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Complex
There are some polynomial
equations
with real
coefficients
that don't have real solutions. Examples of these are 
, 
. Mathematically we express this by saying that
is not an algebraically closed
field.
In order to solve that kind of equation, we have to “extend” our number
system
by adjoining a number 
that has the property
that 
. In this way we extend the field of real numbers
to a field
whose elements are called complex numbers. A formal construction can be seen at [complex numbers] (cf. the field adjunction). The field
is algebraically closed: every polynomial with complex coefficients, and especially every polynomial with real coefficients, (and with positive
degree) has at least one complex zero (which might be real as well).
Any complex number can be written as 
(with
). Here we call 
the real part of 
and 
the imaginary part of .
We write this as
Re
Im
can be written also as 
. Thus, a complex number is real if and only if its imaginary part is equal to zero.
By writing 
as 
we can also look at complex numbers as ordered pairs. With this notation, real numbers are the pairs of the form 
.
The rules of addition
and multiplication
for complex numbers are:
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).
We have also the negatives:
and the multiplicative inverses:
Seeing complex numbers as ordered pairs also let us give
the structure
of vector space
(over
). The norm of is defined as
where
is the conjugate of and it's defined as
. Thus we can also characterize real numbers as those complex numbers such that
.
Conjugation
obeys the following rules:
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The real and imaginary parts of a complex number may be expressed with the conjugate as
Im
The ordered-pair notation lets us visualize complex numbers as points
in the plane; this is called the complex plane, often also the -plane. As well, we can also describe complex numbers with polar coordinates.
If 
is represented in polar coordinates as 
we call the modulus of and 
its argument.
If
, then
and
. So we have the following expression, called the polar form of complex number :
Multiplication of complex numbers can be done in a very neat way using polar coordinates:
Remark. The adjective complex qualifying such nouns as “number”, “root” and “solution” is in the English language ambiguous; it may mean that it is a question of a element belonging to either
or to
, i.e. the word complex may either have its basic sense or mean `non-real'.