Polar Form of Complex Number
The polar form of a complex number is another way of representing complex numbers. The form z = a+bi is the rectangular form of a complex number, where (a, b) are the rectangular coordinates. The polar form of a complex number is represented in terms of modulus and argument of the complex number. It is said Sir Isaac Newton was the one who developed 10 different coordinate systems, one among them being the polar coordinate system.
In this minilesson, we will get an overview of representing the polar form of complex numbers, the magnitude of complex numbers, the argument of the complex number, modulus of the complex number.
What Is Polar Form of Complex Numbers?
In polar form, complex numbers are represented as the combination of the modulus r and argument θ of the complex number. The polar form of a complex number z = x + iy with coordinates (x, y) is given as z = r cosθ + i r sinθ = r (cosθ + i sinθ). The polar form is represented with the help of polar coordinates of real and imaginary numbers in the coordinate system. Consider a complex number A = x + i y in a twodimensional coordinate system:
In the figure above, we have cosθ = x/r; sinθ = y/r ⇒ x = rcosθ, y = rsinθ. Using Pythagoras theorem, we have r^{2} = x^{2} + y^{2 }and tanθ = y/x ⇒ r = √(x^{2} + y^{2 }) and θ = tan^{1} (y/x).
 The horizontal and vertical axes are the real axis and the imaginary axis, respectively.
 r  the length of the vector and θ  the angle made with the real axis, are the real and complex components of the polar form of the complex number.
 There is a point A with coordinates (x, y)
 The distance from the origin (0,0) to point A is given as r.
 The line joining the origin to point A makes an angle θ with the positive xaxis.
 The polar coordinates are given as (r, θ) and rectangular coordinates are given as (x, y).
Equation of Polar Form of Complex Numbers
The polar form of a complex number z = x + iy with coordinates (x, y) is given as z = r cosθ + i r sinθ = r (cosθ + i sinθ). The abbreviated polar form of a complex number is z = rcis θ, where r = √(x^{2} + y^{2}) and θ = tan^{1} (y/x). The components of polar form of a complex number are:
 r  It signifies absolute value or represents the modulus of the complex number.
 Angle θ  It is called the argument of the complex number.
Representation of Polar Form of Complex Number
We write complex numbers in terms of the distance from the origin and a direction (or angle) from the positive horizontal axis. Polar coordinates are expressed as (r, θ). Polar form for a complex number z=a+bi is given by z = r cosθ + i r sinθ, where r = √(a^{2} + b^{2}), a=r cosθ and b=r sinθ
 θ = tan^{1}(b / a) if z lies in the first or fourth quadrant
 θ = tan^{1}(b / a) + 180° if z lies in the second quadrant
 θ = tan^{1}(b / a)  180° if z lies in the third quadrant
Tips and Tricks:
 The values of polar coordinates and rectangular coordinates depend on each other. If we know any two values, the remaining two values can be found easily using the relation established between them.
 The conversion formulas for rectangular to polar coordinates are given as r = √(x^{2} + y^{2}) and θ = tan^{1} (y/x).
Conversion from Rectangular Form to Polar Form of Complex Number
The conversion of complex number z=a+bi from rectangular form to polar form is done using the formulas r = √(a^{2} + b^{2}), θ = tan^{1}(b / a). Consider the complex number z =  2 + 2√3 i, and determine its magnitude and argument. We note that z lies in the second quadrant, as shown below:
Using Pythagoras Theorem, the distance of z from the origin, or the magnitude of z, is z = √((2)^{2} + (2√3)^{2}) = √(4+12) = √16 = 4. Now, let us calculate the angle between the line segment joining the origin to z (OP) and the positive real direction (ray OX). Note that the angle POX' is tan^{1}(2√3/(2)) = tan^{1}(√3) = tan^{1}(√3). Since the complex number lies in the second quadrant, the argument θ =  tan^{1}(√3) + 180° =  60° + 180° = 120°. So, the polar form of complex number z =  2 + 2√3 i will be 4(cos120° + i sin120°)
Product of Polar Form of Complex Number
Let us consider two complex numbers in polar form, z = r_{1}(cos θ_{1} + i sin θ_{1}), w = r_{2}(cos θ_{2} + i sin θ_{2}), Now, let us multiply the two complex numbers:
zw = r_{1}(cos θ_{1} + i sin θ_{1}) × r_{2}(cos θ_{2} + i sin θ_{2})
= r_{1}r_{2 }[(cos θ_{1}cos θ_{2}  sin θ_{1}sin θ_{2}) + i (sin θ_{1}cos θ_{2} + cos θ_{1}sin θ_{2})]
= r_{1}r_{2 }[cos(θ_{1} + θ_{2}) + i sin(θ_{1} + θ_{2})]
Related Topics to Polar Form of Complex Number
Important Notes on Polar Form of Complex Number
 To determine the argument of z, we should plot it and observe its quadrant, and then accordingly calculate the angle which the line joining the origin to z makes with the positive real direction.
 The polar form makes operations on complex numbers easier.
 Modulus of z, z is the distance of z from the origin.
 It is easy to see that for an arbitrary complex number z = x + yi, its modulus will be z = √(x^{2} + y^{2})
 Argument of z, Arg(z), is the angle between the line joining z to the origin and the positive real direction and lies in the interval (π. π].
Examples on Polar Form of Complex Number

