How to transform exponential complex numbers to rectangular form?

Complex numbers are the representation of numerical expressions in the form of a+ib where a and b are real integers and i stands for imaginary numbers. Consider a complex number 2 + 3i, in this expression 2 and 3 are the integers or real numbers whereas, ‘i’ represents the imaginary number.

Forms of the complex number

On the basis of the way of representation, these complex numbers are divided into three different forms. They are rectangular form, polar form, and exponential form.

Forms	Formula
Rectangular form	z = a + ib
Polar form	z = r(cosθ + isinθ)
exponential form	z = reiθ

Types of the complex number 

The standard representation of the complex number is given in the form of a + ib. And, as per this representation complex numbers are classified into four different types. The four types are zero complex numbers, purely real numbers, purely imaginary numbers, and imaginary numbers.

Types of the complex number	Requirement	Example
Zero complex number	a = 0 and b = 0	0 (zero)
Purely real number	a ≠ 0 and b = 0	2, 3, 4, 5, 6, 7
Purely imaginary number	a = 0 and b ≠ 0	-3i, 4i, 7i
Imaginary number	a ≠ 0 and b ≠ 0	(-2 + i)(1 – i)

How to transform exponential complex numbers to rectangular form?

Solution:

Exponential complex numbers are given in the form of re^iθ where r stands for the amplitude of the expression and θ is the phase that is expressed in unit radian.

At the same time, rectangular forms of a complex number are expressed in form of x + iy where x and y are the real integers and ‘i’ represents an imaginary number.

Derivation of rectangular form

There is an exponential form

= re^iθ

Where θ is the phase expressed in unit radians.

Now, firstly determine the value of x and y from the given expression to convert it in rectangular form x + iy

rcos(θ) = x

rsin(θ) = y

Hence, the rectangular form would be

x + iy

Sample Problems

Question 1: Convert the complex number 20e^1.95i into a rectangular form.

Solution:

=> 20e^1.95i

Finding the value of x and y

x = 20 cos(1.95) = -7.4

y = 20 sin(1.95) = 18.58

Rectangular form = -7.4 + 18.58i

Question 2: Convert the complex number 40e^0.95i into a rectangular form.

Solution:

=> 40e^0.95i

Finding the value of x and y

x = 40cos(0.95) = 23.27

y = 40sin(0.95) = -32.54

Rectangular form = 23.27 – 32.54i

Question 3: Convert the complex number 53.45e^1.27i into a rectangular form.

Solution:

=> 53.45e^1.27i

Finding the value of x and y

x = 53.45cos(1.27) = 16

y = 53.45sin(1.27) = 51

Rectangular form = 16 + 51i