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To work with complex numbers in a convenient way, especially for multiplication and division, we often use the polar form. In polar form, a complex number is represented in terms of its magnitude (or absolute value) and its angle (or argument) with respect to the positive x-axis.
It takes the form \( r(\cos(\theta) + i\sin(\theta)) \), where:

• \(r\) is the magnitude of the complex number.
• \(\theta\) is the argument, which is the angle formed with the real axis.

This representation makes it easy to handle operations like multiplication and division, as we'll see in other sections.

The magnitude of a complex number is similar to the length of the vector that represents it on the complex plane. You can think of it as the "distance" from the origin in the complex plane to the point that represents the complex number.
If a complex number is given in standard form as \( a + bi \), its magnitude is calculated using the formula \( \sqrt{a^2 + b^2} \).
In polar form, this magnitude is represented by \( r \). It's simply the coefficient that multiplies the trigonometric functions \(\cos(\theta)\) and \(i\sin(\theta)\).
The magnitude is always a non-negative real number.

The argument of a complex number is crucial for understanding its direction from the origin on the complex plane. It is essentially the angle \( \theta \) that the line, connecting the origin to the complex number, makes with the positive real axis.
Calculating the argument for a complex number \(a + bi\) involves using the \( \tan^{-1} \) function, specifically \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). This angle can be measured in radians or degrees, and it tells us exactly where the number is located in its plane.
In the context of division, the argument plays a significant role as we subtract the argument of the divisor from the dividend to find the resulting argument of their quotient.

Dividing complex numbers becomes much simpler when they are expressed in polar form. If we have two complex numbers, say \( a = r_1 (\cos(\theta_1) + i \sin(\theta_1)) \) and \( b = r_2(\cos(\theta_2) + i \sin(\theta_2)) \), their division can be broken down into two key steps:

• Divide the magnitudes: The magnitude of the resultant quotient \( \frac{a}{b} \) is simply \( \frac{r_1}{r_2} \).
• Subtract the arguments: The argument of \( \frac{a}{b} \) is \( \theta_1 - \theta_2 \).

This results in the quotient also being in polar form: \( \frac{a}{b} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)) \).
This method elegantly simplifies the arithmetic required and keeps the focus on angular relationships, like those found in trigonometry.