Comment - More on Complex Numbers

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

We can add two such numbers by adding their real and imaginary parts separately. Thus (5 + 7i) + (2 - 3i) = 7 + 4i.

We subtract them similarly: (5 + 7i) - (2 - 3i) = 3 + 10i.

We can multiply them as follows: (5 + 7i) * (2 - 3i) = 10 + (14 - 15)i - 21i² = 31 - i.

To do division you make use of the fact that (a + ib) * (a - ib) = a² - (ib)² = a² + b².

Thus you write .

It is common to represent complex numbers by points in the "complex plane". The real part of the complex number (a + ib) is a, its imaginary part is b. We represent it by the point with x coordinate a, and y coordinate b.
The x axis is, in this complex plane, called the real axis, and the y 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 r and called its magnitude, and the angle that a line from it to the origin makes with the x axis at the origin, usually called theta (). To anticipate what we will later see, the relations between these quantities is

x and y can be expressed in terms of r and by

and the wonderful fact

implies that we can write

Exercises: Evaluate

1. (4 + i) / (3 - 2i).

2. (3 + 3i) * (2 - i).

3. Find r given x = 3, y = 4.

4. Find given x = 3, y = -2.

5. Find given x = -2, y = 4.