Comment - More on Complex Numbers

A complex number is the sum of a real number and another real number multiplied by , where is a square root of -1.
Thus it can be written as where and are real.

We can add two such numbers by adding their real and imaginary parts separately. Thus .

We subtract them similarly: .

We can multiply them as follows: .

To do division you make use of the fact that .

Thus you write .

It is common to represent complex numbers by points in the "complex plane". The real part of the complex number is , its imaginary part is . We represent it by the point with coordinate , and coordinate .
The axis is, in this complex plane, called the real axis, and the axis is the imaginary axis. Numbers on the real axis are ordinary real numbers and numbers on the imaginary axis are imaginary numbers.

You can represent a complex number alternatively, by its distance to the origin, usually written as and called its magnitude, and the angle that a line from it to the origin makes with the axis at the origin, usually called theta ( ). To anticipate what we will later see, the relations between these quantities is

and

and can be expressed in terms of and by

and the wonderful fact

implies that we can write

Exercises: Evaluate

1. .

2. .

3. Find given .

4. Find given .

5. Find given .