Complex Numbers and Rotations

Complex Numbers

• Complex numbers are two-dimensional numbers that have both a real (horizontal) and an imaginary (vertical) component.
• They can be visualized as points on a coordinate plane.
• Complex numbers can also be interpreted as vectors.

Multiplication of Complex Numbers

• When two complex numbers are multiplied, their arguments (angles) add together. This results in a rotation.
• Analogy: Addition corresponds to translations, while multiplication corresponds to rotations.

Magnitudes and the Unit Circle

• Complex number multiplication involves the multiplication of magnitudes.
• To preserve measure during rotation, the rotating factor must have a magnitude of one. This is the reason for using the unit circle.

Rotation (Pivot)

• Represent the preimage point as a complex number.
• Locate the angle of rotation in standard position on the unit circle:
  • One ray on the positive x-axis.
  • Rotating counterclockwise.
• Represent the point on the unit circle as a complex number.
• Multiply the complex numbers.
• Convert the product back to a point.

Multiplying Complex Numbers

• Treat the imaginary unit i like the variable x.
• Substitute i^2 = -1. This turns into a negative sign for the number here.

Components and Coordinates

• The real component of the resulting complex number corresponds to the horizontal component, which gives the x-coordinate of the image.
• The imaginary component corresponds to the vertical component, giving the y-coordinate of the image.

Example Rotation

• Rotate the point (6, 3) by 150° counterclockwise about the origin.

Verifying Rotation Result

• The rotated point is approximately (-3, 3\sqrt{3}).