Square Root of Unity, Cube Root of Unity, and Complex Rotations

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!

5 Cube Root of Unity.jpeg

Timestamps

Cube root gives 3 solutions just as square root gives 2 solutions: 0:00
Difference of squares for square root of unity gives 2 solutions: 1:07
Difference of cubes for square root of unity gives 3 solutions: 3:11
2nd cube root has 2 solutions via the PQ quadratic equation: 5:15
2nd cube root solutions are complex or involve imaginary numbers: 7:01
Note that the complex solutions are inversely equal to each other: 8:30
The complex factors are just rotations in the complex plane: 11:40
Solving complex angles via exact trigonometric ratios: 14:16
Complex angles are 120 degrees and 240 degrees: 15:47
Summary of cube roots of unity: 1, complex rotation of 120°, complex rotation of 240°: 16:44