How Do You Write a Complex Number in Polar Form?

To write a complex number in polar form, you convert it from its standard rectangular form (often written as z = x + yi) into a form that uses its distance from the origin and its angle relative to the positive x-axis.

The polar form of a complex number z = x + yi is generally expressed as z = r(cos θ + i sin θ). An abbreviated way to write this, as stated in the reference, is z = rcis θ.

Components of Polar Form

The polar form relies on two key components:

• r (the modulus or absolute value): This represents the distance of the complex number from the origin (0,0) in the complex plane.
• θ (the argument): This represents the angle between the positive x-axis and the line segment connecting the origin to the complex number in the complex plane.

As per the reference, r signifies the absolute value or represents the modulus of the complex number.

Steps to Convert from Rectangular to Polar Form

Here’s how to convert a complex number z = x + yi to its polar form z = rcis θ:

1. Calculate the Modulus (r):

   • The modulus r is found using the formula derived from the Pythagorean theorem:
     r = √(x² + y²)
   • x is the real part of the complex number.
   • y is the imaginary part of the complex number.

2. Calculate the Argument (θ):

   • The argument θ is the angle whose tangent is y/x. The principal value of the argument lies typically in the interval (-π, π] or [0, 2π).
   • Use the inverse tangent function:
     θ = tan⁻¹ (y/x)
   • Important Consideration: The tan⁻¹ (y/x) function on calculators usually gives an angle in the first or fourth quadrant. You must consider the quadrant of the original complex number (x, y) to find the correct angle θ.
     • If x > 0 (Quadrant I or IV), θ = tan⁻¹ (y/x) (or + 2π if you need a positive angle).
     • If x < 0 and y ≥ 0 (Quadrant II), θ = tan⁻¹ (y/x) + π.
     • If x < 0 and y < 0 (Quadrant III), θ = tan⁻¹ (y/x) - π (or + π if using [0, 2π)).
     • If x = 0, θ = π/2 if y > 0, θ = -π/2 (or 3π/2) if y < 0. If x=0 and y=0, the modulus r is 0, and the argument θ is undefined (or can be any angle).

3. Write the Complex Number in Polar Form:

   • Once you have calculated r and θ, write the complex number as:
     z = r(cos θ + i sin θ)
   • Or using the abbreviated form from the reference:
     z = rcis θ

Summary Table

Component	Symbol	Calculation Formula	Description
Modulus	r	√(x² + y²)	Distance from the origin; absolute value.
Argument	θ	tan⁻¹ (y/x)	Angle with the positive x-axis (with quadrant correction).

Example Conversion

Let's convert the complex number z = 1 + i to polar form.

1. Identify x and y:

   • x = 1
   • y = 1

2. Calculate r:

   • r = √(1² + 1²) = √(1 + 1) = √2

3. Calculate θ:

   • The point (1, 1) is in Quadrant I.
   • θ = tan⁻¹ (1/1) = tan⁻¹ (1)
   • In radians, θ = π/4. (Or 45° in degrees).

4. Write in Polar Form:

   • Using the standard form: z = √2 (cos(π/4) + i sin(π/4))
   • Using the abbreviated form: z = √2 cis(π/4)

This shows that the complex number 1 + i can be written as √2 cis(π/4) in polar form, using the calculated modulus r=√2 and argument θ=π/4.

Why Use Polar Form?

Polar form is particularly useful for operations like multiplying, dividing, and taking powers or roots of complex numbers, as these operations become much simpler in polar coordinates compared to rectangular form.