How Do You Write a Complex Number in Polar Form?

A complex number can be written in polar form using its magnitude (or modulus) and angle (or argument).

Here's how to do it:

1. Find the magnitude: The magnitude, denoted by r, is the distance of the complex number from the origin in the complex plane. You can calculate it using the Pythagorean theorem:

   • For a complex number z = a + bi, where a and b are real numbers, the magnitude is r = √(a² + b²).

2. Find the angle: The angle, denoted by θ, is the angle between the positive real axis and the line connecting the origin to the complex number in the complex plane. You can calculate it using trigonometric functions:

   • θ = arctan(b/a) if a is positive.
   • θ = arctan(b/a) + π if a is negative and b is positive.
   • θ = arctan(b/a) - π if a is negative and b is negative.
   • θ = π/2 if a is zero and b is positive.
   • θ = -π/2 if a is zero and b is negative.

3. Write the polar form: The polar form of a complex number z is z = r(cos θ + i sin θ).

Example:

Let's say we have the complex number z = 3 + 4i.

• Magnitude: r = √(3² + 4²) = 5
• Angle: θ = arctan(4/3) ≈ 0.93 radians
• Polar form: z = 5(cos 0.93 + i sin 0.93)

Practical Insights:

• Polar form is useful for representing complex numbers in applications like electrical engineering, signal processing, and fluid dynamics.
• It simplifies complex calculations, especially when dealing with multiplication and division of complex numbers.
• The angle of a complex number in polar form can be expressed in degrees or radians.