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Complex numbers are defined as ordered pairs of reals:

Such pairs can be added and multiplied as follows
- addition:

- multiplication:

with the addition and multiplication is the field of complex numbers. From another of view,
with complex additions and multiplication by real numbers is a 2-dimesional vector space.
To perform basic computations it is convenient to introduce the imaginary unit, i=(0,1).[1] It has the property
Any complex number
can be written as
(this is often called the algebraic form) and vice-versa. The numbers a and b are called the real part and the imaginary part of z, respectively. We denote
and
Notice that i makes the multiplication quite natural:

The square root of number in the denominator in the above formula is called the modulus of z and denoted by
,

We have for any two complex numbers
and

provided 

For
we define also
, the conjugate, by
Then we have


provided 

- Geometric interpretation
Complex numbers may be naturally represented on the complex plane, where
corresponds to the point (x,y), see the fig. 1.
Fig. 1. Graphical representation of a complex number and its conjugate
Obviously, the conjugation is just the symmetry with respect to the x-axis.
- Trigonometric form
As the graphical representation suggests, any complex number z=a+bi of modulus 1 (i.e. a point from the unit circle) can be written as
for some
So actually any (non-null)
can be represented as
where r traditionally stands for |z|.
This is the trigonometric form of the complex number z. If we adopt convention that
then such
is unique and called the argument of z.[2]
Graphically, the number
is the (oriented) angle between the x-axis and the interval containing 0 and z.
- ↑ in some applications it is denoted by j as well.
- ↑ In literature the convention
is found as well.