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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 + i b where a and b are real.

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

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

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

To do division you make use of the fact that ( a + i b ) * ( a − i b ) = a 2 − ( i b ) 2 = a 2 + b 2 .

Thus you write a + i b c + i d = ( a + i b ) ( c − i d ) c 2 + d 2 .

It is common to represent complex numbers by points in the "complex plane". The real part of the complex number ( a + i b ) 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 ( θ ). To anticipate what we will later see, the relations between these quantities is

r 2 = x 2 + y 2 = a 2 + b 2

and

tan ⁡ θ = y x = b a

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

x = r cos ⁡ θ , y = r sin ⁡ θ

and the wonderful fact

cos ⁡ θ + i sin ⁡ θ = exp ⁡ ( i θ )

implies that we can write

x + i y = r exp ⁡ ( i θ )

Exercises: Evaluate

1. 4 + i 3 − 2 i .

2. ( 3 + 3 i ) * ( 2 − i ) .

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

4. Find θ given x = 3 , y = − 2 .

5. Find θ given x = − 2 , y = 4 .

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