Complex number

${\displaystyle \mathbb {C} }$

The letter C with a double line
stands for the set of complex numbers

The complex numbers expand the number range of the real numbers in such a way that the equation becomes solvable. This is achieved by introducing a new imaginary number with the property . This number is called an imaginary unit . In electrical engineering , the letter is used instead in order to avoid confusion with a current strength that is dependent on time (indicated by or ) . ${\displaystyle x^{2}+1=0}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {i} ^{2}=-1}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {j} }$ ${\displaystyle i}$ ${\displaystyle i(t)}$ ${\displaystyle t}$

Complex numbers can be represented in the form , where and are real numbers and is the imaginary unit. The usual calculation rules for real numbers can be applied to the complex numbers shown in this way, whereby can always be replaced by and vice versa. The symbol ( Unicode U + 2102: ℂ, see letter with double bar ) is used for the set of complex numbers . ${\displaystyle a+b\cdot \mathrm {i} }$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {i} ^{2}}$ ${\displaystyle -1}$ ${\displaystyle \mathbb {C} }$

The number range of the complex numbers constructed in this way forms an extension field of the real numbers and has a number of advantageous properties that have proven to be extremely useful in many areas of natural and engineering sciences . One of the reasons for these positive properties is the algebraic closure of the complex numbers. This means that every algebraic equation of positive degree has a solution over the complex numbers, which is not true for real numbers. This property is the content of the fundamental theorem of algebra . Another reason is a relationship between trigonometric functions and the exponential function ( Euler formula ), which can be established using complex numbers. Furthermore, each is on a open set once complex differentiable function there too many times differentiable - unlike in the analysis of the real numbers. The properties of functions with complex arguments are the subject of function theory , also called complex analysis.

definition

The complex numbers can be defined as a number range in the sense of a set of numbers, for which the basic arithmetic operations of addition , multiplication , subtraction and division are explained, with the following properties:

• The real numbers are contained in the complex numbers. This means that every real number is a complex number.
• The associative law and the commutative law apply to the addition and multiplication of complex numbers.
• The distributive law applies.
• For every complex number there is a complex number such that . ${\displaystyle x}$ ${\displaystyle -x}$ ${\displaystyle x+(-x)=0}$
• For every complex number other than zero there is a complex number such  that . ${\displaystyle x}$ ${\displaystyle {\tfrac {1}{x}}}$ ${\displaystyle x\cdot {\tfrac {1}{x}}=1}$
• There is a complex number with the property . ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {i} ^{2}=-1}$
• The complex numbers are minimal among all number ranges with the aforementioned properties.

The last requirement is synonymous with the fact that every complex number can be represented in the form (or in shortened notation or also ) with real numbers and . The imaginary unit is not a real number. The existence of such a number range is demonstrated in the section on the construction of complex numbers . ${\displaystyle a+b\cdot \mathrm {i} }$ ${\displaystyle a+b\,\mathrm {i} }$ ${\displaystyle a+\mathrm {i} \,b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \mathrm {i} }$

Using the terms body and isomorphism, this can be formulated as follows: There are minimal bodies that contain the body of the real numbers and an element with the property . In such a field each element has one and only one representation as with real. The complex numbers are isomorphic to each such field. ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {i} ^{2}=-1}$ ${\displaystyle z}$ ${\displaystyle z=a+b\,\mathrm {i} }$ ${\displaystyle a,b.}$

The coefficients are called the real or imaginary part of . Two notations have been established for this: ${\displaystyle a,b}$ ${\displaystyle a+b\,\mathrm {i} }$

• ${\displaystyle a=\operatorname {Re} {(a+b\,\mathrm {i} )}}$ and ${\displaystyle b=\operatorname {Im} {(a+b\,\mathrm {i} )}}$
• ${\displaystyle a=\Re {(a+b\,\mathrm {i} )}}$ and ${\displaystyle b=\Im {(a+b\,\mathrm {i} )}}$

notation

The notation in the form is also known as the Cartesian or algebraic form (named after René Descartes ) . The term Cartesian is explained by the representation in the complex or Gaussian numerical level (see below). There is also the representation ; However, it does not appear in the standard DIN 1302: 1999 General mathematical symbols and terms . ${\displaystyle a+b\,\mathrm {i} \ }$ ${\displaystyle \left(a,b\right)}$

In electrical engineering , the small i is already used for currents that change over time (see alternating current ) and can lead to confusion with the imaginary unit . Therefore, in accordance with DIN 1302, the letter j can be used in this area. ${\displaystyle \mathrm {i} }$

In physics , a distinction is made between for the amperage with alternating current and for the imaginary unit. Due to the very clear separation, this does not lead to confusion for the attentive reader and is used in this form to a large extent both in the physical-experimental and in the physical-theoretical literature; however, this delicacy cannot be kept by hand. See also: Complex AC bill ${\displaystyle i}$ ${\displaystyle \mathrm {i} }$

Complex numbers can be underlined in accordance with DIN 1304-1 and DIN 5483-3 to distinguish them from real numbers .

Calculating in the algebraic form

addition

Illustrates the addition of two complex numbers in the complex plane

For the addition of two complex numbers with and with applies ${\displaystyle z_{1}=a+b\,\mathrm {i} }$ ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle z_{2}=c+d\,\mathrm {i} }$ ${\displaystyle c,d\in \mathbb {R} }$

${\displaystyle z_{1}+z_{2}=(a+c)+(b+d)\,\mathrm {i} .}$

subtraction

For the subtraction of two complex numbers and (see addition) applies ${\displaystyle z_{1}}$ ${\displaystyle z_{2}}$

${\displaystyle z_{1}-z_{2}=(a-c)+(b-d)\,\mathrm {i} .}$

multiplication

For the multiplication of two complex numbers and (see addition) applies ${\displaystyle z_{1}}$ ${\displaystyle z_{2}}$

