complex

complex

There are some

polynomial equations

with real

coefficients

that don’t have real

solutions

.  Examples of these are

x

2

+

5

=

0

,

x

2

+

x

+

1

=

0

.  Mathematically we express this by saying that

ℝ

is not an algebraically closed field.

In to solve that kind of equation, we have to “extend” our number system

ℝ

by adjoining a number

i

that has the property that

i

2

=

-

1

.  In this way we extend the

field of real

numbers

ℝ

to a field

ℂ

whose elements are called

.  A formal construction can be seen at [

http://planetmath.org/node/471

complex numbers

] (cf. the

field adjunction

).  The field

ℂ

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

z

=

x

+

i

⁢

y

(with

x

,

y

∈

ℝ

). Here we call

x

the

real part

of

z

and

y

the

imaginary part

of

z

.
We write this as

x

=

Re

⁢

(

z

)

,

y

=

Im

⁢

(

z

)

.

Real numbers

are a subset of complex numbers, and a real number

r

can be written also as

r

+

i

⁢

0

.  Thus, a complex number is real if and only if its imaginary part is equal to zero.

By writing

x

+

i

⁢

y

as

(

x

,

y

)

we can also look at complex numbers as ordered pairs.  With this notation, real numbers are the pairs of the form

(

r

,

0

)

.

The rules of addition and multiplication for complex numbers are:

(

a

+

i

⁢

b

)

+

(

x

+

i

⁢

y

)

=

(

a

+

x

)

+

i

⁢

(

b

+

y

)

(

a

,

b

)

+

(

x

,

y

)

=

(

a

+

x

,

b

+

y

)

(

a

+

i

⁢

b

)

⋅

(

x

+

i

⁢

y

)

=

(

a

⁢

x

-

b

⁢

y

)

+

i

⁢

(

a

⁢

y

+

b

⁢

x

)

(

a

,

b

)

⋅

(

x

,

y

)

=

(

a

⁢

x

-

b

⁢

y

,

a

⁢

y

+

b

⁢

x

)

(to see why the last

identity

holds, expand the first

product

and then simplify
by using

i

2

=

-

1

).

We have also the negatives (

http://planetmath.org/OppositeNumber

):

-

(

a

,

b

)

=

(

-

a

,

-

b

)

and the

multiplicative inverses

:

(

a

,

b

)

-

1

=

(

a

a

2

+

b

2

,

-

b

a

2

+

b

2

)

.

Seeing complex numbers as ordered pairs also let us give

ℂ

the structure of

vector space

(over

ℝ

).  The

norm

of

z

=

x

+

i

⁢

y

is defined as

|

z

|

=

x

2

+

y

2

.

Then we have

|

z

|

2

=

z

⁢

z

¯

where

z

¯

is the

conjugate

of

z

=

x

+

i

⁢

y

and it’s defined as

z

¯

=

x

-

i

⁢

y

.  Thus we can also characterize real numbers as those complex numbers

z

such that

z

=

z

¯

.

Conjugation

obeys the following rules:

z

1

+

z

2

¯

=

z

1

¯

+

z

2

¯

z

1

⁢

z

2

¯

=

z

1

¯

⁢

z

2

¯

z

¯

¯

=

z

The real and imaginary parts of a complex number may be expressed with the conjugate as

Re

⁢

(

z

)

=

z

+

z

¯

2

,

Im

⁢

(

z

)

=

z

-

z

¯

2

⁢

i

.

The ordered-pair notation lets us visualize complex numbers as points in the plane; this is called the

complex plane

, often also the

z

-plane

.  As well, we can also describe complex numbers with polar coordinates.

Using this

representation

, we see that the real numbers are located at the abscissa (horizontal) axis, which is then known as the

real axis

.  The ordinate (vertical) axis is known as the

imaginary axis

, since it consists of all complex numbers with real part equal to zero.

If

z

=

a

+

i

⁢

b

is represented in polar coordinates as

(

r

,

t

)

we call

r

the of

z

and

t

its

argument

.

If

r

=

a

+

i

⁢

b

=

(

r

,

t

)

,  then

a

=

r

⁢

sin

⁡

t

and

b

=

r

⁢

cos

⁡

t

.  So we have the following expression, called the

polar form

of complex number

z

:

z

=

a

+

i

⁢

b

=

r

⁢

(

cos

⁡

t

+

i

⁢

sin

⁡

t

)

Multiplication of complex numbers can be done in a very neat way using polar coordinates:

(

r

1

,

t

1

)

⁢

(

r

2

,

t

2

)

=

(

r

1

⁢

r

2

,

t

1

+

t

2

)

.

Remark.

The adjective

complex

qualifying such nouns as “number”, “root” and “solution” is in the English ambiguous; it may mean that it is a question of a element belonging to either

ℂ

or to

ℂ

∖

ℝ

, i.e. the

complex

may either have its basic sense or mean ‘non-real’.

Title

complex

Canonical name

Complex

Date of creation

2013-03-22 11:57:12

Last modified on

2013-03-22 11:57:12

Owner

drini (3)

Last modified by

drini (3)

Numerical id

43

Author

drini (3)

Entry type

Definition

Classification

msc 12D99

Classification

msc 30-00

Synonym

complex number

Related topic

Polynomial

Related topic

ArgandDiagram

Related topic

RealNumber

Related topic

ComplexNumber

Related topic

ComplexConjugate

Related topic

NthRoot

Related topic

RiemannZetaFunction

Related topic

Imaginary

Related topic

ImaginaryUnit

Related topic

Region

Related topic

UnitDisk

Related topic

UpperHalfPlane

Related topic

ZeroesOfAnalyticFunctionsAreIsolated

Related topic

RiemannSphere

Related topic

SquareRoot

Related topic

CardanosFormulae

Related topic

Fundamenta

Defines

complex plane

Defines

z-plane

Defines

real axis

Defines

imaginary axis

Defines

real part

Defines

imaginary part

Defines

conjugate

Defines

argument

Defines

polar form

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