Polar and Exponential Forms of Complex Numbers

Before going to discuss the different forms of complex numbers and conversion between them we have to know about complex numbers. Complex numbers are a part of mathematics represented as a combination of a real part and an imaginary part. A complex number contains the real part as well as the imaginary part where the real part is the constant number and the imaginary part contains the variable “i” with a constant coefficient. Let a+ib be a complex number then a is called a real part and b called an imaginary coefficient.

There are three forms of complex numbers. They are, 

1. General form
2. Polar form
3. Exponential form

General form of a complex number

The general form of the complex number is represented as z = a + ib where a is called as real part and b is called the imaginary part of the complex number. It can also be represented in the diagrammatic form below.

Diagrammatic representation of the complex number

Polar form representation of complex numbers

The Polar form of the complex number is represented as z = r(cos∅ + i sin∅) where rcos∅ is called as real part and rsin∅ is called the imaginary part of the complex number. It can also be represented in the cartesian form below.

Diagrammatic form of polar form of complex numbers

In the above diagram a = rcos∅ and b = rsin∅. In general form, a + ib where a = real part and b = imaginary part, but in polar form there is an angle is included in the cartesian where a=rcos∅ and b=rsin∅ . Here r is the square root of the sum of squares of both a and b and also ∅ can also have a formula which is tan^-1(imaginary part/real part). Therefore r can be represented as a Square root (a² + b²). Therefore ∅ can be represented as tan^-1(b/a) where b is the imaginary part and a is a real part.

Exponential form representation of complex numbers

The Exponential form of the complex number is represented as z = r exp(i∅) where exp(i∅) is also represented as cos∅ + i sin∅. From this, I can say that the Exponential form, polar form, and general form are related closely.

Z = r(cos∅ + i sin∅)

Z = r e^{i ∅}

Z = r angle(∅) [This is a phasor representation of exponential form]

Different Complex number representation

1. In general form Z = a + ib
2. In polar form Z = r(cos∅ + i sin∅)
3. In Exponential form Z = r e^{i ∅}

Conversion of complex numbers

Complex numbers can be converted to convenient polar form or exponential form or general form. How this was converted is shown below.

Converting general form to Polar form

1. Before converting the general form to the Polar form check whether the general form is in the form of a+ib and values of a, and b is known already in the general form.
2. The polar form looks like Z = r(cos∅ + i sin∅).
3. To convert into the above polar form structure we need to know how a, and b values in general form relate to r, ∅.
4. The formulas of r,∅ are r = √(a² + b²), ∅ = tan^-1(b/a).
5. The above formulas in terms of a, and b are derived to convert from general form to polar form so that we can substitute r, ∅ in polar form Z = r(cos∅ + i sin∅). 

Converting general form to Exponential form

1. Before convert general form to an exponential form check whether the general form is in the form of Z = a + ib and values of a, and b are known already in a general form.
2. The exponential form looks like Z = r e^{i ∅}.
3. To convert into the above exponential form structure we need to know how a, and b values in general form relate to r, ∅.
4. The formulas of r, ∅ are r = √(a² + b²), ∅ = tan^-1(b/a).
5. The above formulas in terms of a and b is derived to convert from general form to polar form so that we can substitute r, ∅ in polar form Z = r e^{i ∅}. 

Converting polar form to general form

1. Before converting polar form to general form check whether the polar form is in the form of Z = r(cos∅ + i sin∅) and values of r, ∅ which is known already in polar form.
2. The general form looks like Z = a + ib.
3. To convert into the above general form structure we need to know how r,∅ values in general form relate to a, b.
4. The formulas of a,b are a = rcos∅, b = rsin∅ where r,∅ is known already in polar form.
5. The above formulas in terms of r,∅ is derived to convert from polar form to general form so that we can substitute a, b in general form Z = a + ib.

Converting polar form to exponential form

1. Before converting polar form to exponential form check whether the polar form is in the form of Z = r(cos∅ + i sin∅) and values of r, ∅ which is known already in polar form.
2. The exponential form looks like Z = re^i∅.
3. To convert into the above exponential form structure we need to know r,∅ values only because exponential form is also want r,∅ values.
4. Substitute r,∅ value to Z = re^i∅ to convert from polar form to exponential form.

