Polar Representation of Complex Numbers

Complex numbers, which take the form z = x + yi, can also be represented in a way that highlights their geometric properties. This alternative representation is known as the polar form. The polar representation of a complex number expresses it in terms of its magnitude (modulus) and direction (argument), offering a more intuitive understanding of its position in the complex plane.

In polar form, a complex number is written as z = r(cos⁡θ + isin⁡θ), where:

• r is the modulus, representing the distance from the origin to the point (x, y).
• θ is the argument, indicating the angle between the positive real axis and the line segment joining the origin to (x, y).

This form is particularly useful in various mathematical and engineering applications, such as simplifying the multiplication and division of complex numbers. By converting complex numbers from rectangular to polar form, operations involving these numbers become more manageable and visually intuitive.

Geometrical or Algebraic Form of a Complex Number

The complex number z = (x + i y) is represented by a point P (x, y) on the Argand plane, And every point on the Argand/Complex plane represents a unique complex number. If a complex number is purely real then it's imaginary part Im(z) = 0 and it lies exactly on the real axis (X-axis), whereas a purely imaginary complex number has its real part Re(z) =0 and it lies exactly on the imaginary axis (Y-axis).

Note: If a point P (x, y) represents a complex number z on the Argand plane then the complex number Z = (x + i y) is known as the affix of the point P.

In the above representation of the complex number length of the line segment, OP is equal to the modulus of the complex number and the angle θ is the angle formed by OP in the anticlockwise direction with the positive direction of the X-axis known as the amplitude or argument of the complex number denoted by amp(z) or arg(z).

From the above figure, θ can be calculated as follows

tan(θ) = y / x

⇒ tan(θ) = |Im(z) / Re(z)|

⇒ θ = tan^-1{|Im(z) / Re(z)|}

Note: The angle θ can have infinite values in the multiple of 2π. But the unique value of θ that lies in the range (-π ≤ θ ≤ π) is called the principal argument or principal value of the amplitude. The point to be remembered is the value of the principal argument of a complex number (z) depends on the position of the complex number (z) i.e the quadrant in which the point P representing the complex number (z) lies.

Let's discuss the different cases to find out the value of the principal argument. Let α be the acute angle subtended by OP with the X-axis and θ is the principal argument of the complex number (z). 

Case 1. When the complex number z = (x + i y) lies in the first quadrant i.e. x > 0 & y > 0 then the value of the principal argument (θ = α).

Case 2. When the complex number z = (x + i y) lies in the second quadrant i.e. x < 0 & y > 0 then the value of the principal argument (θ = π - α).

Case 3. When the complex number z = (x + i y) lies in the third quadrant i.e. x < 0 & y < 0 then the value of the principal argument (θ = α - π).

Case 4. When the complex number z = (x + i y) lies in the fourth quadrant i.e. x > 0 & y < 0 then the value of the principal argument (θ = -α).

Polar or Trigonometrical Form of a Complex Number

We have seen the geometrical form of a complex number (z) in which it is represented by (x, y) on the Argand plane, OP = |z| and arg(z) = θ. Now we will use the geometrical form of a complex number to obtain its polar form. In the polar or trigonometrical form, the complex number (z) is represented by (r, θ). where r = |z| & θ = arg(z).

From the above figure, we can find OM = x = |z| cos θ and MP = y = |z| sin θ ;

Now put the values of x & y in z = (x + i y) and we get z = |z| cos θ + i |z| sin θ 

Taking |z| common, z = |z| (cos θ + i sin θ) 

And putting |z| = r ;

The polar form of a complex number (z) is given by

z = r (cos θ + i sin θ) ; 

If we take the general value of the argument arg(z) = 2n π + θ, then the polar form of z is given by

z = r [cos (2n π + θ) + i sin (2n π + θ)] ; where n is an integer.

As we have θ in the expression of the polar form of z then again there will be four different cases depending upon the principal argument values θ  in the four quadrants. Let's discuss them using the results obtained above for the principal argument θ.

• Case 1. When the complex number z lies in the first quadrant then the value of the principal argument (θ = α). So, the polar form of z = r (cos α + i sin α).
• Case 2. When the complex number z lies in the second quadrant then the value of the principal argument (θ = π - α). So, the polar form of z = r [cos (π - α) + i sin (π - α)] or z = r (-cos α + i sin α)
• Case 3. When the complex number z lies in the third quadrant then the value of the principal argument (θ = α - π). So, the polar form of z = r [cos (α - π) + i sin (α - π)] or z = r (-cos α - i sin α)
• Case 4. When the complex number z lies in the fourth quadrant then the value of the principal argument (θ = -α). So, the polar form of z = r [cos (-α) + i sin (-α)] or z = r (cos α - i sin α).

