Complex Numbers were created because mathematicians were getting annoyed at equations like
(which of course has no solution in the reals). They are an extension on the real numbers which includes a new unit called the 'imaginary unit'. Mathematicians define this 'imaginary unit' as
.
All complex numbers (
) take the form:
, where i is this imaginary unit i discussed earlier.
ok, the main operations to know are:
Real part of a Complex Number:Say you had a complex number
, the REAL component of this is the part of z which is not proceeded by an I, so you can say:
 = 3)
More generally: if
, then
 = a )
Imaginary Part of a Complex Number:The imaginary component of a complex number is the bit which is proceeded by an i (which is why it is called the imaginary part):
If
, then:
 = 66)
More generally: if
, then:
 = b)
Complex Conjugate:The complex conjugate of a complex number is given by changing the sign of the imaginary part. Graphically it represents a reflection in the
)
axis (but thats on argand diagrams, so you might not of seen them).
If you have:
then the complex conjugate of z is:
(just change the negative to positive, or positive to negative)
Note that this little bar:
indicates a complex conjugate and is achieved in latex by typing \overline{z}
Addition:
 + (c + di) = (a + c) + (b + d)i)
(i.e. to add two complex numbers, you add the real & imaginary bits of each number)
Subtraction:
 - (c + di) = (a - c) + (b - d)i)
(as you would expect)
Multiplication::
Multiplication is much like expanding brackets (it IS expanding brackets) but you have to remember to simplify numbers etc:
(c + di) = ac + adi + bci + bdi^2 = ac + (ad + bc)i + bd(-1) \mbox{ (as $i^2 = -1$)}\\<br />\implies (a + bi)(c + di) = (ac - bd) + (ad + bc)i)
Division:
Division is weird, so it requires a bit of thought. I'll give you an example just straight up:
Say you wanted to find:
The first step is to multiply both top and bottom lines by the complex conjugate of the bottom line. This is done because if you multiply a complex number with its conjugate, you generally get a real number, which simplifies the process immensely.
(1 - 5i)}{(1 + 5i)(1 - 5i)} = \dfrac{3 - 15i + 2i - 10i^2}{1 - 5i + 5i - 25i^2} = \dfrac{13 - 13i}{1 + 25} = \dfrac{13 - 13i}{26} = \dfrac{1}{2} - \dfrac{1}{2}i)
In General:
(c - di)}{c^2 + d^2} = \dfrac{ac - adi + bci + bd}{c^2 + d^2} = \dfrac{(ac + bd) + (bc - ad)i}{c^2 + d^2})
That should be all you need for complex numbers.
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