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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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)


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)