As we have discussed both the forms of complex numbers and we have also understood that the polar form of a complex number z (r, θ) can easily be obtained from its geometrical/algebraic form z (x, y); Let's see one sample problem on the conversion of the complex number in the algebraic form to the polar form.

Read More,

• Real Numbers
• Imaginary Numbers
• Properties of Real Numbers

Solved Examples on Polar Representation of Complex Numbers

Example 1: If (z = -i) or (z = 0 - i) is a complex number in the algebraic form then its polar form representation is given by

Solution:

As z = (0 - i) ≈ P (0, -1), it lies on the Y-axis (θ = -α)

α = tan-1(Im(z) / Re(z))

α = tan-1(|(-1) / 0|) = tan^-1(|-∞|) = π / 2

θ = -α = -(π / 2)

r = |z| = √(0² + 1²) = 1

z = r (cos θ + i sin θ)

z = 1 [cos(π / 2) - i sin(π / 2)]

Example 2: If z = (1 + i) is a complex number in the algebraic form then its polar form representation is given by

Solution:

As z = (1 + i) ≈ P (1, 1), it lies in the first quadrant then (θ = α)

α = tan^-1(Im(z) / Re(z))

α = tan^-1(1) = π / 4 

θ = α = π / 4

r = |z| = √(1² + 1²) = √2

z = r (cos θ + i sin θ)

z = √2 [cos(π / 4) + i sin(π / 4)]

Example 3: If z = (-1 - √3 i) is a complex number in the algebraic form then its polar form representation is given by

Solution:

As z = (-1 - √3 i) ≈ P (-1, -√3), it lies in the third quadrant then (θ = α - π)

α = tan-1(Im(z) / Re(z))

α = tan-1(|-√3 /(-1)|) = tan-1(√3) = π / 3

θ = (α - π)

θ = (π  / 3) - π = - (2 π)  / 3

r = |z| = √((-1)² + (-√3)²) = √4 = 2

z = r (cos θ + i sin θ)

z = 2 [cos(-(2π) / 3) + i sin((-2π) / 3)]

z = 2 [cos(2π / 3) - i sin(2π / 3)] or

z = 2 [- cos(π / 3) - i sin(π / 3)]

Practice Problems on Polar Representation of Complex Numbers

Problem 1: Express the complex number z = 3 + 4i in polar form.

Problem 2: Convert the complex number z = -5 + 12i to its polar representation.

Problem 3: Find the polar form of the complex number z = -7 - 24i.

Problem 4: Determine the polar representation of the complex number z = 8i.

Problem 5: Express the complex number z = -6 in polar form.

Problem 6: Find the magnitude and angle for the 1−i.

Problem 7: Convert 0+2i to polar form.

Problem 8: Express −2−2i in polar coordinates.

Problem 9: Convert 2−3i to polar form.

Problem 10: Find the polar form of −4+4i.

Read More:

• Introduction to Complex Analysis
• Complex Numbers

Summary

The polar representation of complex numbers provides the powerful alternative to the Cartesian form particularly advantageous for the operations like multiplication, division and exponentiation. By expressing complex numbers in the terms of their magnitude and angle one can simplify many calculations and gain deeper insights into their geometric properties. This form is essential for the understanding and working with the complex numbers in the fields such as the engineering, physics and applied mathematics.

What is the polar representation of a complex number?

The polar representation of a complex number expresses it in terms of its magnitude (or modulus) and angle (or argument). A complex number z = x + yi can be written as z = r(cos⁡θ + isin⁡θ) or z = re^iθ, where rrr is the magnitude and θ is the angle.

How do you find the magnitude of a complex number?

The magnitude (or modulus) of a complex number z = x + yi is found using the formula r = √(x² + y²).

What is Euler's formula?

Euler's formula states that for any real number θ, e^iθ = cos⁡θ + isin⁡θ. This is used in the polar representation of complex numbers as z = re^iθ.

How do you multiply two complex numbers in polar form?

To multiply two complex numbers in polar form z₁ = r₁e^ix and z₂ = r₂ e^iy​, you multiply their magnitudes and add their angles:

z₁z₂ = r₁r₂ e^i(x+y)