Understanding the Roots of Unity
In complex analysis, the "n-th roots of unity" are the complex numbers $z$ such that $z^n = 1$. When we look for the fourth roots of unity, we are solving the equation:
$$z^4 = 1$$
While we know that $1$ and $-1$ are solutions, the complex roots are often missed without a systematic approach like De Moivre's Theorem.
The Mathematical Framework
De Moivre's Theorem states that for any real number $\theta$ and integer $n$:
$$(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$$
To find the roots, we express $1$ in its polar form:
$$1 = 1 \cdot (\cos(0 + 2k\pi) + i \sin(0 + 2k\pi))$$
where $k$ is an integer. By raising this to the power of $1/4$, we get the general formula for the fourth roots:
$$z_k = \cos\left(\frac{2k\pi}{4}\right) + i \sin\left(\frac{2k\pi}{4}\right) = \cos\left(\frac{k\pi}{2}\right) + i \sin\left(\frac{k\pi}{2}\right)$$
where $k = 0, 1, 2, 3$.
Step-by-Step Calculation
We substitute the values of $k$ into the formula to find the four roots:
For $k = 0$: $$z_0 = \cos(0) + i \sin(0) = 1 + 0i = 1$$
For $k = 1$: $$z_1 = \cos\left(\frac{\pi}{2}\right) + i \sin\left(\frac{\pi}{2}\right) = 0 + 1i = i$$
For $k = 2$: $$z_2 = \cos(\pi) + i \sin(\pi) = -1 + 0i = -1$$
For $k = 3$: $$z_3 = \cos\left(\frac{3\pi}{2}\right) + i \sin\left(\frac{3\pi}{2}\right) = 0 - 1i = -i$$
Summary of Results
The fourth roots of unity are $1, i, -1,$ and $-i$.
Geometric Intuition
If you plot these roots on the complex plane, they form the vertices of a square inscribed in the unit circle. The roots are equally spaced by an angle of $90^\circ$ ($ rac{\pi}{2}$ radians) starting from the positive real axis. This symmetry is a beautiful property of the $n$-th roots of unity for any value of $n$.