Complex Numbers/From real to complex numbers
Introduction
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]The real-valued coefficients are referred to as real part or imaginary part of a complex number .
- and
Gaussian Plane
[edit | edit source]With the identification of with we can draw a complex number in plane.
Polar coordinates
[edit | edit source]The following equations show the link between exponential functions and trigonometric functions:
which results from and .
Polar coordinates
[edit | edit source]Exponential function and trigonometry
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]In the relationship between trigonometric functions and Exponential function is defined with following equation:
see Euler's formula.
Difference: complex and real differentiation
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]A complex number and the complex number conjugated to it
Absolute Value
[edit | edit source]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
[edit | edit source]Pythagorean theorem
[edit | edit source]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
[edit | edit source]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
[edit | edit source]The algebraic properties result directly from the definition of the two links.
Addition
[edit | edit source]For the addition of two complex numbers with and
Vector Space - Visualization Addition
[edit | edit source][[File:Komplexe addition.svg |thumb|The addition of two complex numbers in the complex plane]]
Subtraction
[edit | edit source]For the subtraction of two complex numbers and (see addition) applies
Multiplication
[edit | edit source]For the multiplication of two complex numbers and (see addition) applies
Division
[edit | edit source]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:
[edit | edit source]Subtraction example:
[edit | edit source]Multiplication calculation example:
[edit | edit source]Computational Example Division:
[edit | edit source]Learning Activity
[edit | edit source]- 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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]does not have (in contrast to ) no order, i.e., there is complete order relation to two complex numbers.
Links between Representations of Complex Numbers
[edit | edit source]Algebraic Structure - Polar coordinates
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]As above, represents the real part and the imaginary part of the complex number .
Arithmetic operations in the polar form
[edit | edit source]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
[edit | edit source]Trigonometric form - Division
[edit | edit source]Exponential Expression
[edit | edit source]Real part and imaginary part function
[edit | edit source]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
[edit | edit source]- 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
[edit | edit source]- Maxima CAS/complex numbers
- Cauchy-Rieman equations
- Fundamental set of algebra
- Conjugation in ]
- Field and Isomorphism for the mapping between and .
- Complex Analysis
Sources of literature
[edit | edit source]- ↑ Dunham, William (September 1991), "Euler and the fundamental theorem of algebra" (PDF), The College Journal of Mathematics, 22 (4): 282–293, JSTOR 2686228
- ↑ 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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