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.
[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 (2nd 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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