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Complex numbers in rectangular form are written as \( z = a + bi \), where \( a \) represents the real part and \( b \) represents the imaginary part. This form allows for straightforward addition and subtraction of complex numbers. The real part \( a \) and the imaginary part \( b \) are both represented as Cartesian coordinates on the complex plane, meaning that each complex number can be plotted as a point or vector.

When converting from polar coordinates to rectangular form, the expressions for \( a \) and \( b \) use the trigonometric functions cosine and sine respectively:

• \( a = r \cos \theta \)
• \( b = r \sin \theta \)

This conversion is essential for utilizing basic arithmetic operations like addition and multiplication that may be more easily conducted in rectangular form.

Polar coordinates provide another way of representing complex numbers. In this system, a complex number is expressed as \( z = r[\cos \theta + i \sin \theta] \), where \( r \) is the magnitude (or modulus) of the complex number, and \( \theta \) is the argument (or angle).

The magnitude \( r \) indicates how far away the number is from the origin in the complex plane, which is calculated using the formula: \( r = \sqrt{a^2 + b^2} \). On the other hand, the argument \( \theta \) correlates with the direction of the complex number vector and can be found using \( \theta = \tan^{-1} \left(\frac{b}{a}\right) \).

Converting between polar and rectangular forms is important for allowing flexibility. For example, polar form is often more convenient for calculations, such as finding powers and roots of complex numbers, thanks to the simple multiplication and division rules it offers.

Multiplying complex numbers involves distributing each part of one complex number over each part of the other, taking care to apply the rules of imaginary numbers, mainly that \( i^2 = -1 \). This can be more straightforward when numbers are in rectangular form. For example, with two complex numbers, \( z_1 = a_1 + b_1i \) and \( z_2 = a_2 + b_2i \), their product is calculated as follows:

• \( z_1 \cdot z_2 = (a_1 + b_1i)(a_2 + b_2i) \)
• Perform expansion: \( = a_1a_2 + a_1b_2i + b_1i a_2 + b_1b_2i^2 \)
• Since \( i^2 = -1 \), the expression simplifies to: \( = (a_1a_2 - b_1b_2) + (a_1b_2 + b_1a_2)i \)

In the solution of the exercise, translating the polar form into rectangular form first allows applying this process effectively. Additionally, when multiplying numbers already in polar form, the magnitudes are multiplied, and the angles (arguments) are added, simplifying the process considerably. Understanding both these approaches is vital to mastering complex number operations.