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### 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

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/
```
Associated Topics:
High School Imaginary/Complex Numbers

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