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 $\mathbb{R}$ is not an algebraically closed field.

In to solve that kind of equation, we have to “extend” our number system $\mathbb{R}$ by adjoining a number $i$ that has the property that $i^{2}=-1$ . In this way we extend the field of real numbers $\mathbb{R}$ to a field $\mathbb{C}$ whose elements are called . A formal construction can be seen at [http://planetmath.org/node/471complex numbers] (cf. the field adjunction). The field $\mathbb{C}$ 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 $z=x+iy$ (with $x,\,y\in\mathbb{R}$ ). Here we call $x$ the real part of $z$ and $y$ the imaginary part of $z$ . We write this as

x=\mbox{Re}(z),\qquad y=\mbox{Im}(z).

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:

	$\displaystyle(a+ib)+(x+iy)=(a+x)+i(b+y)$		$\displaystyle(a,\,b)+(x,\,y)=(a+x,\,b+y)$
	$\displaystyle(a+ib)\cdot(x+iy)=(ax-by)+i(ay+bx)$		$\displaystyle(a,\,b)\cdot(x,\,y)=(ax-by,\,ay+bx)$

(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}=\left(\frac{a}{a^{2}+b^{2}},\,\frac{-b}{a^{2}+b^{2}}\right).

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

|z|=\sqrt{x^{2}+y^{2}}.

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

Conjugation obeys the following rules:

$\displaystyle\overline{z_{1}+z_{2}}$	$\displaystyle=$	$\displaystyle\overline{z_{1}}+\overline{z_{2}}$
$\displaystyle\overline{z_{1}z_{2}}$	$\displaystyle=$	$\displaystyle\overline{z_{1}}\,\overline{z_{2}}$
$\displaystyle\overline{\overline{z}}$	$\displaystyle=$	$\displaystyle z$

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

\mbox{Re}(z)=\frac{z+\overline{z}}{2},\quad\mbox{Im}(z)=\frac{z-\overline{z}}{% 2i}.

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 $a=r\sin{t}$ and $b=r\cos{t}$ . So we have the following expression, called the polar form of complex number $z$ :

z=a+ib=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 $\mathbb{C}$ or to $\mathbb{C}\!\smallsetminus\!\mathbb{R}$ , 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