number, complex - definitie. Wat is number, complex
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Wat (wie) is number, complex - definitie

ELEMENT OF THE REAL COMMUTATIVE ASSOCIATIVE ALGEBRA ℝ[J] / (J² − 1), I.E. THE REALS WITH AN EXTRA SQUARE ROOT OF +1 ADJOINED
Split-complex; Lorentz number; Split complex; Split complex number; Split-complex plane; Perplex number; Perplex numbers; Hallucinatory number; Split-complex numbers; Double number; Hyperbolic number; Split binarion; Hyperbolic unit
  • \R^2}}

complex number         
  • 1=''i''<sup>2</sup> = −1}}.
  • ''z''<sup>2</sup> + 2 + 2''i''}}}}
  • z}}}} in the complex plane
  • 3 + ''i''}} (red triangle). The red triangle is rotated to match the vertex of the blue one (the adding of both angles in the terms ''φ''<sub>1</sub>+''φ''<sub>2</sub> in the equation) and stretched by the length of the [[hypotenuse]] of the blue triangle (the multiplication of both radiuses, as per term ''r''<sub>1</sub>''r''<sub>2</sub> in the equation).
  • z}}, as a point (black) and its position vector (blue)
  • r}} locate a point in the complex plane.
  • y}}.
  • The Mandelbrot set with the real and imaginary axes labeled.
  • using straightedge and compass]].
  • sin(1/''z'')}}. White parts inside refer to numbers having large absolute values.
  • Addition of two complex numbers can be done geometrically by constructing a parallelogram.
NUMBER THAT CAN BE PUT IN THE FORM A + BI, WHERE A AND B ARE REAL NUMBERS AND I IS CALLED THE IMAGINARY UNIT
Complex numbers; Real part; Imaginary part; Complex Number; Complex field; Complex Numbers; Complex number field; Mod-arg form; Imaginary plane; Complex arithmetic; Wessel diagram; ℂ; C number; Complex addition; Complex division; Polar form; ℜ; ℑ; C numbers; Classification of complex numbers; Complex-valued; Principal argument; Non real numbers; Complex domain; Real and imaginary parts; History of complex numbers; A+ib; Complex value; Complex math; Complex mathematics; Division of complex numbers; Multiplication of complex numbers; Applications of complex numbers; A+bi; Generalizations of complex numbers; Generalization of complex numbers; Complex quantity; Complex square; Matrix representation of complex numbers
<mathematics> A number of the form x+iy where i is the square root of -1, and x and y are real numbers, known as the "real" and "imaginary" part. Complex numbers can be plotted as points on a two-dimensional plane, known as an {Argand diagram}, where x and y are the Cartesian coordinates. An alternative, polar notation, expresses a complex number as (r e^it) where e is the base of natural logarithms, and r and t are real numbers, known as the magnitude and phase. The two forms are related: r e^it = r cos(t) + i r sin(t) = x + i y where x = r cos(t) y = r sin(t) All solutions of any polynomial equation can be expressed as complex numbers. This is the so-called {Fundamental Theorem of Algebra}, first proved by Cauchy. Complex numbers are useful in many fields of physics, such as electromagnetism because they are a useful way of representing a magnitude and phase as a single quantity. (1995-04-10)
Complex number         
  • 1=''i''<sup>2</sup> = −1}}.
  • ''z''<sup>2</sup> + 2 + 2''i''}}}}
  • z}}}} in the complex plane
  • 3 + ''i''}} (red triangle). The red triangle is rotated to match the vertex of the blue one (the adding of both angles in the terms ''φ''<sub>1</sub>+''φ''<sub>2</sub> in the equation) and stretched by the length of the [[hypotenuse]] of the blue triangle (the multiplication of both radiuses, as per term ''r''<sub>1</sub>''r''<sub>2</sub> in the equation).
  • z}}, as a point (black) and its position vector (blue)
  • r}} locate a point in the complex plane.
  • y}}.
  • The Mandelbrot set with the real and imaginary axes labeled.
  • using straightedge and compass]].
  • sin(1/''z'')}}. White parts inside refer to numbers having large absolute values.
  • Addition of two complex numbers can be done geometrically by constructing a parallelogram.
NUMBER THAT CAN BE PUT IN THE FORM A + BI, WHERE A AND B ARE REAL NUMBERS AND I IS CALLED THE IMAGINARY UNIT
Complex numbers; Real part; Imaginary part; Complex Number; Complex field; Complex Numbers; Complex number field; Mod-arg form; Imaginary plane; Complex arithmetic; Wessel diagram; ℂ; C number; Complex addition; Complex division; Polar form; ℜ; ℑ; C numbers; Classification of complex numbers; Complex-valued; Principal argument; Non real numbers; Complex domain; Real and imaginary parts; History of complex numbers; A+ib; Complex value; Complex math; Complex mathematics; Division of complex numbers; Multiplication of complex numbers; Applications of complex numbers; A+bi; Generalizations of complex numbers; Generalization of complex numbers; Complex quantity; Complex square; Matrix representation of complex numbers

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation i2 = −1; every complex number can be expressed in the form a + bi, where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols C {\displaystyle \mathbb {C} } or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation ( x + 1 ) 2 = − 9 {\displaystyle (x+1)^{2}=-9} has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions −1 + 3i and −1 − 3i.

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i2 = −1 combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with {1, i} as a standard basis.

This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.

In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.

https://en.wikipedia.org/wiki/Complex_number
Applications of dual quaternions to 2D geometry         
FOUR-DIMENSIONAL ALGEBRA OVER THE REAL NUMBERS
Draft:Dual-complex numbers; Dual complex number; Dual complex numbers; Semiquaternions; Study's quaternions; Semiquaternion; Dual-complex numbers; Dual-complex number
In this article, we discuss certain applications of the dual quaternion algebra to 2D geometry. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which we will call the planar quaternions.
https://en.wikipedia.org/wiki/Applications_of_dual_quaternions_to_2D_geometry

Wikipedia

Split-complex number

In algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1. {\displaystyle j^{2}=1.} A split-complex number has two real number components x and y, and is written z = x + y j . {\displaystyle z=x+yj.} The conjugate of z is z ∗ = x − y j . {\displaystyle z^{*}=x-yj.} Since j 2 = 1 , {\displaystyle j^{2}=1,} the product of a number z with its conjugate is N ( z ) := z z ∗ = x 2 − y 2 , {\displaystyle N(z):=zz^{*}=x^{2}-y^{2},} an isotropic quadratic form.

The collection D of all split complex numbers z = x + y j {\displaystyle z=x+yj} for x , y ∈ R {\displaystyle x,y\in \mathbb {R} } forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies N ( w z ) = N ( w ) N ( z ) . {\displaystyle N(wz)=N(w)N(z).} This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.

A similar algebra based on R 2 {\displaystyle \mathbb {R} ^{2}} and component-wise operations of addition and multiplication, ( R 2 , + , × , x y ) , {\displaystyle (\mathbb {R} ^{2},+,\times ,xy),} where xy is the quadratic form on R 2 , {\displaystyle \mathbb {R} ^{2},} also forms a quadratic space. The ring isomorphism

relates proportional quadratic forms, but the mapping is not an isometry since the multiplicative identity (1, 1) of R 2 {\displaystyle \mathbb {R} ^{2}} is at a distance 2 {\displaystyle {\sqrt {2}}} from 0, which is normalized in D.

Split-complex numbers have many other names; see § Synonyms below. See the article Motor variable for functions of a split-complex number.

https://en.wikipedia.org/wiki/Split-complex number
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