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A number of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c0241401.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c0241402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c0241403.png" /> are real numbers (cf. [[Real number|Real number]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c0241404.png" /> is the so-called imaginary unit, that is, a number whose square is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c0241405.png" /> (in engineering literature, the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c0241406.png" /> is also used): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c0241407.png" /> is called the real part of the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c0241408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c0241409.png" /> its imaginary part (written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414011.png" />). The real numbers can be regarded as special complex numbers, namely those with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414012.png" />. Complex numbers that are not real, that is, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414013.png" />, are sometimes called imaginary numbers. The complicated historical process of the development of the notion of a complex number is reflected in the above terminology which is mainly of traditional origin.
+
A number of the form $z=x+iy$, where $x$ and $y$ are real numbers (cf.
 +
[[Real number|Real number]]) and $i=\def\i{\sqrt{-1}}\i$ is the so-called imaginary unit,
 +
that is, a number whose square is equal to $-1$ (in engineering
 +
literature, the notation $j=\i$ is also used): $x$ is called the real
 +
part of the complex number $z$ and $ y$ its imaginary part (written
 +
$x=\def\Re{\mathrm{Re}\;}\Re z$, $y=\def\Im{\mathrm{Im}\;}\Im z$). The real numbers can be regarded as special complex
 +
numbers, namely those with $y=0$. Complex numbers that are not real,
 +
that is, for which $y\ne 0$, are sometimes called imaginary numbers. The
 +
complicated historical process of the development of the notion of a
 +
complex number is reflected in the above terminology which is mainly
 +
of traditional origin.
  
Algebraically speaking, a complex number is an element of the (algebraic) extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414014.png" /> of the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414015.png" /> obtained by the adjunction to the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414016.png" /> of a root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414017.png" /> of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414018.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414019.png" /> obtained in this way is called the field of complex numbers or the complex number field. The most important property of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414020.png" /> is that it is algebraically closed, that is, any polynomial with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414021.png" /> splits into linear factors. The property of being algebraically closed can be expressed in other words by saying that any polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414022.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414023.png" /> has at least one root in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414024.png" /> (the d'Alembert–Gauss theorem or fundamental theorem of algebra).
+
Algebraically speaking, a complex number is an element of the
 +
(algebraic) extension $\C$ of the field of real numbers $\R$ obtained by
 +
the adjunction to the field $\R$ of a root $i$ of the polynomial
 +
$X^2+1$. The field $\C$ obtained in this way is called the field of complex
 +
numbers or the complex number field. The most important property of
 +
the field $\C$ is that it is algebraically closed, that is, any
 +
polynomial with coefficients in $\C$ splits into linear factors. The
 +
property of being algebraically closed can be expressed in other words
 +
by saying that any polynomial of degree $n\ge 1$ with coefficients in $\C$
 +
has at least one root in $\C$ (the d'Alembert–Gauss theorem or
 +
fundamental theorem of algebra).
  
The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414025.png" /> can be constructed as follows. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414027.png" /> or complex numbers, are taken to be the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414029.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414030.png" /> in Cartesian rectangular coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414034.png" />. Here the sum of two complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414036.png" /> is the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414037.png" />, that is,
+
The field $\C$ can be constructed as follows. The elements $z=(x,y)$, $z'=(x',y'),\dots$ or
 +
complex numbers, are taken to be the points $z=(x,y)$, $z'=(x',y'),\dots$ of the plane $\R^2$
 +
in Cartesian rectangular coordinates $x$ and $y$, $x'$ and $y',\dots$. Here
 +
the sum of two complex numbers $z=(x,y)$ and $z'=(x',y')$ is the complex number $(x+x',y+y')$,
 +
that is,
 +
$$z+z'=(x,y)+(x',y')=(x+x',y+y'),\label{1}$$
 +
and the product of those complex numbers is the complex
 +
number $(xx'-yy',xy'+x'y)$, that is,
 +
$$zz'=(x,y)(x'y') = (xx'-yy',xy'+x'y).\label{2}$$
 +
The zero element $0=(0,0)$ is the same as the
 +
origin of coordinates, and the complex number $(1,0)$ is the identity of
 +
$\C$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
The plane $\R^2$ whose points are identified with the elements of $\C$ is
 +
called the complex plane. The real numbers $x,x',\dots$ are identified here
 +
with the points $(x,0)$, $(x',0),\dots$ of the $x$-axis which, when referring to the
 +
complex plane, is called the real axis. The points $(0,y)=iy$, $(0,y')=iy',\dots$ are
 +
situated on the $y$-axis, called the imaginary axis of the complex
 +
plane $\C$; numbers of the form $iy,iy',\dots$ are called pure imaginary. The
 +
representation of elements $z,z',\dots$ of $\C$, or complex numbers, as points
 +
of the complex plane with the rules (1) and (2) is equivalent to the
 +
above more widely used form of notating complex numbers:  
 +
$$z=(x,y)=x+iy, z'=(x',y')=x'+iy',\dots,$$
 +
also
 +
called the algebraic or Cartesian form of writing complex
 +
numbers. With reference to the algebraic form, the rules (1) and (2)
 +
reduce to the simple condition that all operations with complex
 +
numbers are carried out as for polynomials, taking into account the
 +
property of the imaginary unit: $ii=i^2=-1$.
  