${\displaystyle z_{1}\cdot z_{2}=(ac+bd\,\mathrm {i} ^{2})+(ad+bc)\,\mathrm {i} =(ac-bd)+(ad+bc)\,\mathrm {i} .}$

division

To divide the complex number by the complex number (see addition) with , you expand the fraction with the complex number conjugated to the denominator . The denominator becomes real (and is the square of the amount of ): ${\displaystyle z_{1}}$ ${\displaystyle z_{2}}$ ${\displaystyle z_{2}\neq 0}$ ${\displaystyle z_{2}}$ ${\displaystyle {\bar {z}}_{2}=c-d\,\mathrm {i} }$ ${\displaystyle c+d\,\mathrm {i} }$

${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {(a+b\,\mathrm {i} )(c-d\,\mathrm {i} )}{(c+d\,\mathrm {i} )(c-d\,\mathrm {i} )}}={\frac {ac+bd}{c^{2}+d^{2}}}+{\frac {bc-ad}{c^{2}+d^{2}}}\mathrm {i} .}$

Sample calculations

Addition:

${\displaystyle (3+2\mathrm {i} )+(5+5\mathrm {i} )=(3+5)+(2+5)\mathrm {i} =8+7\mathrm {i} }$

Subtraction:

${\displaystyle (5+5\mathrm {i} )-(3+2\mathrm {i} )=(5-3)+(5-2)\mathrm {i} =2+3\mathrm {i} }$

Multiplication:

${\displaystyle (3+5\mathrm {i} )\cdot (4+11\mathrm {i} )=(3\cdot 4-5\cdot 11)+(3\cdot 11+5\cdot 4)\mathrm {i} =-43+53\mathrm {i} }$

Division:

${\displaystyle {{\frac {(2+5\mathrm {i} )}{(3+7\mathrm {i} )}}={\frac {(2+5\mathrm {i} )}{(3+7\mathrm {i} )}}\cdot {\frac {(3-7\mathrm {i} )}{(3-7\mathrm {i} )}}={\frac {(6+35)+(15\mathrm {i} -14\mathrm {i} )}{(9+49)+(21\mathrm {i} -21\mathrm {i} )}}={\frac {41+\mathrm {i} }{58}}={\frac {41}{58}}+{\frac {1}{58}}\cdot \mathrm {i} }}$

Other properties

• The body of the complex numbers is the one hand, an upper body of the other hand, a two-dimensional - vector space . The isomorphism is also known as natural identification . Usually one uses it also to be formal with the corresponding complex multiplication to define and then to set. At the same time it is determined: ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} \cong \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathrm {i} :=(0,1)^{\mathrm {T} }}$
  1. The rotation of the complex plane at the origin by the positive angle converts the positive real into the positive-imaginary unit . ${\displaystyle +{\tfrac {\pi }{2}}}$ ${\displaystyle +1}$ ${\displaystyle +\mathrm {i} }$
  2. If the positive-real semiaxis in the complex plane goes to the right, then the positive-imaginary semiaxis is placed upwards. This is in line with the mathematically positive sense of rotation .
• The enlargement of the body is of degree ; more precisely is isomorphic to the quotient ring , wherein the minimal polynomial of over is. It also forms the algebraic conclusion of . ${\displaystyle \mathbb {C} :\mathbb {R} }$ ${\displaystyle [\mathbb {C} :\mathbb {R} ]=2}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} [X]/(X^{2}+1)}$ ${\displaystyle X^{2}+1}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$
• As a vector space owns the basis . In addition, like every body, there is also a vector space about itself, i.e. a one-dimensional vector space with a basis . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \{1,\mathrm {i} \}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \{1\}}$
• ${\displaystyle \mathrm {i} }$and are exactly the solutions of the quadratic equation . In this sense it can (but also ) be understood as “ root of ”. ${\displaystyle -\mathrm {i} }$ ${\displaystyle x^{2}+1=0}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {-i} }$ ${\displaystyle -1}$
• ${\displaystyle \mathbb {C} }$in contrast to is not an ordered body , i. That is, there is no linear order relation compatible with the body structure . From two different complex numbers it is therefore not sensible to determine (in relation to the addition and multiplication in ) which of the two is the larger or the smaller number. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

Amount and Metric

amount

The absolute value of a complex number is the length of its vector in the Gaussian plane and can e.g. B. to ${\displaystyle |z|}$ ${\displaystyle z}$

${\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}$

calculate from their real part and imaginary part . As a length, the absolute value is real and not negative. ${\displaystyle \operatorname {Re} {(z)}=a}$ ${\displaystyle \operatorname {Im} {(z)}=b}$

Example:

${\displaystyle |239+\mathrm {i} |={\sqrt {239^{2}+1^{2}}}={\sqrt {57121+1}}={\sqrt {57122}}=169\cdot {\sqrt {2}}}$

Metric

The metric induced by the distance function provides the complex vector space with its standard topology . She agrees with the product topology of agreement, such as the restriction of to the standard metric on matches. ${\displaystyle d_{\mathbb {C} }(z_{1},z_{2}):=|z_{1}-z_{2}|}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} \times \mathbb {R} }$ ${\displaystyle d_{\mathbb {R} }}$ ${\displaystyle d_{\mathbb {C} }}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

Both spaces as are completely under these metrics. In both spaces, the topological concept of continuity can be expanded to include analytical concepts such as differentiation and integration . ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$

Complex number level

Gaussian plane with a complex number in Cartesian coordinates (a, b) and in polar coordinates (r, φ)

While the set of real numbers can be illustrated by points on a number line , the set of complex numbers can be represented as points in a plane (complex plane, Gaussian plane of numbers). This corresponds to the “double nature” of being a two-dimensional real vector space. The subset of real numbers forms the horizontal axis, the subset of purely imaginary numbers (i.e. with real part 0) forms the vertical axis. A complex number with then has the horizontal coordinate and the vertical coordinate , so it is identified with the pair of numbers . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle z=a+b\,\mathrm {i} }$ ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle (a,b)}$

According to the definition, the addition of complex numbers corresponds to the vector addition, whereby the points in the number plane are identified with their position vectors . In the Gaussian plane, the multiplication is a rotational stretching , which will become clearer after the introduction of the polar form below.

