complex number (idea)

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:
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 iθ 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 + iy = r(cos θ + i sin θ) = re^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( 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.
^[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

• imaginary number
• imaginary numbers
• complex number multiplication
• 1=2 using complex numbers

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