and the product of those complex numbers is the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414039.png" />, that is,
+
The complex numbers $z=(x,y)=x+iy$ and $\bar z=(x,-y)=x-iy$ are called conjugate or complex
 +
conjugates in the plane $\C$; they are symmetrically situated with
 +
respect to the real axis. The sum and the product of two conjugate
 +
complex numbers are the real numbers
 +
$$z+\bar z = 2\Re z,\quad z\bar z=|z|^2,$$
 +
where $|z|=r=\sqrt{x^2+y^2}$ is called the
 +
modulus or absolute value of $z$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
The following inequalities always hold:  
 +
$$|z|-|z'| \le |z+z'|\le |z|+|z'|.$$
 +
A complex number $z$ is
 +
different from 0 if and only if $|z|>0$. The mapping $z\mapsto \bar z$ is an
 +
automorphism of the complex plane of order 2 (that is, $z = \bar{\bar z}$) that
 +
leaves all points of the real axis fixed. Furthermore, $\overline{z+z'} = \bar z + \bar{z'}$, $\bar{zz'} = \bar{z}\bar{z'}$.
  
The zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414041.png" /> is the same as the origin of coordinates, and the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414042.png" /> is the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414043.png" />.
+
The operations of addition and multiplication (1) and (2) are
 +
commutative and associative, they are related by the distributive law,
 +
and they have the inverse operations subtraction and division (except
 +
for division by zero). The latter are expressed in algebraic form as:
  
The plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414044.png" /> whose points are identified with the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414045.png" /> is called the complex plane. The real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414046.png" /> are identified here with the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414048.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414049.png" />-axis which, when referring to the complex plane, is called the real axis. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414051.png" /> are situated on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414052.png" />-axis, called the imaginary axis of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414053.png" />; numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414054.png" /> are called pure imaginary. The representation of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414055.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414056.png" />, or complex numbers, as points of the complex plane with the rules (1) and (2) is equivalent to the above more widely used form of notating complex numbers:
+
$$z-z'=(x+iy)-(x'+iy')=(x-x')+i(y-y'),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414057.png" /></td> </tr></table>
+
$$\frac{z'}{z} = \frac{x'+iy'}{x+iy} = \frac{z\bar z}{|z|^2}
 +
=\frac{xx'+yy'}{x^2+y^2}+i\frac{y'x-x'y}{x^2+y^2},\quad z\ne0.\label{3}$$
 +
Division of a complex number $z'$ by a complex number $z\ne0$ thus
 +
reduces to multiplication of $z'$ by
 +
$$\frac{\bar z}{|z|^2} = \frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}.$$
 +
It is an important question
 +
whether the extension $\C$ of the field of reals constructed above,
 +
with the rules of operation indicated, is the only possible one or
 +
whether essentially different variants are conceivable. The answer is
 +
given by the uniqueness theorem: Every (algebraic) extension of the
 +
field $\R$ obtained from $\R$ by adjoining a root $i$ of the equation
 +
$X^2+1$ is isomorphic to $\C$, that is, only the above rules of operation
 +
with complex numbers are compatible with the requirement that the root
 +
$i$ be algebraically adjoined. This fact, however, does not exclude
 +
the existence of interpretations of complex numbers other than as
 +
points of the complex plane. The following two interpretations are
 +
most frequently employed in applications.
  
also called the algebraic or Cartesian form of writing complex numbers. With reference to the algebraic form, the rules (1) and (2) reduce to the simple condition that all operations with complex numbers are carried out as for polynomials, taking into account the property of the imaginary unit: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414058.png" />.
+
Vector interpretation. A complex number $z=x+iy$ can be identified with the
 +
vector $(x,y)$ with coordinates $x$ and $y$ starting from the origin (see
 +
Fig.).
  
The complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414060.png" /> are called conjugate or complex conjugates in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414061.png" />; they are symmetrically situated with respect to the real axis. The sum and the product of two conjugate complex numbers are the real numbers
+
<img style="border:1px solid;"
 
+
src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024140a.gif" />
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414062.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414063.png" /> is called the modulus or absolute value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414064.png" />.
 
 
 
The following inequalities always hold:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414065.png" /></td> </tr></table>
 
 
 
A complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414066.png" /> is different from 0 if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414067.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414068.png" /> is an automorphism of the complex plane of order 2 (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414069.png" />) that leaves all points of the real axis fixed. Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414071.png" />.
 