Polar shape

The color representation of the complex number level is often used to illustrate complex functions (here: identity). The color codes the argument and the lightness indicates the amount . ${\displaystyle \arg }$ ${\displaystyle |\cdot |}$

If one uses instead of the Cartesian coordinates and polar coordinates and with as the argument function, one can also use the complex number in the following so-called polar form based on the Eulerian relation (also polar representation ) ${\displaystyle a=\operatorname {Re} (z)}$ ${\displaystyle b=\operatorname {Im} (z)}$ ${\displaystyle r=|z|}$ ${\displaystyle \varphi =\arg(z)}$ ${\displaystyle \arg }$ ${\displaystyle z=a+b\,\mathrm {i} }$

${\displaystyle z=r\cdot \mathrm {e} ^{\mathrm {i} \varphi }=r\cdot (\cos \varphi +\mathrm {i} \cdot \sin \varphi )}$

represent that results from and . The representation with the help of the complex exponential function is also called exponential representation (the polar form), the representation using the expression trigonometric representation (the polar form). Because of Euler's relation , both representations are equivalent. Furthermore, there are the abbreviated spellings for them, especially in practice ${\displaystyle a=r\cdot \cos \varphi }$ ${\displaystyle b=r\cdot \sin \varphi }$ ${\displaystyle r\cdot \mathrm {e} ^{\mathrm {i} \varphi }}$ ${\displaystyle r\cdot (\cos \varphi +\mathrm {i} \cdot \sin \varphi )}$

${\displaystyle z=r\cdot \operatorname {cis} \,\varphi =r\cdot \operatorname {E} \,(\varphi )=r\,\angle \,\varphi \,,}$

in which stands for the sum and the representation with the angle operator is referred to as the verso representation . ${\displaystyle \operatorname {cis} \,\varphi }$ ${\displaystyle \cos \varphi +\mathrm {i} \cdot \sin \varphi }$ ${\displaystyle \angle }$

In the complex number plane, the Euclidean vector length (i.e. the distance to the origin 0) and the angle enclosed with the real axis correspond to the number . Usually, however, one calls the amount of (or its module ) (notation ) and the angle the argument (or the phase ) of (notation ). ${\displaystyle r}$ ${\displaystyle \varphi }$ ${\displaystyle z}$ ${\displaystyle r}$ ${\displaystyle z}$ ${\displaystyle r=|z|}$ ${\displaystyle \varphi }$ ${\displaystyle z}$ ${\displaystyle \varphi =\operatorname {arg} (z)}$

Since and can be assigned to the same number , the polar representation is initially ambiguous. Therefore, one usually restricts to the interval , in order to then speak of its main value for instead of the argument itself . However , any argument could be assigned to the number, and in this case it can actually be set to 0 for the purpose of clarity. ${\displaystyle \varphi }$ ${\displaystyle \varphi +2\pi }$ ${\displaystyle z}$ ${\displaystyle \varphi }$ ${\displaystyle (-\pi ;\pi ]}$ ${\displaystyle -\pi <\varphi \leq \pi }$ ${\displaystyle z\neq 0}$ ${\displaystyle z=0}$

The argument of is also the imaginary part of the complex natural logarithm ${\displaystyle z}$

${\displaystyle \ln z=\ln |z|+\mathrm {i} \cdot \arg(z).}$

With the choice of a fully defined branch of the logarithm, an argument function is also determined (and vice versa). ${\displaystyle \mathbb {C} }$

All values form the unit circle of the complex numbers with the amount , these numbers are also called unimodular and form the circle group . ${\displaystyle \mathrm {e} ^{\mathrm {i} \varphi }}$ ${\displaystyle 1}$

The fact that the multiplication of complex numbers (apart from zero) corresponds to rotational stretching can be expressed mathematically as follows: The multiplicative group of complex numbers without the zero can be expressed as the direct product of the group of rotations , the circle group , and the stretching by a factor unequal Zero, the multiplicative group . The first group can be parameterized by the argument , the second corresponds to the amounts. ${\displaystyle \mathbb {C} ^{\times }}$ ${\displaystyle \mathbb {R} ^{+}}$ ${\displaystyle \varphi }$

Complex conjugation

→ Main article : Conjugation (mathematics)

A complex number and the complex number conjugated to it ${\displaystyle z=a+b\,\mathrm {i} }$ ${\displaystyle {\bar {z}}=a-b\,\mathrm {i} }$

If you change the sign of the imaginary part of a complex number , you get the conjugate complex number (sometimes also written). ${\displaystyle b}$ ${\displaystyle z=a+b\,\mathrm {i} ,}$ ${\displaystyle z}$ ${\displaystyle {\bar {z}}=a-b\,\mathrm {i} }$ ${\displaystyle z^{*}}$

The conjugation is an (involutive) body automorphism, since it is compatible with addition and multiplication, i. h., for all true ${\displaystyle \mathbb {C} \to \mathbb {C} ,\,z\mapsto {\bar {z}}}$ ${\displaystyle y,z\in \mathbb {C} }$

${\displaystyle {\overline {y+z}}={\bar {y}}+{\bar {z}},\quad {\overline {y\cdot z}}={\bar {y}}\cdot {\bar {z}}.}$

In the polar representation, the conjugate complex number has the negative angle of exactly the same as the absolute value. The conjugation in the complex number plane can therefore be interpreted as the reflection on the real axis . In particular, the real numbers are mapped to themselves under the conjugation. ${\displaystyle {\bar {z}}}$ ${\displaystyle z.}$

The product of a complex number and its complex conjugate gives the square of its absolute value: ${\displaystyle z=a+b\,\mathrm {i} }$ ${\displaystyle {\bar {z}}}$

${\displaystyle z\cdot {\bar {z}}=(a+b\,\mathrm {i} )(a-b\,\mathrm {i} )=a^{2}+b^{2}=|z|^{2}}$

The complex numbers thus form a trivial example of a C * algebra .

