Question

What are the seventh roots of unity?

Answer 1

Step 1: Define the seventh roots of unity

On the unit circle, there are seventh roots of unity.

Let $Z$ be the roots of unity.

$Z={\left(1\right)}^{\frac{1}{7}}$

Step 2:Rewrite the $1$in polar form.

The polar form of a complex number is $Z=a+ib=r\left(\cos \theta +i\sin \theta \right)$

Where $r=\sqrt{a^2+b^2}$ and $\theta ={\tan }^{-1}\left|\frac{b}{a}\right|$

$Z={\left(cos0+isin0\right)}^{\frac{1}{7}}$

$\Rightarrow$ $Z={\left(cos2n\mathrm{\pi }+isin2n\mathrm{\pi }\right)}^{\frac{1}{7}}$

$\Rightarrow$ $Z=\cos \left(\frac{2n\mathrm{\pi }}{7}\right)+i\sin \left(\frac{2n\mathrm{\pi }}{7}\right)$

$\Rightarrow$ $Z=e^{\frac{2n\mathrm{\pi }i}{7}}$

Where $n=0,1,2,3,4,5,6$

Thus, the required roots are: $1,e^{\frac{2\mathrm{\pi }i}{7}},e^{\frac{4\mathrm{\pi }i}{7}},e^{\frac{6\mathrm{\pi }i}{7}},e^{\frac{8\mathrm{\pi }i}{7}},e^{\frac{10\mathrm{\pi }i}{7}},e^{\frac{12\mathrm{\pi }i}{7}}$

Hence, $1,e^{\frac{2\mathrm{\pi }i}{7}},e^{\frac{4\mathrm{\pi }i}{7}},e^{\frac{6\mathrm{\pi }i}{7}},e^{\frac{8\mathrm{\pi }i}{7}},e^{\frac{10\mathrm{\pi }i}{7}},e^{\frac{12\mathrm{\pi }i}{7}}$ are the seventh roots of unity.

Alternative Method:

The seventh root of unity lies on the unit circle with angles difference of $\frac{2\mathrm{\pi }}{7}$ starting from $\left(1,0\right)$.

Hence, $e^0,e^{\frac{2\mathrm{\pi }i}{7}},e^{\frac{4\mathrm{\pi }i}{7}},e^{\frac{6\mathrm{\pi }i}{7}},e^{\frac{8\mathrm{\pi }i}{7}},e^{\frac{10\mathrm{\pi }i}{7}},e^{\frac{12\mathrm{\pi }i}{7}}$ are the seventh roots of unity.