# Polar Form of a Complex Number

## Definition

The polar form of a complex number is a way of expressing the number in terms of its magnitude and angle relative to the positive real axis. A complex number \( z \) can be written as \( z = x + iy \), where \( x \) and \( y \) are real numbers representing the real and imaginary parts of \( z \), respectively. In the complex plane, this number corresponds to the point \( P \) with Cartesian coordinates \( (x, y) \).

To convert the Cartesian coordinates to polar coordinates, we define \( r \) as the length of the vector \( OP \), which represents the magnitude of \( z \) and is given by \( r = |z| = \sqrt{x^2 + y^2} \). The angle \( \theta \), known as the argument of \( z \), is the angle between the positive real axis and the vector \( OP \).

The relationships between the Cartesian coordinates and the polar coordinates are given by:

So, the complex number \( z \) can also be expressed as:

Choosing Argument

Since argument of a complex number is not unique. If \(\theta\) is the argument, then \(2k\pi+\theta\) is also the argument for \(k\in\mathbb Z\). Therefore, the principal value of the argument, typically within the interval \((-π, π]\), is conventionally chosen for the polar form to maintain consistency and standardization across mathematical texts and computational tools.

Consider a complex number \( z \), represented by point \( P \) on the complex plane. If the distance from the origin to \( P \) is 5, and the angle \( OP \) makes with the positive x-axis is \( \frac{4\pi}{3} \), then \( z \) is given by:

Using the cosine and sine values for the given angle, we can write \( z \) in Cartesian form:

This is the Cartesian form of the complex number whose magnitude is 5 and whose argument (angle from the positive x-axis) is \( \frac{4\pi}{3} \).

Modulus of \(\cos(\theta) + i\sin(\theta)\)

**Some special considerations on polar form:**

\( z = r(\cos\theta - i\sin\theta) \) is not in correct polar form, but we may convert it to correct form by using trigonometric identities.

\( z = r(\cos\theta - i\sin\theta) = r(\cos(-\theta) + i\sin(-\theta)) \) (Supposing \( r > 0 \))

Similarly,

Incorrect polar form | Correct polar form |
---|---|

\( r(-\cos\theta + i\sin\theta) \) | \( r(\cos(\pi - \theta) + i\sin(\pi - \theta)) \) |

\( r(-\cos\theta - i\sin\theta) \) | \( r(\cos(\pi + \theta) + i\sin(\pi + \theta)) \) |

\( r(\sin\theta + i\cos\theta) \) | \( r(\cos(\frac{\pi}{2} - \theta) + i\sin(\frac{\pi}{2} - \theta)) \) |

\( r(\sin\theta - i\cos\theta) \) | \( r(\cos(\theta - \frac{\pi}{2}) + i\sin(\theta - \frac{\pi}{2})) \) |

\( r(-\sin\theta + i\cos\theta) \) | \( r(\cos(\frac{\pi}{2} + \theta) + i\sin(\frac{\pi}{2} + \theta)) \) |

\( r(-\sin\theta - i\cos\theta) \) | \( r(\cos(\frac{\pi}{2} - \theta) + i\sin(\frac{\pi}{2} - \theta)) \) |

Example

**Problem:** Convert the complex number \( -1 - i \) to its polar form.

**Solution:**

Step 1: Find \( \alpha = \tan^{-1}\left(\frac{|y|}{|x|}\right) \). Given \( -1 - i \), we have \( x = -1 \) and \( y = -1 \). The magnitudes are \( |x| = 1 \) and \( |y| = 1 \). Thus,

Step 2: Since \( -1 - i \) lies in the third quadrant, the argument \( \theta \) is \( -\pi + \alpha \). Hence,

Step 3: We note that if \( \theta \) is the argument of a complex number, then \( 2k\pi + \theta \) is also an argument for \( k \in \mathbb{Z} \). We can choose any such argument for the polar form.

Step 4: Find \( r = |z| \). The modulus \( r \) is given by

Thus, the polar form is

or using any argument \( 2k\pi + \theta \),

for \( k \in \mathbb{Z} \).