The sum of a complex number and its complex conjugate results in twice its real part: ${\displaystyle z=a+b\,\mathrm {i} }$ ${\displaystyle {\bar {z}}}$

${\displaystyle z+{\bar {z}}=(a+b\,\mathrm {i} )+(a-b\,\mathrm {i} )=2a=2\,\operatorname {Re} {(z)}}$

The difference between a complex number and its complex conjugate results in the -fold of its imaginary part: ${\displaystyle z=a+b\,\mathrm {i} }$ ${\displaystyle {\bar {z}}}$ ${\displaystyle \mathrm {2i} }$

${\displaystyle z-{\bar {z}}=(a+b\,\mathrm {i} )-(a-b\,\mathrm {i} )=2b\,\mathrm {i} =2\,\mathrm {i} \,\operatorname {Im} {(z)}}$

Conversion formulas

From the algebraic form to the polar form

For is in algebraic form ${\displaystyle z=a+b\,\mathrm {i} }$

${\displaystyle r=|z|={\sqrt {a^{2}+b^{2}}}={\sqrt {z\cdot {\overline {z}}}}.}$

The argument for is arbitrary, but is often set to 0 or left undefined. For can the argument in the interval with the help of a trigonometric inverse function, e.g. with the help of the arccosine ${\displaystyle z=0\quad (\Longleftrightarrow r=0)}$ ${\displaystyle \varphi }$ ${\displaystyle z\neq 0}$ ${\displaystyle \varphi }$ ${\displaystyle (-\pi ;\pi ]}$

${\displaystyle \varphi =\arg(z)={\Biggl \{}{\begin{matrix}\\\\\end{matrix}}{\Biggr .}}$	${\displaystyle \arccos {\frac {a}{r}}}$	For ${\displaystyle b\geq 0}$	${\displaystyle {\Biggl .}{\begin{matrix}\\\\\end{matrix}}{\Biggr \}}\;=\operatorname {arctan2} (a,b)}$
	${\displaystyle -\arccos {\frac {a}{r}}}$	For ${\displaystyle b<0}$

be determined. Methods that use the arctangent are listed in the article Arctangent and Arctangent # Conversion of Planar Cartesian Coordinates into Polar . This also includes the variant of the arctangent function, often called arctan2 , but also atan2 , available in many programming languages and spreadsheets , which receives both values ​​and assigns the result to the appropriate quadrant depending on the sign of and . ${\displaystyle a}$ ${\displaystyle b}$

The calculation of the angle in the interval can in principle be carried out in such a way that the angle is first calculated in the interval as described above and then increased by, if it is negative: ${\displaystyle \varphi }$ ${\displaystyle [0,2\pi )}$ ${\displaystyle (-\pi ,\pi ]}$ ${\displaystyle 2\pi }$

${\displaystyle \varphi '=\arg(z)={\begin{cases}\varphi +2\pi &{\text{falls}}\ \varphi <0\\\varphi &{\text{sonst}}\end{cases}}}$

(see polar coordinates ).

From the polar form to the algebraic form

${\displaystyle a=\operatorname {Re} (z)=r\cdot \cos \varphi }$

${\displaystyle b=\operatorname {Im} (z)=r\cdot \sin \varphi }$

As above, represents the real part and the imaginary part of that complex number. ${\displaystyle a}$ ${\displaystyle b}$

Arithmetic operations in polar form

The following operands are to be linked with one another using arithmetic operations:

${\displaystyle z_{1}=r_{1}\cdot (\cos \varphi _{1}+\mathrm {i} \cdot \sin \varphi _{1})=r_{1}\cdot \mathrm {e} ^{\mathrm {i} \varphi _{1}}}$

${\displaystyle z_{2}=r_{2}\cdot (\cos \varphi _{2}+\mathrm {i} \cdot \sin \varphi _{2})=r_{2}\cdot \mathrm {e} ^{\mathrm {i} \varphi _{2}}}$

When multiplying, the amounts and are multiplied with each other and the associated phases or are added. In division, the amount of the dividend is divided by the amount of the divisor and the phase of the divisor is subtracted from the phase of the dividend. There is also a somewhat more complicated formula for addition and subtraction: ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$ ${\displaystyle \varphi _{1}}$ ${\displaystyle \varphi _{2}}$

Trigonometric shape

Multiplying two complex numbers is equivalent to adding the angles and multiplying the amounts.

• ${\displaystyle {z_{1}\cdot z_{2}=r_{1}\cdot r_{2}\cdot \left[\cos(\varphi _{1}+\varphi _{2})+\mathrm {i} \cdot \sin(\varphi _{1}+\varphi _{2})\right]}}$

Dividing two complex numbers is equivalent to subtracting the angles and dividing the amounts.

• ${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\cdot \left[\cos(\varphi _{1}-\varphi _{2})+\mathrm {i} \cdot \sin(\varphi _{1}-\varphi _{2})\right]}$

• ${\displaystyle z_{1}\pm z_{2}=t\cdot (\cos \chi +\mathrm {i} \cdot \sin \chi )}$

  with   and the arctan2 function. ${\displaystyle t:={\sqrt {r_{1}^{2}+r_{2}^{2}\pm 2r_{1}r_{2}\cos(\varphi _{1}-\varphi _{2})}},}$
  ${\displaystyle \chi :=\operatorname {arctan2} \left(r_{1}\cos \varphi _{1}\pm r_{2}\cos \varphi _{2},r_{1}\sin \varphi _{1}\pm r_{2}\sin \varphi _{2}\right)}$
  

Exponential form

• ${\displaystyle z_{1}\cdot z_{2}=r_{1}\cdot r_{2}\cdot \mathrm {e} ^{\mathrm {i} (\varphi _{1}+\varphi _{2})}}$
• ${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\cdot \mathrm {e} ^{\mathrm {i} (\varphi _{1}-\varphi _{2})}}$
• ${\displaystyle z_{1}\pm z_{2}=t\cdot \mathrm {e} ^{\mathrm {i} \chi }}$with and as above. ${\displaystyle t}$ ${\displaystyle \chi }$

Arithmetic operations 3rd stage

The arithmetic operations of the third level include exponentiation , extraction of the root (square root) and logarithmization .

