$ z_1=1 $</p> <p>$ z_2=\cos 120^{\circ}+i \sin 120^{\circ} $</p> <p>$ z_2^3=\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)^3 $</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/13736c15-3a6d-42ad-0916-636a21696000/public"></p> <!----> </div>

### <Success>$n^{th} \text{roots of unity}$</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$ z_2^3=\cos 360^{\circ}+i \sin 360^{\circ} $ (By De Moivre formula)</p> <p>$ =1 $</p> <p>$z_3^3=\left(z_2 \times z_2\right)^3=1 $</p> <p>$n=4 ; \quad z^4=1 $</p> <p>$ z_1=1, z_2=i \quad z_3=-1 \quad z_4=-i$</p> <p>$n^{t h} \text { roots of unity } $</p> <p>$z^n=1 $</p> <p>$z^n=\cos 2 k \pi+i \sin 2 k \pi$</p> </div>

### <Success>$n^{th} \text{roots of unity}$</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$(\cos \theta+i \sin \theta)^n=\cos 2 k \pi+i \sin 2 k \pi \quad k \in \mathbb{Z} $</p> <p>$\cos n \theta+i \sin n \theta=\cos 2 k \pi+i \sin 2 k \pi \quad \quad k = 0,1,2,3…..$</p> <p>For $\theta=\frac{2 k \pi}{n}, k=0,1,2, \cdots$.</p> <p>the equation (1) holds.</p> <p>Define $\theta_k=\frac{2 k \pi}{n}, k=0,1, \ldots, n-1$.</p> <p>& $ z_k=\cos \theta_k+i \sin \theta_k, k=0,1, \ldots, n-1 $</p> <p>$\Rightarrow \quad \left(z_k\right)^n=1, k=0,1, \ldots, n-1 $</p> </div>

### <Success>$n^{th} \text{roots of unity}$</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>Claim : {$z_k$ : $k \in$ $\Z$} = {$z_k: k=0,1, \ldots n-1$}</p> <p>Let $r \in$ $\Z$ $\quad \theta_r=\frac{2 r \pi}{n} \rightarrow z_r$</p> <p>By quotient reminder theorem .</p> <p>$r=q n+k \text { where } q \in Z, \quad k \in{0,1, \ldots n-1} $</p> <p>$\theta_r=\frac{2 \pi(q n+k)}{n}=2 \pi q+\frac{2 k \pi}{n}=2 \pi q+\theta_k $</p> <p>$ z_r=\cos \left(2 \pi q+\frac{2 k \pi}{n}\right)+i \sin \left(2 \pi q+\frac{2 k \pi}{n}\right) $</p> <p>$z_r=\cos \left(\frac{2 k \pi}{n}\right)+i \sin \left(\frac{2 k \pi}{n}\right) $</p> <p>$ z_r=z_k$</p> </div>

### <Success>$n^{th} \text{roots of unity}$</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>The $n^{th}$ roots of unity $:z^n=1$</p> <p>$ z_k = \cos(\frac{2k\pi}{n})+ i \sin\frac {2k\pi}{n}, k=0,1,…n-1.$</p> <p>$ n=2 \quad , \quad z^2=1$</p> <p>$ z_0 =1\quad z_1=\cos\pi +i\sin\pi$</p> <p>$ z_1=-1$</p> <p>$ n=3 \quad , \quad z^3=1$</p> <p>$ z_0 =1\quad z_1=\cos\frac{2\pi}{3} +i\sin\frac {2\pi}{3}$</p> <p>$ z_2=\cos\frac{4\pi}{3} +i\sin\frac {4\pi}{3}.$</p> </div>

### <Success>$n^{th} \text{roots of unity}$</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <p>$ \theta_1 =\frac{2\pi} {n}$</p> <p>$ \theta_2 =\frac{2\pi} {n}+\frac{2\pi} {n}$</p> <p>$ n=8 $</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/cad6dbae-183a-4eec-e62e-6c9f70b58d00/public"></p> <!----> </div>

#### <Success>Remark</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p><strong>(1)</strong> The geometric image of the $n^{th}$ roots of unity are the vertices of a regular polygon with n sides inscribed in the unit circle with one of the vertices at 1.</p> <p><strong>(2)</strong> $z_1 =\cos\frac {2\pi}{n} + i\sin\frac{2\pi}{n}$</p> <p>$z_1^2 =\cos\frac {4\pi}{n} + i\sin\frac{4\pi}{n}$</p> <p>$z_1^k = \cos\frac {2k\pi}{n} + i\sin\frac{2k\pi}{n} ,\quad$ (By De Moivre’s theorem )</p> <p>$z_1^k =z_k$</p> </div>

#### <Success>Remark</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$ \text{The}\ n^{th} \text {root of unity}$</p> <p>$\{z_0,z_1,z_2 …,z_{n-1}\} =\{z_1^0,z_1^1,z_1^2,….z_1^{n-1}\}$</p> </div>

### <Success>Thank you</Success> <div> </div>