Converting exponential form to general form

1. Before converting exponential form to general form check whether the exponential form is in the form of Z = re^i∅ and values of r,∅ is known already in exponential form.
2. The general form looks like Z = a + ib.
3. To convert into the above general form structure we need to know how r,∅ values in general form relate to a,b.
4. Formulas for a,b derived from Z = re^i∅ = r(cos∅ + isin∅) where a = rcos∅, b = rsin∅. Since e^i∅ = cos∅ + isin∅ we know it already in trigonometry.
5. The above formulas in terms of r,∅ is derived to convert from exponential to general form so that we can substitute a, b in general form Z = a + ib. 

Converting exponential form to polar form

1. Before converting exponential form to polar form check whether the exponential form is in the form of Z = re^i∅  and values of r, ∅ is known already in exponential form.
2. The polar form looks like Z = r(cos∅ + isin∅).
3. To convert into the above polar form structure we need to know r,∅ values only because the polar form is also want r,∅ values.
4. Substitute r, ∅ value to Z = r(cos∅ + isin∅) to convert from exponential form to polar form.

Sample Questions

Question 1: Convert 2 + i 9 into polar form.

Solution:

Let Z = 2 + i 9 

Z is in the form of a + ib

Where a = 2 and b = 9

Polar form of complex number Z = r (cos∅ + i sin∅)

Compare a + ib with polar form r cos∅ + i rsin∅

Here r = √(a² + b²)

r = √(2² + 9²) 

r = √(4+81) 

r = square root(85)

r = 9.2

And ∅ has formula which is tan(b/a)

∅ = tan^-1(b/a) = tan^-1(9/2)

∅ = 77°

From this r,∅  we can represent general form 2 + i9 into polar form Z = 9.2(cos 77° + i sin 77°)

Question 2: Convert the polar form (r, ∅) = (-1,0) into general form.

Solution:  

Given that polar form coordinates (r, ∅) = (-1, 0)

General form or rectangular form of complex number Z = a + ib

Where a = rcos∅, b = r sin∅

From the given polar form in question a = -1 × cos(0) and b = -1 × sin(0)

a = -1, b = 0 [cos(0) = 1 and sin(0) = 0]

General form Z = a + ib = -1 + i 0.

Question 3: Convert the exponential form 2eⁱ⁸⁰ into general form as well as polar form.

Solution:      

Given that exponential form 2eⁱ⁹⁰

2 eⁱ⁸⁰ is in the form of  r e^i∅

r e^i∅ is represented in polar form as r(cos∅ + isin∅)

Where r=2 and ∅=80 by comparing

Substitute r,∅ in polar form r(cos∅+isin∅) we get polar form as  2(cos80+i sin80)

In the above polar form a=2 cos80 and b=2 sin80 by comparing general form and polar form 

a = 2 cos80 = 0 .17 and b = 2 sin80 = 0.98

General form a + ib = 0.17 + i 0.98.

Question 4: Convert the polar form (r, ∅) = (1, 90) into general form.

Solution:

Given that polar form coordinates (r, ∅) = (1, 89)

General form or rectangular form of complex number Z = a + ib

Where a = rcos∅, b = r sin∅

From the given polar form in question a = 1× cos(89) and b = 1 × sin(89)

a = 0.017, b = 0.99 [cos(89) = 0.017 and sin(89) = 0.99]

General form Z = a + ib = 0.017 + i 0.99   

Question 5: Convert the polar form (r, ∅) = (4, 45°) into the Exponential form.

Solution:  

Given that polar form coordinates (r,∅)=(4,45)

To convert into Exponential form we have the formula  r e^i∅

Where r = 4 and ∅ = 45

Therefore Exponential form r e^i∅ = 4eⁱ⁴⁵

Question 6: Convert Z = 7 + i9 into Exponential form.

Solution:    

To convert into exponential form we have the formula re^i∅

Compare Z = 7 + i9 with Z = a + ib then a = 7 and b = 9

Where r = √(a² + b²)

r = √(7 × 7+ 9 × 9)

r = √(130)

r = 11.4

Where ∅ = tan^-1(b/a) = tan^-1(9/7)

∅ = 52.12°

Therefore Exponential form re^i∅  = 11.4 ei^52.12