What is this?
This writeup on complex numbers shows several ways of representing complex numbers.

Representations

Complex numbers, when first introduced, are generally in rectangular form, which is
a + bi
where a and b are both real numbers, and i2 (i squared) is defined to be equal to -1.[1] (this indicates a note at the end of this writeup) If, instead of a and b, we use x and y instead, it could help to plot the point on a complex plane, or Argand diagram. For example, 2 + 1.5i could be plotted like this: [2]

```
y

^
|
2 +               (2, 1.5)
|
1.5| - - - - - - - #   2 + 1.5i
|               .
1 +               .
|               .
|               .
|               .
---+---+---+---+---+---+---+--> x
|       1       2       3
|
```

However, in many cases, it is easier to work with complex numbers using polar form. One may recall this from geometry or pre-calculus: instead of the rectangular coordinates (x, y), the polar coordinates (r, θ) may be used. (θ is the lower case Greek letter theta) From the slightly modified diagram

```
y

^
|
2 +
|
1.5|             _/#
|     r    _--  |
1 +       __/     |
|     _/        | y
|  _--          |
| / t   x       |
---+===============+---+---+--> x
|       1       2       3
|
```
(t, above, is used in place of θ) one could see that
x = r cos θ
y = r sin θ
The original complex number was in the form x + yi, so by substituting in the values,
x + iy
= r cos θ + ir sin θ
= r(cos θ + i sin θ)
It can also be seen that
x2 + y2 = r2
tan θ = y/x
which allows us to write 2 + i1.5 in this alternate form: [3]
r = sqrt( 22 + 1.52 ) = 2.5
θ = tan-1 ( 1.5/2 ) = arctan ( 1.5/2 ) = approximately 0.6435 (radians) [4] [5]
so
2 + i1.5 = 2.5(cos 0.6435 + i sin 0.6435)
(2, 1.5)rect = (2.5, 0.6435)polar
It is easier in reverse. Given (2.5, 0.6435) in polar coordinates,
x = 2.5 cos 0.6435 = 2.0000
y = 2.5 sin 0.6435 = 1.5000
shows that rectangular coordinates are about (2, 1.5).

A third way to represent a complex number is known as Euler's formula:
eiθ = cos θ + i sin θ
(the constant e (2.718...) raised to the (i times theta) power)
An HTML-ized proof of this would be ugly, so you'll have to be content with how to derive it:

• take the power series for e, only use iθ as the power ( eiθ instead of ex )
• reduce the powers of i, so only every other term will have a factor of i
• rearrange the terms to factor out the i in one set of terms
• note that the terms with the factor of i is the power series for sine, and the remaining terms are the power series for cosine

When multiplying and dividing complex numbers, the algebra may become unwieldy, so a simpler form is used: z to represent the complex number.
This relates the 4 ways to represent a complex number:
z = x + iy = r(cos θ + i sin θ) = reiθ
z is the complex number; Re z = x, which is the real part of z; Im z = y, which is the imaginary part of z; the absolute value or modulus of z is | z | = mod z = r = sqrt( x2 + y2 ); the angle of z = θ

Side note:
When working with current, j is used instead of i to indicate the imaginary number. This is because I is used to indicate a fixed current, and i is used to indicate a variable current.

Notes
[1] There is difference between saying "i squared is defined to be equal to -1" and "i is defined to be equal to the square root of -1". The former is correct, the latter is incorrect; this subtlety is often taken advantage of in false proofs that involve i.
[2] Since pictures can't be inserted in writeups, diagrams will be ASCII art.
[3] A square root symbol cannot be done easily in [X]HTML, so square root will be indicated by sqrt.