Potencies

Natural exponents

For natural numbers , the calculated th power in the polar form to ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle z=r\mathrm {e} ^{\mathrm {i} \varphi }}$

${\displaystyle z^{n}=r^{n}\cdot \mathrm {e} ^{\mathrm {i} n\varphi }=r^{n}\cdot (\cos n\varphi +\mathrm {i} \cdot \sin n\varphi )}$

(see the set of de Moivre ) or the algebraic form by means of the binomial theorem to ${\displaystyle z=a+b\,\mathrm {i} }$

${\displaystyle z^{n}=\sum _{k=0, \atop k{\text{ gerade}}}^{n}{\binom {n}{k}}(-1)^{\frac {k}{2}}a^{n-k}b^{k}+\mathrm {i} \sum _{k=1, \atop k{\text{ ungerade}}}^{n}{\binom {n}{k}}(-1)^{\frac {k-1}{2}}a^{n-k}b^{k}.}$

Any complex exponent

The general definition of a complex base and exponent power is ${\displaystyle z\neq 0}$ ${\displaystyle \omega }$

${\displaystyle z^{\omega }:=\mathrm {e} ^{\omega \cdot \ln z},}$

where stands for the main value of the complex logarithm (see below), so the formula also provides a main value. In the case, however, all possible results agree with this main value and the function becomes clear. ${\displaystyle \ln(z)}$ ${\displaystyle \omega \in \mathbb {Z} }$

root

→ Main article : Roots from complex numbers

Logarithms

The complex natural logarithm (unlike the real on ) ambiguous. A complex number is called the logarithm of the complex number if ${\displaystyle \mathbb {R} ^{+}}$ ${\displaystyle w}$ ${\displaystyle z}$

${\displaystyle \mathrm {e} ^{w}=z.}$

With , any number with any is also a logarithm of . One therefore works with main values , i. H. with values ​​of a certain strip of the complex plane. ${\displaystyle w}$ ${\displaystyle w+2m\pi \mathrm {i} }$ ${\displaystyle m\in \mathbb {Z} }$ ${\displaystyle z}$

The principal value of the natural logarithm of the complex number

${\displaystyle z=r\mathrm {e} ^{\mathrm {i} \varphi }\in \mathbb {C} ^{\times }}$

is

${\displaystyle \ln z=\ln r+\mathrm {i} \varphi }$

with and . In other words: The main value of the natural logarithm of the complex number is ${\displaystyle r>0}$ ${\displaystyle -\pi <\varphi \leq \pi }$ ${\displaystyle z\in \mathbb {C} ^{\times }}$

${\displaystyle \ln z=\ln |z|+\mathrm {i} \,\arg(z),}$

wherein the principal value of the argument of is. ${\displaystyle \arg(z)}$ ${\displaystyle z}$

The finite subgroups

All elements of a finite subgroup of the multiplicative unit group are roots of unit . Among all the orders of group elements there is a maximum, for example . Since is commutative, an element with this maximum order then also creates the group, so that the group is cyclic and exactly from the elements ${\displaystyle \mathbb {C} ^{\times }=\mathbb {C} \setminus \{0\}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \mathbb {C} }$

${\displaystyle \exp \left({2\pi \mathrm {i} k \over n}\right),\quad k=0,1,\dotsc ,n-1}$

consists. All elements lie on the unit circle .

The union of all finite subgroups is a group that is isomorphic to the torsion group . It lies close to its completion , the already mentioned circle group , which can also be understood as a 1-sphere and is too isomorphic. ${\displaystyle \mathbb {Q} /\mathbb {Z} }$ ${\displaystyle \mathbb {R} /\mathbb {Z} }$

Pragmatic calculation rules

The easiest way to perform the calculations is as follows:

• Addition and subtraction of complex numbers are performed component-wise (in the algebraic form).
• The multiplication of complex numbers can advantageously be carried out in algebraic form or in exponential form (multiplication of the amounts and addition of the arguments (angles)), depending on the specification.
• When dividing complex numbers, their amounts are divided in exponential form and their arguments (angles) subtracted, or in algebraic form the quotient is expanded with the conjugate denominator.
• When a complex number is raised to the power of a real exponent, its magnitude is raised to the power and its argument (angle) is multiplied by the exponent; the use of the algebraic form (with Newton's binomial theorem ) is more cumbersome in most cases (especially for higher powers).
• When square rooting a complex number with a real exponent, its amount is square rooted and its argument (angle) is divided by the exponent. This creates the first solution. With a -th root, solutions arise that are distributed at an angle of around the origin of the Gaussian plane. See root (math) . A square root can also be calculated quite easily in Cartesian form. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle 2\pi /n}$
• When multiplying in algebraic form, one of the four multipliers can be saved using the following procedure. However, three additional additions or subtractions are necessary and the calculation is more difficult to parallelize.

${\displaystyle p_{1}=(a-b)(c+d)}$

${\displaystyle p_{2}=bc}$

${\displaystyle p_{3}=ad}$

${\displaystyle \operatorname {Re} (z_{1}z_{2})=p_{1}+p_{2}-p_{3}}$

${\displaystyle \operatorname {Im} (z_{1}z_{2})=p_{2}+p_{3}}$

Construction of complex numbers

In this section it is demonstrated that there actually exists a field of complex numbers which satisfies the properties required in the above definition. Different constructions are possible, but apart from isomorphism they lead to the same body. ${\displaystyle \mathbb {C} }$

Pairs of real numbers

The construction initially makes no reference to the imaginary unit : In the 2-dimensional real vector space of the ordered real number pairs , in addition to the addition ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle z=(a,b)}$

${\displaystyle (a,b)+(c,d):=(a+c,b+d)}$

(this is the usual vector addition) a multiplication by

${\displaystyle (a,b)\cdot (c,d):=(a\cdot c-b\cdot d,a\cdot d+b\cdot c)}$

Are defined.

After this definition one writes and becomes a body, the body of complex numbers. The imaginary unit is then defined by. ${\displaystyle \mathbb {C} =\mathbb {R} ^{2}}$ ${\displaystyle (\mathbb {C} ,+,\cdot )}$ ${\displaystyle \mathrm {i} :=(0,1)}$

Since a basis of the form, it can be used as a linear combination ${\displaystyle \{(1,0),(0,1)\}=\{1,\mathrm {i} \}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle z}$

${\displaystyle z=1\cdot (a,0)+\mathrm {i} \cdot (b,0)=a+\mathrm {i} b}$

represent.