 
 
The operations of addition and multiplication (1) and (2) are commutative and associative, they are related by the distributive law, and they have the inverse operations subtraction and division (except for division by zero). The latter are expressed in algebraic form as:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414072.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414073.png" /></td> </tr></table>
 
 
 
Division of a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414074.png" /> by a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414075.png" /> thus reduces to multiplication of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414076.png" /> by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414077.png" /></td> </tr></table>
 
 
 
It is an important question whether the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414078.png" /> of the field of reals constructed above, with the rules of operation indicated, is the only possible one or whether essentially different variants are conceivable. The answer is given by the uniqueness theorem: Every (algebraic) extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414079.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414080.png" /> by adjoining a root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414081.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414082.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414083.png" />, that is, only the above rules of operation with complex numbers are compatible with the requirement that the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414084.png" /> be algebraically adjoined. This fact, however, does not exclude the existence of interpretations of complex numbers other than as points of the complex plane. The following two interpretations are most frequently employed in applications.
 
 
 
Vector interpretation. A complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414085.png" /> can be identified with the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414086.png" /> with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414088.png" /> starting from the origin (see Fig.).
 
 
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024140a.gif" />
 
  
 
Figure: c024140a
 
Figure: c024140a
  
In this interpretation, addition and subtraction of complex numbers is carried out according to the rules of addition and subtraction of vectors. However, multiplication and division of complex numbers, which must be performed according to (2) and (3), do not have immediate analogues in vector algebra (see [[#References|[4]]], [[#References|[5]]]). The vector interpretation of complex numbers is immediately applicable, for example, in electrical engineering in the description of alternating sinusoidal currents and voltages.
+
In this interpretation, addition and subtraction of complex numbers is
 
+
carried out according to the rules of addition and subtraction of
Matrix interpretation. The complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414089.png" /> can be identified with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414090.png" />-matrix of special type
+
vectors. However, multiplication and division of complex numbers,
 
+
which must be performed according to (2) and (3), do not have
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414091.png" /></td> </tr></table>
+
immediate analogues in vector algebra (see
 
+
[[#References|[4]]],
where the operations of addition, subtraction and multiplication are carried out according to the usual rules of matrix algebra.
+
[[#References|[5]]]). The vector interpretation of complex numbers is
 
+
immediately applicable, for example, in electrical engineering in the
By using polar coordinates in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414092.png" />, that is, the radius vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414093.png" /> and polar angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414094.png" />, called here the argument of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414095.png" /> (sometimes also called the phase of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414096.png" />), one obtains the trigonometric or polar form of a complex number:
+
description of alternating sinusoidal currents and voltages.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414097.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414098.png" /></td> </tr></table>
 
 
 
The argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c02414099.png" /> is a many-valued real-valued function of the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140100.png" />, whose values for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140101.png" /> differ by integral multiples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140102.png" />; the argument of the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140103.png" /> is not defined. One usually takes the principal value of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140104.png" />, defined by the additional condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140105.png" />. The [[Euler formulas|Euler formulas]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140106.png" /> transform the trigonometric form (4) into the exponential form of a complex number:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
 
 
 
The forms (4) and (5) are particularly suitable for carrying out multiplication and division of complex numbers:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140108.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140109.png" /></td> </tr></table>
 
 
 
Under multiplication (or division) of complex numbers the moduli are multiplied (or divided) and the arguments are added (or subtracted). Raising to a power or extracting a root is carried out according to the so-called de Moivre formulas:
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140110.png" /></td> </tr></table>
+
Matrix interpretation. The complex number $w=u+iv$ can be identified with a
 +
$(2\times 2)$-matrix of special type
 +
$$w=\begin{pmatrix}\phantom{-}u&v\\ -v&u\end{pmatrix}$$
 +
where the operations of addition,
 +
subtraction and multiplication are carried out according to the usual
 +
rules of matrix algebra.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140111.png" /></td> </tr></table>
+
By using polar coordinates in the complex plane $\C$, that is, the
 +
radius vector $r=|z|$ and polar angle $\def\phi{\varphi}\phi=\arg z$, called here the argument of $z$
 +
(sometimes also called the phase of $z$), one obtains the
 +
trigonometric or polar form of a complex number:  
 +
$$z=r(\cos\phi + i\sin\phi),\label{4}$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140112.png" /></td> </tr></table>
+
$$r\cos\phi = \Re z,\quad r\sin\phi=\Im z.$$
 +
The argument $\phi=\arg z$ is a many-valued real-valued function of the
 +
complex number $z\ne 0z\ne 0$, whose values for a given $z$ differ by integral
 +
multiples of $2\pi i$; the argument of the complex number $z=0$ is not
 +
defined. One usually takes the principal value of the argument $\phi = \def\Arg{\mathrm{Arg}} \Arg z$,
 +
defined by the additional condition $-\pi < \Arg z \le pi$. The
 +
[[Euler formulas|Euler formulas]] $e^{\pm i\phi} = \cos\phi\pm i\sin\phi$ transform the trigonometric form
 +
(4) into the exponential form of a complex number:  
 +
$$z=re^{i\phi}\label{5}$$
 +
The forms (4)
 +
and (5) are particularly suitable for carrying out multiplication and
 +
division of complex numbers:
 +
$$zz'=rr'[\cos(\phi+\phi')+i\sin(\phi+\phi')]=rr'e^{i(\phi+\phi')},$$
  
