How can a complex number be viewed in rectangular form?Can you work on an example?

A complex number can be viewed as a point in the Argand
diagram (complex plane).

Complex plane is a modified
Cartesian plane, where the real part of the complex number is represented along x axis
and the imaginary part is represented along y axis.

The
rectangular form of a complex number is usually called the algebraic form of the complex
number.

z = x + y*i

x is the
real part of the complex number z - Re(z)

y is the
imaginary part of the complex number z - Im(z)

i =
sqrt(-1)

Also, the Argand diagram helps to determine the
displacement of a point with respect to origin of the complex
plane.

The distance is called the modulus of the complex
number:

|z| = sqrt[Re(z)^2 +
Im(z)^2]

|z| = sqrt(x^2 +
y^2)

The argument of a complex number is the angle made to
real axis, x-axis.

tan a = y/x =
Re(z)/Im(z)

The geometric form of a complex number
is:

z = |z|(cos a + i*sin
a)

Example:

We
have the complex number z = 1 +
i

The given form is the
rectangular form or the algebraic form.

The
real part is:

Re(z) = 1

The
imaginary part is:

Im(z) =
1

Now, we'll write z in polar
form:

z = |z|(cos a + i*sin
a)

We'll calculate the modulus and the argument of the
complex number z:

|z| = sqrt(1^2 +
1^2)

|z| = sqrt 2

tan a 
=y/x

tan a = 1/1

tan a =
1

a = 45
degrees

The polar form
is:

z = sqrt 2(cos 45 + i*sin
45)