First properties

• The figure is a body embedding of in , on the basis of which we identify the real number with the complex number . ${\displaystyle \mathbb {R} \to \mathbb {C} ,\,a\mapsto (a,0)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a}$ ${\displaystyle (a,0)}$

Regarding the addition:

• the number the neutral element (the zero element) in and ${\displaystyle 0=(0,0)}$ ${\displaystyle \mathbb {C} }$
• the number the inverse element in . ${\displaystyle -z=(-a,-b)}$ ${\displaystyle \mathbb {C} }$

Regarding the multiplication is:

• the number the neutral element (the one element) of and ${\displaystyle 1=(1,0)}$ ${\displaystyle \mathbb {C} }$
• the inverse ( reciprocal ) to be . ${\displaystyle z=(a,b)\neq (0,0)}$ ${\displaystyle z^{-1}=\left({\frac {a}{a^{2}+b^{2}}},\,{\frac {-b}{a^{2}+b^{2}}}\right)}$

Reference to the representation in the form a + b i

By the imaginary unit is set; for this applies what corresponds to the above embedding . ${\displaystyle \mathrm {i} :=(0,1)}$ ${\displaystyle \mathrm {i} ^{2}=(-1,0)}$ ${\displaystyle -1\in \mathbb {R} }$

Each complex number has the unique representation of the shape ${\displaystyle z=(a,b)\in \mathbb {C} }$

${\displaystyle z=(a,b)=(a,0)+(0,b)=a\cdot (1,0)+b\cdot (0,1)=a+b\,\mathrm {i} }$

with ; this is the usual notation for complex numbers. ${\displaystyle a,b\in \mathbb {R} }$

Polynomials: adjunction

Another construction of the complex numbers is the factor ring

${\displaystyle \mathbb {R} [X]/(X^{2}+1)}$

of the polynomial ring in an indeterminate over the real numbers. The number corresponds to the image of the indeterminate , the real numbers are identified with the constant polynomials. ${\displaystyle \mathrm {i} }$ ${\displaystyle X}$

This construction principle can also be used in other contexts, one speaks of adjunction .

Matrices

The set of - matrices of form ${\displaystyle 2\times 2}$

${\displaystyle Z={\begin{pmatrix}a&-b\\b&a\end{pmatrix}}=a{\begin{pmatrix}1&0\\0&1\end{pmatrix}}+b{\begin{pmatrix}0&-1\\1&0\end{pmatrix}}=a\cdot E+b\cdot I}$ With ${\displaystyle a,b\in \mathbb {R} }$

also forms a model of the complex numbers. The real unit and the imaginary unit are represented by the identity matrix or the matrix . Therefore: ${\displaystyle 1}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle E}$ ${\displaystyle I}$

${\displaystyle \operatorname {Re} (Z)=a}$

${\displaystyle \operatorname {Im} (Z)=b}$

${\displaystyle I^{2}=-E}$

${\displaystyle \operatorname {abs} (Z)={\sqrt {a^{2}+b^{2}}}={\sqrt {\det Z}}}$

This set is a subspace of the vector space of the real matrices. ${\displaystyle 2\times 2}$

Real numbers correspond to diagonal matrices ${\displaystyle {\begin{pmatrix}a&0\\0&a\end{pmatrix}}.}$

The linear mappings belonging to the matrices are, provided that and not both are zero, rotational extensions in space . It is exactly the same rotational stretching as in the interpretation of the multiplication with a complex number in the Gaussian plane of numbers . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle a+b\mathrm {i} }$

history

The term “complex numbers” was introduced by Carl Friedrich Gauß ( Theoria residuorum biquadraticorum, 1831), the origin of the theory of complex numbers goes back to the Italian mathematicians Gerolamo Cardano ( Ars magna, Nuremberg 1545) and Rafael Bombelli ( L'Algebra, Bologna 1572; probably written between 1557 and 1560).

The impossibility of a naive root extraction of this kind was noticed and emphasized very early on in the treatment of quadratic equations, e.g. B. in the algebra of Muhammed ibn Mûsâ Alchwârizmî , written around 820 AD . But mathematical research did not stop at the most obvious and incontestable conclusion that this type of equation was unsolvable. ${\displaystyle x^{2}=-1\Rightarrow x=\pm {\sqrt {-1}}}$

In a certain sense, the Italian Gerolamo Cardano (1501–1576) went beyond this in his 1545 book Artis magnae sive de regulis algebraicis liber unus . There he deals with the task of finding two numbers whose product is 40 and whose sum is 10. He emphasizes that the equation to be used for this

${\displaystyle x(10-x)=40}$

${\displaystyle x^{2}-10x+40=0}$

Has no solution but adds some remarks by going into the solution

${\displaystyle x_{1,2}=-{\frac {p}{2}}\pm {\sqrt {{\frac {p^{2}}{4}}-q}}}$

the general normalized quadratic equation

${\displaystyle x^{2}+px+q=0}$

for and the values ​​−10 and 40 are inserted. So if it were possible, the resulting expression ${\displaystyle p}$ ${\displaystyle q}$

${\displaystyle {\sqrt {25-40}}={\sqrt {-15}}}$

to give a meaning, in such a way that one could calculate with this sign according to the same rules as with a real number, then the expressions would

${\displaystyle 5+{\sqrt {-15}}}$

${\displaystyle 5-{\sqrt {-15}}}$

in fact each represent a solution.

