Name: Square Root of Unity, Cube Root of Unity, and Complex Rotations
Uploaded: 2025-02-18T23:01:22+00:00
Duration: 16 min 56 s
Description: 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 fir

7 months ago

Health & Science Education math science education cubic equation cubic formula complex numbers imaginary numbers cube root of unity mathematics

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!

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