Complex Numbers and Rotations

Flashcards covering complex numbers, their properties, and their use in rotations.

Complex Numbers

Two-dimensional numbers with real (horizontal) and imaginary (vertical) components.

Complex Numbers

Points in the coordinate plane can be represented as these.

Complex Numbers and Multiplication

When two of these numbers multiply, their arguments add together, resulting in a rotation.

Rotations and Complex Numbers

Addition is to translations as multiplication is to these.

Magnitudes

Complex number multiplication also has these multiply to preserve measure, requiring the rotating factor to have a magnitude of one.

Unit Circle

We use this to make complex number multiplication preserve measure.

Rotation Pivot

Represent the preimage point as a complex number, locate the angle of rotation, represent that point on the unit circle as a complex number, multiply the complex numbers, and convert the product back to a point.

Multiplying Complex Numbers

Treat 'i' like 'x' and substitute i^2 = -1.

Real Component

The horizontal component, making it the image's x-coordinate.

Imaginary Component

The vertical component, making it the image's y-coordinate.