Mathematical Investigations III
Name:
Mathematical Investigations III - A View of the World
More Explorations with Complex Numbers
Exploration. This section is optional. Work on it if you are ahead of the rest of the class.
E1. Compute the magnitudes of the given complex numbers in each row.
a) | i |
| 3  4i |
| (i )  (3  4i) |
b) |1  i |
|2i|
| (1  i)  (2  i) |
c) | 3  i |
| 2  6i |
| (3  i)  (2  6i) |
What appears to be happening? State a general pattern.
Can you state a formula for | (a  bi )  (c  di ) | ?
Can you prove that your formula is correct? A direct computation is messy, but it will get you
there!
Poly 8x.1
Rev. S11
Mathematical Investigations III
Name:
E2. You may have learned already that if x is a real number, then x2  x . Is that equation true
even when x is allowed to be a complex number?
E3. For a complex number z = a + bi, try computing z  z . How should the relationship
be fixed for complex numbers?
x2  x
E4. Using the concept that |z| represents the distance of z from the origin in the complex plane,
what equation would graph the unit circle on an Argand diagram in the complex plane?
What equation would produce a unit circle centered at 1 ?
What equation would produce a unit circle centered at i ?
What equation would produce a circle of radius r centered at a  bi ?
Poly 8x.2
Rev. S11