For the square root of negative numbers and more generally for all of any real number and a positive real number composite numbers ${\displaystyle \alpha }$ ${\displaystyle \beta }$

${\displaystyle \alpha +{\sqrt {-\beta }}}$ or ${\displaystyle \alpha -{\sqrt {-\beta }}}$

The term imaginary number has been used since the middle of the 17th century , originally from René Descartes , who in his La Géométrie (1637) used it to express the difficulty of understanding complex numbers as non-real solutions to algebraic equations. In the 17th century, John Wallis made the first advances in a geometric interpretation of complex numbers. In 1702 Gottfried Wilhelm Leibniz called it a fine and wonderful refuge for the human spirit, almost a hybrid between being and non-being. The introduction of the imaginary unit as a new number is attributed to Leonhard Euler . He achieved valuable new knowledge by calculating with imaginary numbers, for example he published Euler's formula in his Introduction to Analysis in 1748 and for the first time explicitly published the formula of Abraham de Moivre (late 17th century, who in turn had it from Isaac Newton ) , but even Euler still had great difficulties in understanding and classifying complex numbers, although he routinely calculated them. ${\displaystyle \mathrm {i} }$

The geometrical interpretation was first made by the Danish surveyor Caspar Wessel (published in 1799 in the treatises of the Royal Danish Academy of Sciences , but only known to other circles about a hundred years later), by Jean-Robert Argand (in an obscure private print 1806, which was published by Legendre Knowledge came and became known to broader circles in 1813) and Gauss (unpublished) discovered. Gauss explicitly mentions the representation in a letter to Friedrich Bessel dated December 18, 1811. According to Argand, the geometric representation on the numerical level is sometimes also called the Argand diagram .

Augustin-Louis Cauchy is considered the founder of complex analysis in a paper on integration in the complex that was submitted to the French Academy in 1814, but was not published until 1825. In 1821, in his textbook Cours d'analysis, he defined a function of a complex variable in the complex number plane and proved many fundamental theorems of function theory .

Based on the philosophical ideas of Immanuel Kant , William Rowan Hamilton found in 1833 a logically flawless justification of the complex numbers as an ordered pair of real numbers. He interpreted the complex number as a pair of numbers and defined addition or multiplication by: ${\displaystyle a+b\cdot \mathrm {i} }$ ${\displaystyle (a,b)}$

${\displaystyle {\begin{aligned}(a_{1},b_{1})+(a_{2},b_{2})&=(a_{1}+a_{2},b_{1}+b_{2})\\(a_{1},b_{1})(a_{2},b_{2})&=(a_{1}a_{2}-b_{1}b_{2},a_{1}b_{2}+a_{2}b_{1})\end{aligned}}}$

Today these things do not cause any conceptual or actual difficulties. Due to the simplicity of the definition, the already explained meaning and applications in many fields of science, the complex numbers are in no way inferior to the real numbers. The term “imaginary” numbers, in the sense of imagined or unreal numbers, has developed over the centuries into a crooked but retained designation.

meaning

Complex numbers in physics

Complex numbers play a central role in basic physics. In quantum mechanics , the state of a physical system is understood as an element of a ( projective ) Hilbert space over the complex numbers. Complex numbers are used to define differential operators in the Schrödinger equation and the Klein-Gordon equation . For the Dirac equation one needs a range extension of the complex numbers, the quaternions . Alternatively, a formulation with Pauli matrices is possible, but these have the same algebraic structure as the quaternions.

Complex numbers play an important role in physics and technology as a calculation aid. In particular, the treatment of differential equations for oscillation processes can be simplified, since the complicated relationships in connection with products of sine or cosine functions can be replaced by products of exponential functions, with only the exponents having to be added. For example, in the complex alternating current calculation, suitable imaginary parts are inserted into the real output equations, which are then ignored when evaluating the calculation results. As a result, harmonic oscillations (real) are added to circular movements in the complex plane in the interim calculation , which have more symmetry and are therefore easier to deal with.

In optics , the refractive and absorbing effects of a substance are summarized in a complex, wavelength-dependent permittivity (dielectric constant) or the complex refractive index , which in turn is attributed to the electrical susceptibility .

In fluid dynamics , complex numbers are used to explain and understand planar potential flows . Any complex function of a complex argument always represents a planar potential flow - the geometric location corresponds to the complex argument in the Gaussian plane, the streaming potential to the real part of the function, and the streamlines to the isolines of the imaginary part of the function with the opposite sign. The vector field of the flow velocity corresponds to the complex conjugate first derivative of the function. By experimenting with different superimpositions of parallel flow, sources, sinks, dipoles and eddies one can show the flow around different contours. These flow patterns can be distorted by conformal mapping  - the complex argument is replaced by a function of the complex argument. For example, the flow around a circular cylinder (parallel flow + dipole) can be distorted into the flow around an airfoil-like profile ( Joukowski profile ) and the role of the supporting vortex on an aircraft wing can be studied. As useful as this method is for learning and understanding, it is generally insufficient for accurate calculation.

Complex numbers in electrical engineering

In electrical engineering, the representation of electrical quantities with the help of complex numbers is widespread. It is used when calculating variables that change sinusoidally over time, such as electric and magnetic fields. With the representation of a sinusoidal alternating voltage as a complex quantity and corresponding representations for resistors, capacitors and coils, the calculations of the electrical current , the active and the reactive power in a circuit are simplified . The coupling given by differential quotients or integrals changes into a coupling by trigonometric functions; the calculation of the relationships can thus be made much easier. The interaction of several different sinusoidal voltages and currents, which can have their zero crossings at different times, can also be easily represented in a complex calculation. For more information on this subject, see the article on the complex AC bill .

In the last few years, digital signal processing has grown in importance, the foundation of which is the calculation with complex numbers.

Body theory and algebraic geometry

The field of complex numbers is the algebraic closure of the field of real numbers.

Any two algebraically closed fields with the same characteristic and the same degree of transcendence over their prime field (which is determined by the characteristic) are ( ring-theoretical ) isomorphic . For a field of characteristic 0 with uncountable degree of transcendence, this is equal to the cardinality of the body. In terms of body theory, the complex numbers form the only algebraically closed body with characteristic 0 and the cardinality of the continuum . A construction of the field of complex numbers is also possible purely algebraically with the help of this statement, for example as an extension of the algebraic closure of the rational numbers by many transcendent elements. Another construction provides an ultra product : To do this, form its algebraic closure for every finite field and form the ultra product with respect to any free ultrafilter . From Łoś's theorem it follows that this ultra-product is an algebraically closed field with characteristic 0, the cardinality of the continuum follows from set-theoretical considerations. ${\displaystyle \aleph }$ ${\displaystyle \aleph }$

Under the catchphrase Lefschetz principle , various sentences are summarized that allow results of the algebraic geometry , which are proven using the complex numbers, to be transferred to other algebraically closed bodies with characteristic 0 (which is decisive for the completeness of the theory of algebraically closed bodies with characteristic 0). The consideration of the complex case offers the advantage that topological and analytical methods can be used to obtain algebraic results. The above ultra-product construction allows the transfer of results in the case of a characteristic other than 0 to the complex numbers.

