complex numbers - definitie. Wat is complex numbers
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Wat (wie) is complex numbers - definitie

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
  • 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.

complex number         
<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         

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.

Num.         
  • [[Priest]], [[Levite]], and furnishings of the [[Tabernacle]]
  • [[Balaam]] and the Angel (illustration from the 1493 ''[[Nuremberg Chronicle]]'')
FOURTH BOOK OF THE BIBLE
Num.; Numbers (book of Bible); Numbers, Book of; Book of numbers; Book of Num.; Book Of Numbers; The Book of Numbers; Numbers 30; Numbers 32; Numbers 6; Numbers 16; Numbers 34; Numbers 26; Numbers 27; Numbers 36; Numbers 35; Numbers 22; Numbers 24; Numbers 28; Numbers 3; Numbers 29; Numbers 14; Numbers 7; Numbers 4; Numbers 23; Numbers 17; Numbers 19; Numbers 12; Numbers 20; Numbers 8; Numbers 18; Numbers 9
¦ abbreviation Numbers (in biblical references).

Wikipedia

Complex number

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 i 2 = 1 {\displaystyle i^{2}=-1} ; every complex number can be expressed in the form a + b i {\displaystyle 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 + b i {\displaystyle 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 + 3 i {\displaystyle -1+3i} and 1 3 i {\displaystyle -1-3i} .

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i 2 = 1 {\displaystyle i^{2}=-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.