Polar Representation of Complex Numbers

Date: 11/04/2002 at 23:48:20
From: Christie Shroyer
Subject: Pre-Calculus - Complex numbers

A) The polar representation of a complex number z=a+ib is 
   z=(cos(x) + sin(x)). Express the quantities r and x 
   in terms of a and b.

B) Represent the quantities r and x geometrically.

C) Give two examples of complex numbers and find their corresponding 
   polar representation.

D) What is/are the value(s) of x for a number that is purely real.

E) What is/are the value(s) of x for a number that has real part 
   equal to zero.

Date: 11/05/2002 at 12:44:21
From: Doctor Peterson
Subject: Re: Pre-Calculus - Complex numbers

Hi, Christie.

You copied something wrong in the first question; it should be

    z = r[cos(x) + i sin(x)]

What you are calling x, more often denoted by the Greek letter theta, 
is the angle from the positive x axis to the number z; r is the 
distance from the origin to z:

                        |
                        |       z
                       b+------+
                        |     /|
                        |    / |
                        |  r/  |
                        |  /   |
                        | /    |
                        |/ x   |
     -------------------+------+-------------
                        |      a
                        |
                        |
                        |
                        |
                        |
                        |
                        |
                        |

You can use trigonometry to show that, as indicated in the problem, 
the real part a will be r cos(x) and the imaginary part b will be 
r sin(x). Similarly, you can find r and x from the picture, using the 
Pythagorean theorem and the definition of a trig function.

You can pick any numbers you want for part (c); try 1+i and 1-i for 
starters, and then try one where x is 60 degrees.

A purely real number, with b=0, will lie on the x-axis (positive or 
negative). What will the angle x be in my picture? (There are two 
answers.)

A purely imaginary number, with a=0, will lie on the y-axis. What 
will the angle be?

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/