where the first of these is also applicable for negative integer exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140113.png" />. Geometrically, multiplication of a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140114.png" /> by a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140115.png" /> reduces to rotating the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140116.png" /> over the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140117.png" /> (anti-clockwise if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140118.png" />) and subsequently multiplying its length by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140119.png" />; in particular, multiplication by a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140120.png" />, which has modulus one, is merely rotation over the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140121.png" />. Thus, complex numbers can be interpreted as operators of a special type (affinors, cf. [[Affinor|Affinor]]). In this connection, the mixed vector-matrix interpretation of multiplication of complex numbers is sometimes useful:
+
$$\frac{z}{z'}=\frac{r}{r'}[\cos(\phi-\phi')+i\sin(\phi-\phi')]
 +
=\frac{r}{r'}e^{i(\phi-\phi')},\quad r>0$$
 +
Under multiplication (or division) of complex numbers the moduli
 +
are multiplied (or divided) and the arguments are added (or
 +
subtracted). Raising to a power or extracting a root is carried out
 +
according to the so-called de Moivre formulas:
 +
$$z^n = r^n(\cos n\phi + i\sin n\phi) = r^n e^{in\phi},$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140122.png" /></td> </tr></table>
+
$$z^{1/n} = r^{1/n}\Big(\cos\frac{\phi+2k\pi}{n}+i\sin\frac{\phi+2k\pi}{n}\Big)
 +
=r^{1/n}e^{i(\phi+2k\pi)/n},$$
  
in which the multiplicand is treated as a matrix-vector and the multiplier as a matrix-operator.
+
$$k=0,\dots,n-1,$$
 +
where the first of these is also applicable for negative integer
 +
exponents $n$. Geometrically, multiplication of a complex number $z$
 +
by a complex number $z'=r'e^{i\phi'}$ reduces to rotating the vector $z$ over the
 +
angle $\phi'$ (anti-clockwise if $\phi'>0$) and subsequently multiplying its
 +
length by $|z'|=r'$; in particular, multiplication by a complex number $z'=e^{i\phi'}$,
 +
which has modulus one, is merely rotation over the angle $\phi'$. Thus,
 +
complex numbers can be interpreted as operators of a special type
 +
(affinors, cf.
 +
[[Affinor|Affinor]]). In this connection, the mixed vector-matrix
 +
interpretation of multiplication of complex numbers is sometimes
 +
useful:
 +
$$(x,y)\begin{pmatrix}\phantom{-}u&v\\ -v&u\end{pmatrix}=(xu-yv,xv+yu),$$
 +
in which the multiplicand is treated as a matrix-vector
 +
and the multiplier as a matrix-operator.
  
The bijection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140123.png" /> induces on the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140124.png" /> the topology of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140125.png" />-dimensional real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140126.png" />; this topology is compatible with the field structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140127.png" /> and thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140128.png" /> is a topological field. The modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140129.png" /> is the Euclidean norm of the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140130.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140131.png" /> endowed with this norm is a complex one-dimensional Euclidean space, also called the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140133.png" />-plane. The topological product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140134.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140135.png" /> times, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140136.png" />) is a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140137.png" />-dimensional Euclidean space. For a satisfactory analysis of functions it is usually necessary to consider their behaviour in the complex domain. This is due to the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140138.png" /> is algebraically closed. Even the behaviour of such elementary functions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140139.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140142.png" /> can be properly understood only when they are regarded as functions of a complex variable (see [[Analytic function|Analytic function]]).
+
The bijection $(x,y)\mapsto x+iy$ induces on the field $\C$ the topology of the
 +
$2$-dimensional real vector space $\R^2$; this topology is compatible
 +
with the field structure of $\C$ and thus $\C$ is a topological
 +
field. The modulus $|z|$ is the Euclidean norm of the complex number
 +
$z={x,y}$, and $\C$ endowed with this norm is a complex one-dimensional
 +
Euclidean space, also called the complex $z$-plane. The topological
 +
product $\C^n=\C\times\cdots\times\C$ ($n$ times, $n\ge 1$) is a complex $n$-dimensional Euclidean
 +
space. For a satisfactory analysis of functions it is usually
 +
necessary to consider their behaviour in the complex domain. This is
 +
due to the fact that $\C$ is algebraically closed. Even the behaviour
 +
of such elementary functions as $z^n$, $\cos z$, $\sin z$, $e^z$ can be properly
 +
understood only when they are regarded as functions of a complex
 +
variable (see
 +
[[Analytic function|Analytic function]]).
  
