Represent complex numbers on the complex plane in rectangular and polar form.
Given a complex number in rectangular or polar form, identify its real and imaginary component.
Quick Lesson Plan
This lesson covers very similar skills to yesterday’s lesson except that the points now don’t just represent locations but complex numbers. If your students already have familiarity with graphing complex numbers, you could skip this lesson and be sure to add questions about complex numbers to the homework you assign for Lesson 8.1. Or you can use this lesson as is and just consider it a review day and an opportunity for students to solidify skills!
The overlay of the polar grid on top of the Cartesian grid will help students accurately graph a polar point and then both estimate and precisely determine the real and imaginary component of the complex number represented by that point.
The magnitude of a complex number is not specifically addressed in the AP Precalculus course framework, but it provides a nice connection to the polar representation of the complex number.
What does the x-coordinate represent?
What does the y-coordinate represent?
(Question 4) Does 4 represent the real component and 5π/6 the imaginary component? Why or why not?
How would we find the real component? How would we find the imaginary component? Where have we done this before?
Note that we use the horizontal and vertical displacement approach, rather than the right triangle and adjust approach in the sample answers. Either approach is completely fine, and students may have a preference for one over the other, but we feel the approach shown provides better support and intuition around the conversion formulas, since the angle in the formula is the angle measured in standard position, not the angle found in the reference triangle.
There are a few nuances in vocabulary that you will want to make sure to use correctly.
A complex number is a single entity. Whenever students are asked to write the complex number in either polar or Cartesian/rectangular form, they are asked to write this entity (__+i____), not give a set of coordinates!
A complex number can be represented by a point in the polar plane. Students can be given the coordinates of this point in either polar or Cartesian coordinates.
The real component of a complex number is graphed as a horizontal position (x-coordinate) and the imaginary component is graphed as a vertical position (y-coordinate).
There is no such thing as a complex point.
While a point representing a complex number in Cartesian coordinates can immediately identify the number’s real and imaginary component, the polar coordinates of a number can be used to determine the real and imaginary component of the number, though some work is required.