Complex Numbers/From real to complex numbers

Introduction

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This page about Complex Numbers/From real to complex numbers can be displayed as Wiki2Reveal slides. Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides. The following aspects of Complex Numbers/From real to complex numbers are considered in detail:

  • (1) Real numbers embedded in plane of complex numbers
  • (2) geometric aspects of algebraic operations in  

Extension of the number range

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The complex numbers   extend the number range of the real numbers in such a way that the equation   can be solved. The equation   has no solution in  . The solubility is accomplished by introducing a new imaginary number   with the property  . This number   is referred to as imaginary unit.

Algebraic expression for complex numbers

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Complex numbers can be defined in the form  , where   are respectively real numbers and   is the imaginary unit. The identification with a vector   can be used to represent complex numbers in a coordinate system (Gaußian number level).

Real part and imaginary part

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The real-valued coefficients   are referred to as real part or imaginary part of a complex number  .

  •   and
  •  

Gaussian Plane

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With the identification of   with   we can draw a complex number in plane.

 

Polar coordinates

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The following equations show the link between exponential functions and trigonometric functions:

 

which results from   and  .

Polar coordinates

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Exponential function and trigonometry

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The representation with the aid of the complex e-function   also means exponential representation (the polar form), the representation by means of the expression   geometric representation (the polar form).


Characteristics

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The set of the complex numbers forms a extension of the field of the real numbers. The set of complex numbers is a field with   and has geometric and some algebraic properties that are not valid in field of real values  

Fundamental theorem of algebra

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The complex numbers are algebraically completed[1][2]. This theorem states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Remark - Polynomials with real-valued coefficients

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The theorem is also applicable for polynomials with real-valued coefficients, since every real number is a complex number with its imaginary part equal to zero.

Example

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In   the following algebraic equations with a polynomial with real-valued coefficient has no solution in   but two solutions in the complex numbers.

 

(see [[w:en:Fundamental theorem of algebra).

Trigonometry and exponential function

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In   the relationship between trigonometric functions and Exponential function is defined with following equation:

 

see Euler's formula.

Difference: complex and real differentiation

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On an open set all complex differentiable functions can also be differentiated in the real number. In calculus on the real numbers the following function

 

can be differentiated 2x and the thrid derivation does not exist. If we consider   then   is just a continuous function on   but in none of the points   complex differentiable.

Partial relationship between real and complex numbers

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The real numbers can be considered as a subset of the complex numbers in the sense of a subset of complex numbers. In this context a real number   is identified with the complex number  . In the Gaussian plane, the real numbers corresponded to the points on the   axis.

Complex conjugation

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Changing the [[w:en:sign (mathematics) |sign]] of the imaginary part   of a complex number   leads to the complex conjugation   of a complex number  .   can be created by reflection of   at the x axis of the plane.


Computing the Conjugation

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The conjugation   is a [[w:en:Involution (Mathematics)|(involutoric)] body automorphism, since it is compatible with addition and multiplication, i.e., for all  

 

Geometric representation of conjugation

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In the polar representation, the conjugated complex number   has an unchanged distance to the coordinate origin (i.e.  ) and has the negative angle of  . The conjugation in the complex numerical plane can therefore be interpreted as the 'mirror at the real axis'. In particular, under conjugation exactly the real numbers are mapped onto themselves.

Geometric representation of conjugation

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A complex number   and the complex number conjugated to it  

 
Conjugation in Gaussian Plane

Absolute Value

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The Absolute Value   of a complex number   is the length of its vector in the Gaussian plane or complex plane. The

 

calculate from their real part   and imaginary part  . As a length, the amount is real and not negative.

Example - Absolute Value of a complex number

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Pythagorean theorem

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The real part   and the imaginary part   of a complex number can be interpreted as the the catheti of right triangle where the length of hypotenuse is geometrically the absolute value of the complex number.

 

Characteristics

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In   the following properties apply:

  • (AG/KG) The Assoziative Law and Commutative Law apply to the addition and multiplication of complex numbers.
  • (DG) The Distributive Law applies.
  • '(NE) O and 1 are the neutral elements of the addition resp. of the multiplication.
  • (IE ) For every complex number   there is a complex number   with  .
  • (IE ) For each complex number different from zero   exists a complex number   with  

Calculation - algebraic form

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The algebraic properties result directly from the definition of the two links.

Addition

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For the addition of two complex numbers   with   and  

 

Vector Space - Visualization Addition

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[[File:Komplexe addition.svg |thumb|The addition of two complex numbers in the complex plane]]

Subtraction

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For the subtraction of two complex numbers   and   (see addition) applies

 

Multiplication

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For the multiplication of two complex numbers   and   (see addition) applies

 

Division

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For the division of the complex number   by the complex number   with   the multiplication with the complex conjugate denominator for the numerator and denominator of the fraction. This results in a real valued denominator as the square of the absolute value of  ):

 

Computation Example Addition:

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Subtraction example:

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Multiplication calculation example:

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Computational Example Division:

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Learning Activity

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  • Be given  . Solve the equation:
 
with   and  
  • Two complex numbers are the same when they match the real part and imaginary part. This creates a equation system with two equations and the two unknowns  

Complex numbers as a real vector space

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The body   of the complex numbers is on the one hand an upper body of  , on the other hand a two-dimensional  ve:en:vector space Isomorphism   is also referred to as natural identification.

Base of vector space

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As  -vector space owns   the base  . In addition,   is like each body also a vector space over itself, i.e. a one-dimensional  -vector space with base  .

Order - complex numbers

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  does not have (in contrast to  ) no order, i.e., there is complete order relation to two complex numbers.

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Algebraic Structure - Polar coordinates

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While the set   of the real numbers can be illustrated by points on a number line, the set   corresponds to as a two-dimensional real vector space  .

 

Points - Vectors

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According to the definition, the addition of complex numbers corresponds to the vector addition, the points in the number plane being identified with their [[w:en:location vector]en. The multiplication is a w:en:rotational stretching in the outer plane, which will become clearer after the introduction of the polar form (see Geogebra example).

Conversion formulae: algebraic shape into the polar shape

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For   is in algebraic form

 

For   the argument can be defined with 0, but usually remains undefined. For  , the argument   in the interval   can be used with the aid of a triArgonometric reversal function, e.

 

to be determined.

Conversion formulae: Polar form into algebraic form

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As above,   represents the real part and   the imaginary part of the complex number  .

Arithmetic operations in the polar form

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By arithmetic operations, the following operands are to be linked to one another:

 
 

In the case of multiplication, the absolute values   and   are multiplied for the product and the angles   and   are added. For the division/fraction, the absolute values are divided   and the angles are substracted.

 

Trigonometric Form - Multiplication

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Trigonometric form - Division

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Exponential Expression

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  •  
  •  

Real part and imaginary part function

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Be   This defines the real part function and imaginary part function as a relative image as follows.

  •   with   and  
  •   for all  

(see also Cauchy-Riemann differential equations)

Literature

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  • Paul Nahin: An imaginary tale. The story of  . Princeton University Press, 1998.
  • Reinhold Remmert: complex numbers. In D. Ebbinghaus et al. (Eds.): Numbers. Springer, 1983.

See also

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Sources of literature

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  1. Dunham, William (September 1991), "Euler and the fundamental theorem of algebra" (PDF), The College Journal of Mathematics, 22 (4): 282–293, JSTOR 2686228
  2. Campesato, Jean-Baptiste (November 4, 2020), "14 - Zeroes of analytic functions" (PDF), MAT334H1-F – LEC0101, Complex Variables, University of Toronto, retrieved 2024-09-05

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