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Complex numbers are often expressed in rectangular form. This is a straightforward way to represent them as \(a + bi\). Here, \(a\) is the real part of the complex number, while \(b\) is the imaginary part. Such a format is valuable because it clearly distinguishes between real and imaginary components, helping in operations like addition and subtraction.

In the exercise, the complex number is \(-3 + 3i\). Both parts are clearly labeled, helping us manage the complex number with ease. Simply put, rectangular form displays numbers in a two-dimensional plane. For instance:

• \(-3\) is the value on the real axis,
• \(3i\) is the value on the imaginary axis.

Converting between forms allows us to apply various mathematical techniques more conveniently.

Polar form provides a different way of representing complex numbers which is particularly helpful in multiplication and raising powers. In polar form, a complex number is represented as \(r(\cos \theta + i\sin \theta)\), often noted as \(r\,\text{cis} \,\theta\).

The conversion to polar form involves:

• Finding the modulus \(r\): calculated as \(\sqrt{a^2 + b^2}\),
• Determining the argument \(\theta\): found using \(\tan^{-1}(b/a)\).

It's like viewing a point on a circle by its radius and angle instead of its x and y coordinates. For example, in the exercise \(-3 + 3i\) translates to polar form as \(3\sqrt{2} \text{cis} \,\frac{3\pi}{4}\). This shows how far out and in which direction the number is on the complex plane.

De Moivre's Theorem is a powerful tool for dealing with complex numbers, especially when raising them to a power. It states that \((r\,\text{cis} \,\theta)^n = r^n\,\text{cis}(n\theta)\). This formula simplifies the process of finding powers and roots of complex numbers in polar form.

In the provided exercise, we apply De Moivre’s Theorem to find \((-3 + 3i)^7\). By converting to polar form first, we can plug values directly into the theorem:

• Raise the modulus \(r\) to the power: \((3\sqrt{2})^7\)
• Multiply the angle \(\theta\) by the power: \(7 \times \frac{3\pi}{4}\)

This methodology drastically reduces complexity, as multiplication with polar coordinates is simply easier. After calculations, the final result is simplified back into rectangular form for clarity.

The modulus of a complex number, also known as its magnitude, is a measure of its size on the complex plane. It is represented as \(|z|\). To find it for a complex number \(a + bi\), you calculate \(\sqrt{a^2 + b^2}\).

In our example, the modulus for \(-3 + 3i\) is found as \(\sqrt{(-3)^2 + (3)^2} = \sqrt{18} = 3\sqrt{2}\). This gives the length of the vector from the origin to the point \(-3, 3\).

Understanding the modulus is key to navigating the complex plane and is crucial for converting between rectangular and polar forms. It provides a tangible sense of the 'length' of a complex number, much like how absolute value gives the distance of a real number from zero. This modulus becomes quite relevant when utilizing formulas like De Moivre's Theorem.