Technologically Embodied
Geometric Functions
Complex Multiplication Part 2: Rotate a Complex Number

In this investigation you will explore the effect of rotation of complex numbers about the origin, and describe your results numerically.

On the first three pages, you will rotate real, imaginary, and complex numbers by pi/2, by pi, and by 3 pi/2 radians.

On the fourth page you'll rotate a complex number by an arbitrary angle.

On all four pages, you will use a point on the unit circle to determine the angle of rotation.

As a result of your investigation, you will describe the numeric effect of rotations on both the rectangular (x + yi) and the polar (r, θ) forms of expressing the complex number. You will also describe how to multiply a complex number by i.

• Page 1: Rotate a complex number. Describe how rotation changes the modulus and argument of the original number being rotated.
• Page 2: Rotate a complex number by pi/2, pi, and 3 pi/2 radians. Describe the numeric effect of each rotation on the rectangular (x + yi) form of the original number.
• Page 3: (Optional) Rotate a complex number v by an arbitrary angle. Describe the rectangular (x + yi) form of the rotated image in terms of x_v, y_v, and phi.
• Conclusion1: Based on your results, how can you use rotation to multiply a complex number by i?
• Conclusion2: (Optional) How does your result on the page 3 investigation relate to the rectangular coordinates of phi?

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