Apparently, imaginary quantities first occurred in the celebrated work The great art, or the rules of algebra by G. Cardano, 1545, who regarded them as useless and unsuitable for applications. R. Bombelli (1572) was the first to realize the value of the use of imaginary quantities, in particular for the solution of the [[Cubic equation|cubic equation]] in the so-called irreducible case (when the real roots are expressed in terms of cube roots of imaginary quantities, cf. [[Cardano formula|Cardano formula]]). He gave some of the simplest rules of operation with complex numbers. In general, expressions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140144.png" />, appearing in the solution of quadratic and cubic equations were called "imaginary" in the 16th century and 17th century. However, even for many of the great scholars of the 17th century, the algebraic and geometric nature of imaginary quantities was unclear and even mystical. It is known, for example, that I. Newton did not include imaginary quantities within the notion of number, and that G. Leibniz said that "complex numbers are a fine and wonderful refuge of the divine spirit, as if it were an amphibian of existence and non-existence" .
+
Apparently, imaginary quantities first occurred in the celebrated work
 +
The great art, or the rules of algebra by G. Cardano, 1545, who
 +
regarded them as useless and unsuitable for applications. R. Bombelli
 +
(1572) was the first to realize the value of the use of imaginary
 +
quantities, in particular for the solution of the
 +
[[Cubic equation|cubic equation]] in the so-called irreducible case
 +
(when the real roots are expressed in terms of cube roots of imaginary
 +
quantities, cf.
 +
[[Cardano formula|Cardano formula]]). He gave some of the simplest
 +
rules of operation with complex numbers. In general, expressions of
 +
the form $a+b\i$, $b\ne 0$, appearing in the solution of quadratic and cubic
 +
equations were called "imaginary" in the 16th century and 17th
 +
century. However, even for many of the great scholars of the 17th
 +
century, the algebraic and geometric nature of imaginary quantities
 +
was unclear and even mystical. It is known, for example, that
 +
I. Newton did not include imaginary quantities within the notion of
 +
number, and that G. Leibniz said that "complex numbers are a fine and
 +
wonderful refuge of the divine spirit, as if it were an amphibian of
 +
existence and non-existence" .
  
The problem of expressing the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140145.png" />-th roots of a given number was mainly solved in the papers of A. de Moivre (1707, 1724) and R. Cotes (1722). The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140146.png" /> was proposed by L. Euler (1777, published 1794). It was he who in 1751 asserted that the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024140/c024140147.png" /> is algebraically closed; J. d'Alembert (1747) came to the same conclusion. The first rigorous proof of this fact is due to C.F. Gauss (1799), who introduced the term "complex number" in 1831. The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the "Argand diagramArgand diagram" , came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel.
+
The problem of expressing the $n$-th roots of a given number was
 +
mainly solved in the papers of A. de Moivre (1707, 1724) and R. Cotes
 +
(1722). The symbol $i=\i$ was proposed by L. Euler (1777, published
 +
1794). It was he who in 1751 asserted that the field $\C$ is
 +
algebraically closed; J. d'Alembert (1747) came to the same
 +
conclusion. The first rigorous proof of this fact is due to C.F. Gauss
 +
(1799), who introduced the term "complex number" in 1831. The complete
 +
geometric interpretation of complex numbers and operations on them
 +
appeared first in the work of C. Wessel (1799). The geometric
 +
representation of complex numbers, sometimes called the "Argand
 +
diagramArgand diagram" , came into use after the publication in 1806
 +
and 1814 of papers by J.R. Argand, who rediscovered, largely
 +
independently, the findings of Wessel.
  
