**What is this?**

This writeup on **complex number**s shows several ways of representing complex numbers.

**Representations**

Complex numbers, when first introduced, are generally in rectangular form, which is

`a` + `b``i`

where `a` and `b` are both real numbers, and `i`^{2} (`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 + 1.5`i` 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`+

`y`

`i`, so by substituting in the values,

`x`+

`i`

`y`

=

`r`cos

`θ`+

`i`

`r`sin

`θ`

=

`r`(cos

`θ`+

`i`sin

`θ`)

It can also be seen that

`x`

^{2}+

`y`

^{2}=

`r`

^{2}

tan

`θ`=

`y`/

`x`

which allows us to write 2 +

`i`1.5 in this alternate form:

^{[3]}

`r`= sqrt(

`2`

^{2}+

`1.5`

^{2}) = 2.5

`θ`= tan

^{-1}( 1.5/2 ) = arctan ( 1.5/2 ) = approximately 0.6435 (radians)

^{[4]}

^{[5]}

so

2 +

`i`1.5 = 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:

`e`^{iθ} = 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`i``θ`as the power (`e`^{iθ}instead of`e`^{x}) - 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` + `i``y` = `r`(cos `θ` + `i` sin `θ`) = `r``e`^{iθ}

`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( `x`^{2} + `y`^{2} );
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.

^{[4]} Degrees bad, radians good.

^{[5]} Future approximate decimals will omit "approximately".

**Thanks To**

- my brain - without it, I would have been unable to write this
__Mathematical Methods in the Physical Sciences__(2^{nd}edition), by Mary L. Boas of DePaul University, published by John Wiley & Sons, Inc., copyright 1983 and 1966 - reminded me of the complex math and terms I've forgotten- Dave E. - for reminding me of notation when using current

**See Also**

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