Animated Solution for Mathematics - Complex Numbers: Let $z_1$ and $z_2$ be $n^{th}$ roots of unity which subtend a right angle at the origin. Then $n$ must be of the form

The $n^{th}$ roots of unity are defined as the solutions to the equation $z^n=1$. These points lie on the unit circle in the complex plane and form the vertices of a regular $n$-sided polygon centered at the origin.

Each root can be represented using Euler's formula as:

where $k\in \{0,1,2,\dots ,n-1\}$. Each root corresponds to an angular position ${\theta }_k=\frac{2\pi k}{n}$ measured from the positive real axis.

For two roots $z_1$ and $z_2$ to subtend a right angle at the origin, the absolute difference between their angular positions must be exactly $90^{\circ }$, or $\frac{\pi }{2}$ radians. We express this condition as:

Substituting the expression for the arguments ${\theta }_k=\frac{2\pi k}{n}$, we obtain:

We simplify the equation by dividing both sides by $\pi$:

Rearranging this expression to solve for $n$ yields:

Since $k_1$ and $k_2$ are integers, their difference $|k_1-k_2|$ must also be an integer. Let $m=|k_1-k_2|$, where $m$ is a positive integer.

The relationship $n=4m$ demonstrates that for two vertices of a regular $n$-gon to subtend a right angle at the center, $n$ must be a multiple of $4$.

If $n$ is not a multiple of $4$, it is impossible to select two vertices that form a $90^{\circ }$ angle with the origin. This result highlights the rigid geometric constraints imposed by the algebraic structure of the roots of unity.