The purely arithmetic theory of complex numbers as pairs of real numbers was introduced by W. Hamilton (1837). He found a generalization of complex numbers, namely the quaternions (cf. [[Quaternion|Quaternion]]), which form a non-commutative algebra. More generally, it was proved at the end of the 19th century that any extension of the notion of number beyond the complex numbers requires sacrificing some property of the usual operations (primarily commutativity). See also [[Hypercomplex number|Hypercomplex number]]; [[Double and dual numbers|Double and dual numbers]]; [[Cayley numbers|Cayley numbers]].
+
The purely arithmetic theory of complex numbers as pairs of real
 +
numbers was introduced by W. Hamilton (1837). He found a
 +
generalization of complex numbers, namely the quaternions (cf.
 +
[[Quaternion|Quaternion]]), which form a non-commutative algebra. More
 +
generally, it was proved at the end of the 19th century that any
 +
extension of the notion of number beyond the complex numbers requires
 +
sacrificing some property of the usual operations (primarily
 +
commutativity). See also
 +
[[Hypercomplex number|Hypercomplex number]];
 +
[[Double and dual numbers|Double and dual numbers]];
 +
[[Cayley numbers|Cayley numbers]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh,   "Higher algebra" , MIR (1972) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Kostrikin,   "Introduction to algebra" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.I. Markushevich,   "Theory of functions of a complex variable" , '''1–2''' , Chelsea (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B.V. Shabat,   "Introduction of complex analysis" , '''1''' , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.A. [M.A. Lavrent'ev] Lawrentjew,   B.V. [B.V. Shabat] Schabat,   "Problems in hydrodynamics and their mathematical models" , Moscow (1973) (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A. Hurwitz,   R. Courant,   "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1968)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> G.H. Hardy,   "A course of pure mathematics" , Cambridge Univ. Press (1955)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> N. Bourbaki,   "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD>
 +
<TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[2]</TD>
 +
<TD valign="top"> A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[3]</TD>
 +
<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1–2''' , Chelsea (1977) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[4]</TD>
 +
<TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1''' , Moscow (1976) (In Russian)</TD>
 +
</TR><TR><TD valign="top">[5]</TD>
 +
<TD valign="top"> M.A. [M.A. Lavrent'ev] Lawrentjew, B.V. [B.V. Shabat] Schabat, "Problems in hydrodynamics and their mathematical models" , Moscow (1973) (In Russian)</TD>
 +
</TR><TR><TD valign="top">[6]</TD>
 +
<TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1968)</TD>
 +
</TR><TR><TD valign="top">[7]</TD>
 +
<TD valign="top"> G.H. Hardy, "A course of pure mathematics" , Cambridge Univ. Press (1955)</TD>
 +
</TR><TR><TD valign="top">[8]</TD>
 +
<TD valign="top"> N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)</TD>
 +
</TR></table>

Revision as of 21:57, 22 November 2011

A number of the form $z=x+iy$, where $x$ and $y$ are real numbers (cf. Real number) and $i=\def\i{\sqrt{-1}}\i$ is the so-called imaginary unit, that is, a number whose square is equal to $-1$ (in engineering literature, the notation $j=\i$ is also used): $x$ is called the real part of the complex number $z$ and $ y$ its imaginary part (written $x=\def\Re{\mathrm{Re}\;}\Re z$, $y=\def\Im{\mathrm{Im}\;}\Im z$). The real numbers can be regarded as special complex numbers, namely those with $y=0$. Complex numbers that are not real, that is, for which $y\ne 0$, are sometimes called imaginary numbers. The complicated historical process of the development of the notion of a complex number is reflected in the above terminology which is mainly of traditional origin.

Algebraically speaking, a complex number is an element of the (algebraic) extension $\C$ of the field of real numbers $\R$ obtained by the adjunction to the field $\R$ of a root $i$ of the polynomial $X^2+1$. The field $\C$ obtained in this way is called the field of complex numbers or the complex number field. The most important property of the field $\C$ is that it is algebraically closed, that is, any polynomial with coefficients in $\C$ splits into linear factors. The property of being algebraically closed can be expressed in other words by saying that any polynomial of degree $n\ge 1$ with coefficients in $\C$ has at least one root in $\C$ (the d'Alembert–Gauss theorem or fundamental theorem of algebra).

The field $\C$ can be constructed as follows. The elements $z=(x,y)$, $z'=(x',y'),\dots$ or complex numbers, are taken to be the points $z=(x,y)$, $z'=(x',y'),\dots$ of the plane $\R^2$ in Cartesian rectangular coordinates $x$ and $y$, $x'$ and $y',\dots$. Here the sum of two complex numbers $z=(x,y)$ and $z'=(x',y')$ is the complex number $(x+x',y+y')$, that is, $$z+z'=(x,y)+(x',y')=(x+x',y+y'),\label{1}$$ and the product of those complex numbers is the complex number $(xx'-yy',xy'+x'y)$, that is, $$zz'=(x,y)(x'y') = (xx'-yy',xy'+x'y).\label{2}$$ The zero element $0=(0,0)$ is the same as the origin of coordinates, and the complex number $(1,0)$ is the identity of $\C$.

The plane $\R^2$ whose points are identified with the elements of $\C$ is called the complex plane. The real numbers $x,x',\dots$ are identified here with the points $(x,0)$, $(x',0),\dots$ of the $x$-axis which, when referring to the complex plane, is called the real axis. The points $(0,y)=iy$, $(0,y')=iy',\dots$ are situated on the $y$-axis, called the imaginary axis of the complex plane $\C$; numbers of the form $iy,iy',\dots$ are called pure imaginary. The representation of elements $z,z',\dots$ of $\C$, or complex numbers, as points of the complex plane with the rules (1) and (2) is equivalent to the above more widely used form of notating complex numbers: $$z=(x,y)=x+iy, z'=(x',y')=x'+iy',\dots,$$ also called the algebraic or Cartesian form of writing complex numbers. With reference to the algebraic form, the rules (1) and (2) reduce to the simple condition that all operations with complex numbers are carried out as for polynomials, taking into account the property of the imaginary unit: $ii=i^2=-1$.

