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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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