Question

Find the four complex fourth roots of unity ($1$) and plot them.

Short answer

The four complex fourth roots of unity ($1$) are $1,i,-1,-i$. On plotting them,

Step-by-step solution

Step 1. Given information.

Consider the given question,

$1$can be written as the complex number $1+0i$.

Rewrite $1$in the form $r\left[\cos \theta +i\sin \theta \right]$.

The magnitude of $1$,

$$\begin{gathered} r=\sqrt{1^2+0^2} \\ r=\sqrt{1} \\ r=1\left(1+0i\right) \\ r=1\left[\cos 0°+i\sin 0°\right] \end{gathered}$$

Step 2. Determine the complex roots of 1.

Consider the given question,

n distinct complex roots ofw is given by,

$z_k=\sqrt[n]{r}·\left[\cos \left(\frac{{\theta }_0}{n}+\frac{2k\mathrm{\pi }}{n}\right)+i\sin \left(\frac{{\theta }_0}{n}+\frac{2k\mathrm{\pi }}{n}\right)\right]$,

where $w=r\left[\cos \left({\theta }_0\right)+i\sin \left({\theta }_0\right)\right]$is a complex number, $n\ge 2$and role="math" localid="1646920906566" $k=0,1,2,....\left(n-1\right)$.

The complex roots of $1=1·\left[\cos 0°+i\sin 0°\right]$are given below,

role="math" localid="1646921219672" $$\begin{gathered} z_k=\sqrt[4]{r}·\left[\cos \left(\frac{0°}{4}+\frac{360°k}{4}\right)+i\sin \left(\frac{0°}{4}+\frac{360°k}{4}\right)\right] \\ {z}_k=\sqrt[4]{1}·\left[\cos \left(0°+90°k\right)+i\sin \left(0°+90°k\right)\right] \\ {z}_k=\cos \left(0°+90°k\right)+i\sin \left(0°+90°k\right) \end{gathered}$$

where$k=0,1,2,3$

Step 3. Substitute k=0,1,2 in zk.

Substitute $k=0$in $z_k$,

role="math" localid="1646921362659" $$\begin{gathered} z_0=\cos \left(0°+90°·0\right)+i\sin \left(0°+90°·0\right) \\ {z}_0=\cos \left(0°\right)+i\sin \left(0°\right) \\ {z}_0=1 \end{gathered}$$

Substitute $k=1$in $z_k$,

role="math" localid="1646921387044" $$\begin{gathered} z_1=\cos \left(0°+90°·1\right)+i\sin \left(0°+90°·1\right) \\ {z}_1=\cos \left(90°\right)+i\sin \left(90°\right) \\ {z}_1=i \end{gathered}$$

Step 4. Substitute k=2,3 in zk.

Substitute $k=2$in $z_k$,

$$\begin{gathered} z_2=\cos \left(0°+90°·2\right)+i\sin \left(0°+90°·2\right) \\ {z}_2=\cos \left(180°\right)+i\sin \left(180°\right) \\ {z}_2=-1 \end{gathered}$$

Substitute $k=3$in $z_k$,

$$\begin{gathered} z_3=\cos \left(0°+90°·3\right)+i\sin \left(0°+90°·3\right) \\ {z}_3=\cos \left(270°\right)+i\sin \left(270°\right) \\ {z}_3=-i \end{gathered}$$

Therefore, the four complex roots of unity are$1,i,-1,-i$.

Step 5. Plot the four complex fourth roots of unity 1. 

The four complex roots has the same magnitude, $1$. So, they are at the same distance from the origin.

Also, the arguments of the cube roots are $0°,90°,180°,270°$and the argument between two consecutive pairs is $\frac{360°}{4}=90°$.

On plotting the four complex roots, we get,