# Complex Number

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A Complex Number is a number composed of a real number and an imaginary number.

**Context:**- It can (typically) be expressed in the form [math]\displaystyle{ a + bi }[/math] or [math]\displaystyle{ a + ib }[/math], where [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are real numbers and [math]\displaystyle{ i }[/math] is the imaginary unit (some fields such Electric Engineering represent the imaginary unit with a [math]\displaystyle{ j }[/math]) .
- It can be also expressed in the polar or trigonometric form [math]\displaystyle{ z=r(cos\theta+isin\theta) }[/math] with [math]\displaystyle{ z }[/math] being a non-zero complex number, [math]\displaystyle{ r }[/math] and [math]\displaystyle{ \theta }[/math] are real numbers. In complex analysis [math]\displaystyle{ r }[/math] is called the modulus and [math]\displaystyle{ \theta }[/math] is the argument.
- It can be a member of a Complex Vector (and a complex matrix).

**See:**Cubic Equations, Imaginary Unit, Number Line, Complex Plane, Field Extension, Complex Exponential Equations.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/complex_number Retrieved:2015-2-21.
- A
**complex number**is a number that can be expressed in the form*a*+*bi*, where*a*and*b*are real numbers and*i*is the imaginary unit, that satisfies the equation*x*^{2}= −1, that is, i^{2}= −1*. In this expression,*a*is the*real part*and*b*is the*imaginary part*of the complex number. Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane (also called Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number*a*+*bi*can be identified with the point (*a*,*b*) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone. As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century.*

- A

### 2004

- (Geoff Mannall and Michael Kenwood, 1988) ⇒ Geoff Mannall and Michael Kenwood (2004), "Further Pure Mathematics. Vol. 1." Heinemann ⇒ http://books.google.ca/books?id=QblcleAdQSQC

### 1999

- (Wolfram Math, 1999) ⇒ http://mathworld.wolfram.com/ComplexNumber.html Retrieved:2015-21-11.
- QUOTE: The complex numbers are the field [math]\displaystyle{ C }[/math] of numbers of the form [math]\displaystyle{ x+iy }[/math], where [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are real numbers and [math]\displaystyle{ i }[/math] is the imaginary unit equal to the square root of [math]\displaystyle{ -1, \sqrt{-1} }[/math]. When a single letter[math]\displaystyle{ z=x+iy }[/math] is used to denote a complex number, it is sometimes called an "affix." In component notation, [math]\displaystyle{ z=x+iy }[/math] can be written [math]\displaystyle{ (x,y) }[/math]. The field of complex numbers includes the field of real numbers as a subfield.