The complex numbers $z=(x,y)=x+iy$ and $\bar z=(x,-y)=x-iy$ are called conjugate or complex conjugates in the plane $\C$; they are symmetrically situated with respect to the real axis. The sum and the product of two conjugate complex numbers are the real numbers $$z+\bar z = 2\Re z,\quad z\bar z=|z|^2,$$ where $|z|=r=\sqrt{x^2+y^2}$ is called the modulus or absolute value of $z$.

The following inequalities always hold: $$|z|-|z'| \le |z+z'|\le |z|+|z'|.$$ A complex number $z$ is different from 0 if and only if $|z|>0$. The mapping $z\mapsto \bar z$ is an automorphism of the complex plane of order 2 (that is, $z = \bar{\bar z}$) that leaves all points of the real axis fixed. Furthermore, $\overline{z+z'} = \bar z + \bar{z'}$, $\bar{zz'} = \bar{z}\bar{z'}$.

The operations of addition and multiplication (1) and (2) are commutative and associative, they are related by the distributive law, and they have the inverse operations subtraction and division (except for division by zero). The latter are expressed in algebraic form as:

$$z-z'=(x+iy)-(x'+iy')=(x-x')+i(y-y'),$$

$$\frac{z'}{z} = \frac{x'+iy'}{x+iy} = \frac{z\bar z}{|z|^2} =\frac{xx'+yy'}{x^2+y^2}+i\frac{y'x-x'y}{x^2+y^2},\quad z\ne0.\label{3}$$ Division of a complex number $z'$ by a complex number $z\ne0$ thus reduces to multiplication of $z'$ by $$\frac{\bar z}{|z|^2} = \frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}.$$ It is an important question whether the extension $\C$ of the field of reals constructed above, with the rules of operation indicated, is the only possible one or whether essentially different variants are conceivable. The answer is given by the uniqueness theorem: Every (algebraic) extension of the field $\R$ obtained from $\R$ by adjoining a root $i$ of the equation $X^2+1$ is isomorphic to $\C$, that is, only the above rules of operation with complex numbers are compatible with the requirement that the root $i$ be algebraically adjoined. This fact, however, does not exclude the existence of interpretations of complex numbers other than as points of the complex plane. The following two interpretations are most frequently employed in applications.

Vector interpretation. A complex number $z=x+iy$ can be identified with the vector $(x,y)$ with coordinates $x$ and $y$ starting from the origin (see Fig.).

Figure: c024140a

In this interpretation, addition and subtraction of complex numbers is carried out according to the rules of addition and subtraction of vectors. However, multiplication and division of complex numbers, which must be performed according to (2) and (3), do not have immediate analogues in vector algebra (see [4], [5]). The vector interpretation of complex numbers is immediately applicable, for example, in electrical engineering in the description of alternating sinusoidal currents and voltages.

Matrix interpretation. The complex number $w=u+iv$ can be identified with a $(2\times 2)$-matrix of special type $$w=\begin{pmatrix}\phantom{-}u&v\\ -v&u\end{pmatrix}$$ where the operations of addition, subtraction and multiplication are carried out according to the usual rules of matrix algebra.

By using polar coordinates in the complex plane $\C$, that is, the radius vector $r=|z|$ and polar angle $\def\phi{\varphi}\phi=\arg z$, called here the argument of $z$ (sometimes also called the phase of $z$), one obtains the trigonometric or polar form of a complex number: $$z=r(\cos\phi + i\sin\phi),\label{4}$$

$$r\cos\phi = \Re z,\quad r\sin\phi=\Im z.$$ The argument $\phi=\arg z$ is a many-valued real-valued function of the complex number $z\ne 0z\ne 0$, whose values for a given $z$ differ by integral multiples of $2\pi i$; the argument of the complex number $z=0$ is not defined. One usually takes the principal value of the argument $\phi = \def\Arg{\mathrm{Arg}} \Arg z$, defined by the additional condition $-\pi < \Arg z \le pi$. The Euler formulas $e^{\pm i\phi} = \cos\phi\pm i\sin\phi$ transform the trigonometric form (4) into the exponential form of a complex number: $$z=re^{i\phi}\label{5}$$ The forms (4) and (5) are particularly suitable for carrying out multiplication and division of complex numbers: $$zz'=rr'[\cos(\phi+\phi')+i\sin(\phi+\phi')]=rr'e^{i(\phi+\phi')},$$

$$\frac{z}{z'}=\frac{r}{r'}[\cos(\phi-\phi')+i\sin(\phi-\phi')] =\frac{r}{r'}e^{i(\phi-\phi')},\quad r>0$$ Under multiplication (or division) of complex numbers the moduli are multiplied (or divided) and the arguments are added (or subtracted). Raising to a power or extracting a root is carried out according to the so-called de Moivre formulas: $$z^n = r^n(\cos n\phi + i\sin n\phi) = r^n e^{in\phi},$$

