Explanation:
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend
the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for
the real part and adding a vertical axis to plot the imaginary part.
A complex
number can be visually represented as a pair of numbers forming a vector on a
diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the square root of -1.
The complex plane
A complex
number plotted as a point (red) and position vector (blue) on an Argand diagram; a + bi is the rectangular expression of the point.
A complex
number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram named after Jean-Robert Argand. The numbers
are conventionally plotted using the real part as the horizontal component, and
imaginary part as vertical (see Figure 1). These two values used to identify a
given complex number are therefore called its Cartesian, rectangular, or algebraic form.
The defining
characteristic of a position vector is that it has magnitude and direction.
These are emphasized in a complex number's polar form and it turns out notably that
the operations of addition and multiplication take on a very natural geometric
character when complex numbers are viewed as position vectors: addition
corresponds to vector addition while multiplication corresponds to multiplying
their magnitudes and adding their arguments (i.e. the angles they make with the x axis). Viewed in this way the
multiplication of a complex number by i corresponds to rotating a
complex number anticlockwise through 90° about the origin.
Polar
form
Figure 2: The
argument φ and modulus r locate a point on an Argand
diagram; r (cosφ + isinφ) or re^{i}^{φ} are polar expressions of the point.
Absolute value and argument
Another way of encoding points in the complex plane other than
using the x- and y-coordinates is to use the
distance of a point P to O,
the point whose coordinates are (0, 0) (origin), and the angle of the line
through P and O.
This idea leads to the polar form of complex numbers.
The absolute
value (or modulus or magnitude)
of a complex number z = x+yi is
If z is a real number
(i.e., y = 0), then r = |x|. In general, by Pythagoras' theorem, r is the distance of the
point P representing the complex number z to the origin.
The argument or phase of z is the angle to the
real axis, and is written as arg
(z). As with the modulus, the argument can
be found from the rectangular form x + iy:
0 \\
\arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\
\arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\
\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\
-\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\
\mbox{indeterminate } & \mbox{if } x = 0 \mbox{ and } y = 0.
\end{cases}" *>
The value of φ must always be expressed in radians. It can change by any
multiple of 2π and still give the same angle. Hence, the arg function is
sometimes considered as multivalued. (Our solved example in
mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Complex_number
The
above explanation is copied from Wikipedia, the free encyclopedia and is
remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported
License.