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Example:Finding Argument and Modulus

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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 reiφ 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:



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)




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.