In this video I demonstrate that the square root of 1 (or unity) has two solutions while the cube root of unity has three solutions, two of which are rotations in the complex plane. I show this by first determining the factors using the difference of squares and cubes formulas. The difference of squares gives two factors or solutions: +1 and - 1. The difference of cubes gives the factor +1 and another factor in the form of a quadratic equation, which yields an additional 2 factors. Applying the quadratic formula and simplifying results in a square of a negative number, hence we obtain an imaginary number. This means yields 2 factors that are complex numbers, which I demonstrate are just rotations of 120° and 240° counterclockwise in the complex plane. These solutions will be required for obtaining the solutions of the cubic formula!
Notes: Step 4.1: Cube Root of Unity
Written notes — mirrored from Cubic Formula Proof, which walks through the complete derivation.
The cube root gives 3 solutions just as the square root gives 2 solutions.
We can find these solutions by factoring the square and cube root of unity via difference of squares and difference of cubes formula, respectively.
Note also that these complex factors are just rotations in the complex plane:
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- Full written notes: Cubic Formula Proof
- Playlist: Cubic Formula Proof YouTube playlist
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