$$z^{1/n} = r^{1/n}\Big(\cos\frac{\phi+2k\pi}{n}+i\sin\frac{\phi+2k\pi}{n}\Big) =r^{1/n}e^{i(\phi+2k\pi)/n},$$

$$k=0,\dots,n-1,$$ where the first of these is also applicable for negative integer exponents $n$. Geometrically, multiplication of a complex number $z$ by a complex number $z'=r'e^{i\phi'}$ reduces to rotating the vector $z$ over the angle $\phi'$ (anti-clockwise if $\phi'>0$) and subsequently multiplying its length by $|z'|=r'$; in particular, multiplication by a complex number $z'=e^{i\phi'}$, which has modulus one, is merely rotation over the angle $\phi'$. Thus, complex numbers can be interpreted as operators of a special type (affinors, cf. Affinor). In this connection, the mixed vector-matrix interpretation of multiplication of complex numbers is sometimes useful: $$(x,y)\begin{pmatrix}\phantom{-}u&v\\ -v&u\end{pmatrix}=(xu-yv,xv+yu),$$ in which the multiplicand is treated as a matrix-vector and the multiplier as a matrix-operator.

The bijection $(x,y)\mapsto x+iy$ induces on the field $\C$ the topology of the $2$-dimensional real vector space $\R^2$; this topology is compatible with the field structure of $\C$ and thus $\C$ is a topological field. The modulus $|z|$ is the Euclidean norm of the complex number $z={x,y}$, and $\C$ endowed with this norm is a complex one-dimensional Euclidean space, also called the complex $z$-plane. The topological product $\C^n=\C\times\cdots\times\C$ ($n$ times, $n\ge 1$) is a complex $n$-dimensional Euclidean space. For a satisfactory analysis of functions it is usually necessary to consider their behaviour in the complex domain. This is due to the fact that $\C$ is algebraically closed. Even the behaviour of such elementary functions as $z^n$, $\cos z$, $\sin z$, $e^z$ can be properly understood only when they are regarded as functions of a complex variable (see Analytic function).

Apparently, imaginary quantities first occurred in the celebrated work The great art, or the rules of algebra by G. Cardano, 1545, who regarded them as useless and unsuitable for applications. R. Bombelli (1572) was the first to realize the value of the use of imaginary quantities, in particular for the solution of the cubic equation in the so-called irreducible case (when the real roots are expressed in terms of cube roots of imaginary quantities, cf. Cardano formula). He gave some of the simplest rules of operation with complex numbers. In general, expressions of the form $a+b\i$, $b\ne 0$, appearing in the solution of quadratic and cubic equations were called "imaginary" in the 16th century and 17th century. However, even for many of the great scholars of the 17th century, the algebraic and geometric nature of imaginary quantities was unclear and even mystical. It is known, for example, that I. Newton did not include imaginary quantities within the notion of number, and that G. Leibniz said that "complex numbers are a fine and wonderful refuge of the divine spirit, as if it were an amphibian of existence and non-existence" .

The problem of expressing the $n$-th roots of a given number was mainly solved in the papers of A. de Moivre (1707, 1724) and R. Cotes (1722). The symbol $i=\i$ was proposed by L. Euler (1777, published 1794). It was he who in 1751 asserted that the field $\C$ is algebraically closed; J. d'Alembert (1747) came to the same conclusion. The first rigorous proof of this fact is due to C.F. Gauss (1799), who introduced the term "complex number" in 1831. The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the "Argand diagramArgand diagram" , came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel.

The purely arithmetic theory of complex numbers as pairs of real numbers was introduced by W. Hamilton (1837). He found a generalization of complex numbers, namely the quaternions (cf. Quaternion), which form a non-commutative algebra. More generally, it was proved at the end of the 19th century that any extension of the notion of number beyond the complex numbers requires sacrificing some property of the usual operations (primarily commutativity). See also Hypercomplex number; Double and dual numbers; Cayley numbers.

References

[1] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)
[2] A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian)
[3] A.I. Markushevich, "Theory of functions of a complex variable" , 1–2 , Chelsea (1977) (Translated from Russian)
[4] B.V. Shabat, "Introduction of complex analysis" , 1 , Moscow (1976) (In Russian)
[5] M.A. [M.A. Lavrent'ev] Lawrentjew, B.V. [B.V. Shabat] Schabat, "Problems in hydrodynamics and their mathematical models" , Moscow (1973) (In Russian)
[6] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1968)
[7] G.H. Hardy, "A course of pure mathematics" , Cambridge Univ. Press (1955)
[8] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
How to Cite This Entry:
Complex number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complex_number&oldid=19688
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article