Spectral Theory and Functional Analysis

Many of the results of spectral theory apply to complex vector spaces to a greater extent than to real ones. So z. B. complex numbers as eigenvalues ​​of real matrices (then in each case together with the conjugate-complex eigenvalue). This is explained by the fact that the characteristic polynomial of the matrix always breaks down into linear factors due to the algebraic closeness of over the complex numbers . In contrast, there are real matrices without real eigenvalues, while the spectrum of any bounded operator on a complex (at least one-dimensional) Banach space is never empty. In spectral theory on Hilbert spaces , sentences that in the real case only apply to self-adjoint operators can often be transferred to normal operators in the complex case . ${\displaystyle \mathbb {C} }$

The complex numbers also play a special role in other parts of functional analysis . For example, the theory of C * -algebras is mostly operated in the complex, the harmonic analysis deals with representations of groups on complex Hilbert spaces.

Function theory and complex geometry

The study of differentiable functions on subsets of complex numbers is the subject of function theory . In many respects it is more rigid than real analysis and allows fewer pathologies. Examples are the statement that every function that can be differentiated in a domain is already differentiable as often as desired, or the identity theorem for holomorphic functions .

Function theory often also allows conclusions to be drawn about purely real statements, for example some integrals can be calculated using the residual theorem . An important area of ​​application of these methods is analytical number theory , which traces statements about whole numbers back to statements about complex functions, often in the form of Dirichlet series . A prominent example is the connection between the prime number theorem and Riemannian ζ function . In this context, the Riemann Hypothesis plays a central role.

The above-mentioned rigidity of holomorphic functions is even more evident in global questions, i.e. H. when studying complex manifolds . So on a compact complex manifold there are no non-constant global holomorphic functions; Statements like Whitney's embedding theorem are therefore wrong in the complex. This so-called "analytical geometry" (not to be confused with the classical analytical geometry of René Descartes!) Is also closely linked to algebraic geometry , many results can be transferred. The complex numbers are also sufficiently large in a suitable sense to capture the complexity of algebraic varieties over arbitrary fields of characteristic  0 (Lefschetz principle).

literature

• Paul Nahin: An imaginary tale. The story of . ${\displaystyle {\sqrt {-1}}}$Princeton University Press, 1998.
• Reinhold Remmert: Complex numbers. In D. Ebbinghaus u. a. (Ed.): Numbers. Springer, 1983.

Related topics

• Gaussian numbers and Eisenstein numbers are a generalization of the whole numbers to the complex numbers.
• Hypercomplex numbers generalize the algebraic structure of the complex numbers.
• Complex-valued functions map complex numbers to complex numbers.

Web links

Commons : Complex Numbers  - collection of pictures, videos and audio files

Wikibooks: Imaginary and Complex Numbers  - a compact introduction

Wikibooks: Complex Numbers  - Learning and Teaching Materials

• Writings, documentations, videos
  • History of Complex Numbers
  • A technical paper that gives an introduction to complex numbers
  • Calculating with complex numbers
  • Dimensions: a math film. Embedding the complex numbers in the representation of higher dimensions (also chaos theory), especially Chapters 5 and 6
• Direct execution programs
  • Java applet for geometric interpretation
  • Java class, for computing complex numbers
  • Complex number calculator

Individual evidence

1. ↑ Eberhard Freitag, Rolf Busam: Function Theory 1: With suggestions for solutions to 420 exercises . 4th edition. Springer, Berlin 2007, ISBN 978-3-540-31764-7 .
2. ↑ When using the symbol, it is made even clearer than it might be when using , that the same solution of (the same "sign") must be used for each occurrence . Nevertheless, all algebraic statements remain valid if is replaced by everywhere . ${\displaystyle \mathrm {i} }$ ${\displaystyle {\sqrt {-1}}}$ ${\displaystyle \mathrm {i} ^{2}+1=0}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle -\mathrm {i} }$
3. ^ Ehrhard Behrends: Analysis . 6th edition. tape 1 . Springer Spectrum, Wiesbaden 2015, ISBN 978-3-658-07122-6 , doi : 10.1007 / 978-3-658-07123-3 .
4. ↑ Helmuth Gericke : History of the number concept . Bibliographisches Institut, Mannheim 1970, p. 57-67 .
5. ↑ Remmert: Complex numbers. In: Ebbinghaus u. a .: Numbers. Springer 1983, p. 48.
6. ↑ Nahin: An imaginary tale. P.56.
7. ^ Stillwell: Mathematics and its History. Springer, p. 287.
8. ^ Morris Kline : Mathematical thought from ancient to modern times. Oxford University Press, 1972, Volume 2, p. 631. The letter is reproduced in Volume 8 of the Works, p. 90. Gauss uses the complex number level essentially in his proof of the fundamental theorem of algebra from 1816.
   Felix Klein : History of mathematics in the 19th century. P. 28.
9. ^ Heinz-Wilhelm Alten : 4000 years of algebra. History, cultures, people . Springer, Berlin a. a. 2003, ISBN 3-540-43554-9 , pp. 310 .
10. ↑ Hence the fact that there are uncountably many “wild” automorphisms of ; see Paul B. Yale: Automorphisms of the Complex Numbers. maa.org (PDF; 217 kB). ${\displaystyle \mathbb {C} }$
11. ^ H. Schoutens: The Use of Ultraproducts in Commutative Algebra . (PDF) Springer , 2010, p. 16.
12. ^ Gerhard Frey , Hans-Georg Rück: The Strong Lefschetz Principle in Algebraic Geometry . In: manuscripta mathematica . tape 55 , 1986, pp. 385 ( online ).
13. ↑ Frey, Rück, p. 389.
14. ↑ Dirk Werner : Functional Analysis . 7th edition. Springer, 2011, ISBN 978-3-642-21016-7 , pp. 261 .