Chapter 12
Complex Numbers and
PolarCoordinates
C
omplex numbers are unreal. Yes, that’s the truth. A complex number has a term with a
multiple of i, and i is the imaginary number equal to . Many of the algebraic rules
that apply to real numbers also apply to complex numbers, but you have to be careful because
many rules are different for these numbers. The problems in this chapter will help you under-
stand their particular properties.
You’ll also work on graphing complex numbers. Polar coordinates are quite different from
the usual (x,y) points on the Cartesian coordinate system. Polar coordinates bring together
both angle measures and distances, all in one neat package. With the polar coordinate
system, you can graph curves that resemble flowers and hearts and other elegant shapes.
The Problems You’ll Work On
In this chapter, you’ll work on complex numbers and polar coordinates in the following ways:
✓ Simplifying powers of i into one of four values
✓ Adding and subtracting complex numbers by combining like parts
✓ Multiplying complex numbers and simplifying resulting powers of i
✓ Dividing complex numbers by multiplying by a conjugate
✓ Interpreting graphs of basic polar coordinates
✓ Graphing polar equations such as cardioids and lemniscates
What toWatch Out For
When working with complex numbers and polar coordinates, some challenges will include
the following:
✓ Multiplying imaginary numbers correctly
✓ Choosing the correct conjugate and simplifying the difference of squares correctly
when dividing complex numbers
