Polar Form of Complex Numbers

Understanding magnitude and argument in the complex plane

Introduction to Polar Representation

In the article on geometric representation, we learned that every complex number can be represented as a vector in the complex plane. This vector is uniquely determined by its real and imaginary parts (rectangular form).

However, there is another equally valid way to describe a complex number: using its magnitude (length) and argument (angle). This is called the polar form.

Key Insight:

Any vector can be uniquely described by either:

  • Rectangular Form: Using horizontal (real) and vertical (imaginary) components
  • Polar Form: Using magnitude (distance from origin) and angle (direction)

Components of Polar Form

A complex number in polar form is completely described by two parameters:

The Magnitude (Modulus)

Magnitude Definition:

The magnitude \(r\) (or \(|z|\)) is the distance from the origin to the point representing the complex number in the complex plane.

Magnitude Formula:
\(\displaystyle r = |z| = \sqrt{a^2 + b^2}\)

The Argument (Angle)

Argument Definition:

The argument \(\varphi\) (or \(\theta\)) is the angle between the positive real axis and the vector representing the complex number.

The angle is measured:

  • Counterclockwise: Positive angles
  • Clockwise: Negative angles
Angle Measurement:

Angles are typically expressed in radians (0 to 2π) or degrees (0° to 360°). The principal argument \(\text{arg}(z)\) is usually taken to be in the range \((-\pi, \pi]\).

Polar Form Definition

The polar form of a complex number is expressed in several equivalent notations:

Polar Form Notations:

For a complex number with magnitude \(r\) and argument \(\varphi\):

Ordered Pair
\(\displaystyle z = (r, \varphi)\)

Magnitude and argument as a pair
Trigonometric Form
\(\displaystyle z = r(\cos\varphi + i\sin\varphi)\)

Using trigonometric functions
Exponential Form
\(\displaystyle z = re^{i\varphi}\)

Using Euler's formula
CIS Notation
\(\displaystyle z = r\,\text{cis}(\varphi)\)

Compact trigonometric form

Polar Form in the Complex Plane:

Polar form representation in complex plane

Vector with magnitude \(r\) and angle \(\varphi\) from the positive real axis

Converting Between Forms

From Rectangular to Polar Form

To convert a complex number from rectangular form \(z = a + bi\) to polar form \(z = r(\cos\varphi + i\sin\varphi)\):

1Calculate the magnitude
\(\displaystyle r = |z| = \sqrt{a^2 + b^2}\)
2Calculate the argument
\(\displaystyle \varphi = \text{arctan}\left(\frac{b}{a}\right)\)
Important: Quadrant Consideration

The angle depends on which quadrant the complex number is in. Use the two-argument arctangent function (often called \(\text{atan2}(b, a)\)) for correct results in all quadrants.

From Polar to Rectangular Form

To convert from polar form to rectangular form:

Conversion Formula:
\(\displaystyle a = r\cos\varphi \quad \text{and} \quad b = r\sin\varphi\)

Conversion Examples

Example 1: Rectangular to Polar

Convert \(z = 1 + i\) to polar form

Step 1: Calculate magnitude

\(\displaystyle r = \sqrt{1^2 + 1^2} = \sqrt{2}\)

Step 2: Calculate argument

\(\displaystyle \varphi = \arctan\left(\frac{1}{1}\right) = \arctan(1) = \frac{\pi}{4} = 45°\)

Polar form: \(z = \sqrt{2}\left(\cos 45° + i\sin 45°\right)\)

Example 2: Polar to Rectangular

Convert \(z = 2(\cos 60° + i\sin 60°)\) to rectangular form

Calculate real part:

\(\displaystyle a = 2\cos 60° = 2 \cdot \frac{1}{2} = 1\)

Calculate imaginary part:

\(\displaystyle b = 2\sin 60° = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}\)

Rectangular form: \(z = 1 + \sqrt{3}i\)

Example 3: Standard Angles

Common polar form examples
  • \(z = 1 = 1(\cos 0° + i\sin 0°)\)
  • \(z = i = 1(\cos 90° + i\sin 90°)\)
  • \(z = -1 = 1(\cos 180° + i\sin 180°)\)
  • \(z = -i = 1(\cos 270° + i\sin 270°)\)

Advantages of Polar Form

The polar form is particularly useful for certain operations:

Multiplication
Magnitudes multiply
Arguments add
Division
Magnitudes divide
Arguments subtract
Powers
De Moivre's Theorem
Raise magnitude to power, multiply argument
Roots
Easy to extract roots
Especially useful for finding all roots

Polar Form Summary

Rectangular Form Magnitude Argument Polar Form (CIS)
\(1 + i\) \(\sqrt{2}\) \(45°\) or \(\frac{\pi}{4}\) \(\sqrt{2}\,\text{cis}(45°)\)
\(1 + \sqrt{3}i\) \(2\) \(60°\) or \(\frac{\pi}{3}\) \(2\,\text{cis}(60°)\)
\(-1 + i\) \(\sqrt{2}\) \(135°\) or \(\frac{3\pi}{4}\) \(\sqrt{2}\,\text{cis}(135°)\)
\(3 + 4i\) \(5\) \(\arctan(\frac{4}{3}) \approx 53.13°\) \(5\,\text{cis}(53.13°)\)

Key Points to Remember

  • Polar form describes a complex number by magnitude and argument
  • Magnitude \(r = |z| = \sqrt{a^2 + b^2}\)
  • Argument \(\varphi = \text{arg}(z)\) is the angle from the positive real axis
  • Multiple equivalent notations: \((r, \varphi)\), \(r(\cos\varphi + i\sin\varphi)\), \(r\,\text{cis}(\varphi)\), \(re^{i\varphi}\)
  • Polar form simplifies multiplication, division, and root extraction
  • Always consider the correct quadrant when calculating the argument
























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