Complex Number

​
argument (phase) from (-π,π]
phasor form
●
This Demonstration shows the basic elements of representations of a complex number in a two-dimensional Cartesian coordinate system or in a polar coordinate system.

Details

A complex number can be visually represented using a two-dimensional Cartesian coordinate system as an ordered pair
(x,y)
of real numbers on the complex plane. The representation of a complex number in terms of its Cartesian coordinates in the form
z=x+iy
, where
i
is the imaginary unit, is called the algebraic form of that complex number. The coordinate
x=Re(z)
is called the real part and
y=Im(z)
the imaginary part of the complex number, respectively. The absolute value (or the modulus) of a complex number
z=x+iy
is defined by
abs(z)=
2
x
+
2
y
.
Alternatively to the Cartesian system, the polar coordinate system may used. In polar coordinates
(r,φ)
, the radial coordinate
r=abs(z)
and the angular coordinate
φ=arg(z)
, where
φ∈[0,2π)
, is called the argument (or angle) of the complex number
z
. A complex number
z
is then represented in the trigonometric form
z=r(cos[φ]+isin[φ])
, or in the exponential form
z=abs(z)
iarg[z]

. In technical applications the argument is often chosen from the interval
φ∈(-π,π]
and it is called phase. The corresponding exponential form is then called phasor form.
(Author was supported by project 1ET200300529 of the program Information Society of the National Research Program of the Czech Republic and by the Institutional Research Plan AV0Z10300504; the Demonstration was submitted 2008-10-01.)

External Links

Complex Number (Wolfram MathWorld)
Phasor (Wolfram MathWorld)

Permanent Citation

Štefan Porubský
​
​"Complex Number"​
​http://demonstrations.wolfram.com/ComplexNumber/​
​Wolfram Demonstrations Project​
​Published: October 8, 2008