A number consisting of two parts, called real and imaginary. Complex numbers are written in the form a + bi, where a is the real part, b is the size of the imaginary part, and i is the square root of -1. If the size of the imaginary part is 0 (i.e. b = 0), the number is equal to a real number.

Unlike real numbers, complex numbers can produce negative numbers when squared; because of this, all polynomials have complex roots even though some of them may lack real roots. Another use of complex numbers is to represent a vector in two-dimensional space. In this case, the x-axis is the real part of the number, and the y-axis is the imaginary part.

Arithmetic on complex numbers obeys the same laws of algebra real numbers do.

     (a + bi) + (c + di) = (a + c) + (b + d)i
     (a + bi) - (c + di) = (a - c) + (b - d)i
     (a + bi) * (c + di) = ac + adi + bci + bdi2
                         = (ac - bd) + (ad + bc)i
You can't do any serious data engineering or spectrum analysis without using imaginary/complex numbers. FFTs - in fact all Fourier/Laplace/Z Transforms use two dimensional representation of numbers. The whole idea is in some ways related to vectors, in the sense that an imaginary number can also be expressed as a length (amplitude) and an angle (phase).
Let's define Z to be a complex number where the real part is called real(Z), the imaginary imag(Z). Amplitude is then named mod(Z) and angle arg(Z) (modulo and argument). They can be calculated using:
mod(Z) = ( real(Z)^2 + imag(Z)^2 ) ^ 1/2 . The length, just like in trigonometry
arg(Z) = abs(real(Z) * atan(imag(Z) / real(Z)) . The angle, again just like in trigonometry

Z kan then be written as mod(Z) < arg(Z) (Not really a less than sign in the middle. It means angle) - eg. 12<pi/2 is the same as 0+12i (where i is the square root of -1)
    Quite convenient for some things.
  • Multiplying two complex numbers A and B. A * B = mod(A) * mod(B) < arg(A) + arg(B)
  • Dividing two complex numbers A and B. A / B = mod(A) / mod(B) < arg(A) - arg(B)
  • Taking the power (natural number) of a complex number A^n = mod(A)^n < n * arg(A)
  • Taking the squareroot of a complex number: A^1/2 = mod(A)^1/2 < arg(A) / 2 and the same result multiplyed by 1<pi
The rest of the rules are a bit too complex for HTML (good excuse, eh?), but I guess you get the point. Good thing to know is that if you take the nth root of a complex number, you get n valid results (just like -1 is in a sense the squareroot of 1, which complex numbers actually take into account: The squareroot of 1<0 is 1<0 and 1< pi which is = -1)
In a frequency spectrum (of eg. a sound), all frequencies are represented by an amplitude and a phase. Sounds familiar? It is! Read more in the transformation links!
Our first year mechanical engineering maths lecturer told us:
Working with complex numbers is like apples and oranges. If you have an apple and an orange, that's what you have, one apple and one orange.
The problem with this analogy is that if you square an apple you don't get minus one orange.

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

Yay! Ack! Huh? No! Boo!
You may /msg N-Wing if you have any { complaints | fears | praises | corrections | errors | etc. } { about | of | for | in | etc. } this writeup. You can also /msg me if you didn't understand the previous sentence.

An important property of complex numbers is the fact that two complex numbers are only equal if the real parts are equal and the imaginary parts are equal, ie:

if a + bi = c + di, then:

a=c

and

b=d

Proof:

a + bi = c + di

=> (a + bi) - (c + di) = 0

=> (a - c) + (b - d)i = 0

=> (a - c) = 0 and (b - d) = 0

=> a = c and b = d

This is useful in using De Moivre's Theorem to find Cos(nx) in terms of Cos(x), among other things (it can also be used to find the square root of a complex number)

The set of complex numbers - denoted by C - is the algebraic closure of the real numbers, i.e. the smallest field contaning the reals such that every polynomial equation is solvable. This is probably the most important property of the complex numbers.

This does not imply that it is convenient to define C as the algebraic closure of the reals. The following definition is somewhat easier to work with.

Definition:

C is the smallest field (up to an isomorphism) containing R as a subfield such that the equation X2 + 1 = 0 has a solution in C.

Proof of existence:

Let C0 be R2 together with addition and multiplication defined in the following way:

(a, b) + (c, d) = (a+c, b+d)
(a, b) * (c, d) = (ac - bd, ad + bc)

It is easily verified that C0 has all the required properties of a field. The subfield R x {0} of C0 is isomorphic to R, and the element i = (0, 1) is a root of the equation X2 + 1 = 0.

Thus we have shown that there exists a field with the desired properties. Now we show that C0 is also the smallest such field.

Let F be any other field containing R such that X2 + 1 = 0 has a root in F. Take such a root and call it i1. Let C1 be the subfield {a + i1b: a, b ∈ R} of F. The representation a + i1b of the elements of C1 is unique, for

a + i1b = c + i1d ⇒
(a - c)2 = -(b - d)2
a = c and b = d

Therefore the mapping f: C0 -> C1, f(a, b) = a + i1b is bijective. f preserves the field structure, so f is an isomorphism.

Thus any field containing R and a root of X2 + 1 = 0 also contains (a subfield isomorphic to) C0. C0 is therefore the smallest such field, so we may take C = C0.


Through the proof we have also learnt about the structure of C, which allows us to make some more definitions.

Definition:

Every z ∈ C has a unique representation a + ib where a, b ∈ R.
We call a, b the real and imaginary parts of z, and denote them by Re z, Im z respectively.
The complex conjugate of z is z* = a - ib.
The modulus of z is |z| = sqrt(a2 + b2).


With these elementary definitions we are in a position to build up a theory of complex numbers. In particular we can develop complex analysis, which probably provides the easiest proof of the fundamental theorem of algebra, i.e. that C is algebraically closed. That way C is shown to be the algebraic closure of R.

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