Example 1: The distance of point B from the origin is 4 units and the angle made with the positive xaxis is π/3. Find the polar coordinates of point B using the formula for the polar form of complex numbers.
Solution:
Distance of point B from the origin, r = 4 units
Angle made with the positive xaxis, θ = π/3
The polar coordinates of complex number at point B are (4, π/3)
Answer: Polar coordinates of complex number at point B are (4, π/3)

Example 2: Determine the modulus and argument of z = 1 + 6i using the formula for the polar form of complex numbers.
Solution:
Using the formula for modulus, we have z = √(1^{2} + 6^{2}) = √(1 + 36) = √37
Since the real part and imaginary part of the complex number z = 1 + 6i are positive, z lies in the first quadrant.
The argument of z is given by θ = tan^{1}(6 / 1) = tan^{1}6 = 80.54°
Answer: The modulus and argument of z = 1 + 6i are √37 and 80.54°, respectively.

Example 3: Find the modulus and argument of z = 1  3i using the formula for the polar form of complex numbers.
Solution:
The modulus of z = 1  3i is z = √(1^{2} + (3)^{2}) = √(1+9) = √10
Now, since the real part is positive and the imaginary part is negative, z lies in the fourth quadrant.
The angle θ is given by
θ = tan^{1}(3 / 1) = tan^{1} (3) = 71.565°
The significance of the minus sign is in the direction in which the angle needs to be measured.
Answer: The modulus and argument of z = 1  3i are √10 and 71.565°, respectively.
FAQs on Polar Form of Complex Number
What is the Polar Form of Complex Number?
In polar form, complex numbers are represented as the combination of the modulus r and argument θ of the complex number. The polar form of a complex number z = x + iy with coordinates (x, y) is given as z = r cosθ + i r sinθ = r (cosθ + i sinθ), where r = √(x^{2} + y^{2}) and θ = tan^{1} (y/x)
How do you Convert the Polar Form of Complex Number to Rectangular form?
The conversion formulas for polar to rectangular coordinates are given as x = r cosθ, y = r sinθ
What is the Argument in Polar Form of Complex Number?
The angle formed between the positive xaxis and the line joining a point with coordinates (x, y) of the complex number to the origin is called the argument of the complex number.
What is Rectangular Form and Polar form of Complex Number?
The polar form of a complex number is z = r(cosθ+isinθ), whereas it rectangular form is z=a+bi, where r = √(a^{2} + b^{2}) and θ = tan^{1} (b/a)
What is the Difference Between the Argument and Principal Argument in the Polar Form of a Complex Number?
The argument of a complex number is generally represented as (2nπ + θ), where n is an integer whereas, the value of the principal argument is such that π < θ ≤ π.
Can the Argument in the Polar Form of a Complex Number be Negative?
Yes, the argument of a complex number can be negative, such as for 5+3i.
How are Magnitude and Argument in Polar Form of Complex Number related?
The length of the line segment that is the real axis is called the modulus of the complex number z. The angle measured from the positive real axis to the line segment is called the argument of the complex number, arg(z).
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