There are some polynomial equations with real coefficientsMathworldPlanetmath that don’t have real solutions.  Examples of these are  x2+5=0,  x2+x+1=0.  Mathematically we express this by saying that is not an algebraically closed field.

In to solve that kind of equation, we have to “extend” our number system by adjoining a number i that has the property that  i2=-1.  In this way we extend the field of real numbers to a field whose elements are called .  A formal construction can be seen at [ numbersMathworldPlanetmathPlanetmath] (cf. the field adjunction).  The field is algebraically closedMathworldPlanetmath: every polynomialMathworldPlanetmathPlanetmath 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+iy (with x,y). Here we call x the real part of z and y the imaginary part of z.  We write this as


Real numbers are a subset of complex numbers, and a real number r can be written also as r+i0.  Thus, a complex number is real if and only if its imaginary part is equal to zero.

By writing x+iy 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+ib)+(x+iy)=(a+x)+i(b+y) (a,b)+(x,y)=(a+x,b+y)
(a+ib)(x+iy)=(ax-by)+i(ay+bx) (a,b)(x,y)=(ax-by,ay+bx)

(to see why the last identityPlanetmathPlanetmathPlanetmathPlanetmath holds, expand the first productMathworldPlanetmath and then simplify by using  i2=-1).

We have also the negatives (  -(a,b)=(-a,-b)  and the multiplicative inversesMathworldPlanetmath:


Seeing complex numbers as ordered pairs also let us give the structure of vector space (over ).  The norm of  z=x+iy  is defined as


Then we have  |z|2=zz¯  where z¯ is the conjugate of z=x+iy and it’s defined as  z¯=x-iy.  Thus we can also characterize real numbers as those complex numbers z such that z=z¯.

ConjugationMathworldPlanetmath obeys the following rules:

z1+z2¯ = z1¯+z2¯
z1z2¯ = z1¯z2¯
z¯¯ = z

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


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+ib  is represented in polar coordinates as  (r,t)  we call r the of z and t its argument.

If  r=a+ib=(r,t),  then  a=rsint  and  b=rcost.  So we have the following expression, called the polar form of complex number z:


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 ambiguous; it may mean that it is a question of a element belonging to either or to , 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