Question

Let z = e^2pi i/n. Then z^n = 1, and z is called an nth root of unity. There are n nth roots of unity, equispaced around the unit circle; they have the form z = e^2 pi i (k/n), where k = 0, 1, 2, ..., n - 1. Of course 1 is an nth root of unity, for every n. Draw the unit circle for the four 4th roots of unity. The angle difference (in radians) between adjacent 4th roots is Draw the unit circle for the six 6th roots of unity. The angle difference (in radians) between adjacent 6th roots is Draw the unit circle for the eight 8th roots of unity. The angle difference (in radians) between adjacent 8th roots is

Answer 1

Answer:

• The angle difference between 4th roots of unity is π/2
• The angle difference between 6th roots of unity is π/3
• The angle difference between 8th roots of unity is π/4

Step-by-step explanation:

In general, the angle difference between two consecutive nth roots of unity is 2π/n. That is because we are splitting the circumference of length 2π with n equispaced points, then the arc length between two consecutive points on the unit circumference is 2π/n. We are measuring in radians, so this arc length is the same as the angle difference.

To plot the points on the unit circle, we can use the cartesian representation of complex numbers. For the 4th roots of unity, using Euler's formula we obtain

$z_1=e^{\frac{2\pi i}{4}\cdot0}=1$

$z_2=e^{\frac{2\pi i}{4}\cdot 1}=e^{\frac{\pi i}{2}}=\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2})=i$,

$z_3=e^{\frac{2\pi i}{4}\cdot 2}=e^{\pi i}=\cos(\pi)+i\sin(\pi)=-1$,  

$z_4=e^{\frac{2\pi i}{4}\cdot 3}=e^{\frac{3\pi i}{2}}=\cos(\frac{3\pi}{2})+i\sin(\frac{3\pi}{2})=-i$,

These points are plotted on the attached figure. Similarly, for 6th roots of unity:

$z_1=1$

$z_2=e^{\frac{\pi i}{3}}=\frac{1}{2}+\frac{\sqrt{3}}{2} i$,

$z_3=e^{\frac{2\pi i}{3}}=-\frac{1}{2}+\frac{\sqrt{3}}{2} i$,  

$z_4=e^{\frac{3\pi i}{3}}=-1$,

$z_5=e^{\frac{4\pi i}{3}}=-\frac{1}{2}-\frac{\sqrt{3}}{2} i$,

$z_6=e^{\frac{5\pi i}{3}}=\frac{1}{2}-\frac{\sqrt{3}}{2} i$,  

And for 8th roots of unity:

$z_1=1$

$z_2=e^{\frac{\pi i}{4}}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i$,

$z_3=e^{\frac{2\pi i}{4}}=i$  

$z_4=e^{\frac{3\pi i}{4}}=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i$,

$z_5=e^{\frac{4\pi i}{4}}=-1$,

$z_6=e^{\frac{5\pi i}{4}}=-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} i$,

$z_7=e^{\frac{6\pi i}{4}}=-i$  

$z_8=e^{\frac{7\pi i}{4}}=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} i$