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`?
This web site is based in part upon work supported by the National Science Foundation under KCP Technologies Award ID 0918733, with grant period September 1, 2009 through August 31, 2013. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.