91Ó°ÊÓ

Complex numbers can be represented in various forms, and one of them is the polar form. Polar form is incredibly useful when dealing with operations like multiplication, division, and exponentiation. In polar form, a complex number is expressed as:

• \( z = r(\cos \theta + i\sin \theta) \)

where:

• \( r \) is the magnitude (or modulus) of the complex number
• \( \theta \) is the argument (or angle) measured from the positive x-axis

To convert a complex number from its rectangular form \( a + bi \) to polar form, you calculate the magnitude using

• \( r = \sqrt{a^2 + b^2} \)

and the argument using

• \( \theta = \arctan\left(\frac{b}{a}\right) \).

In the exercise, the complex number \( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i \) was converted to polar form, resulting in \( 1(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}) \), indicating it lies on the unit circle with an angle of \( \frac{\pi}{4} \) radians.

De Moivre’s theorem is a powerful tool for raising complex numbers in polar form to a power. The theorem states:

• For any complex number \( z = r(\cos \theta + i\sin \theta) \) and integer \( n \),

• The expression \( z^n \) is equal to \[ r^n(\cos(n\theta) + i\sin(n\theta)) \].

This theorem is particularly handy when dealing with roots and powers because it simplifies the computation significantly. In the given exercise, De Moivre's Theorem is applied to compute \((z^4)\). Given that the original angle \( \theta = \frac{\pi}{4} \) and \( r = 1 \), it makes the problem straightforward. By applying the theorem, raising \( (\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}) \) to the fourth power results in \( \cos \pi + i\sin \pi \). This further simplifies, showing the power of De Moivre's theorem in transforming complex number calculations.

Rectangular form is perhaps the most familiar way to express complex numbers, written as \( a + bi \), where \( a \) and \( b \) are real numbers. Here:

• \( a \) is the real part of the complex number
• \( b \) is the imaginary part of the complex number

Complex numbers in rectangular form are convenient for addition and subtraction. However, for exponentiation and multiplication, it's often easier to switch to polar form first, as seen in the exercise. When converting back from polar form, you end up with rectangular coordinates by evaluating

• \( z = r(\cos \theta + i\sin \theta) = a + bi \)

In the final step of the exercise, after using De Moivre's theorem, the solution was elegantly returned to rectangular form: \( \cos \pi + i\sin \pi = -1 \), ensuring an intuitive understanding of the result.

The magnitude (or modulus) of a complex number \( z = a + bi \) is a measure of its "size" or its distance from the origin in the complex plane. It is denoted by \( |z| \) and calculated as:

• \( |z| = \sqrt{a^2 + b^2} \)

In polar form, the magnitude is represented as \( r \). It's essential for converting complex numbers into their polar form, as it determines the "length" of the vector representing the complex number.In the problem, the magnitude of \( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i \) is calculated to be 1, indicating that this complex number resides on the unit circle. The unit circle is particularly significant as any complex number with a magnitude of 1 implies that all rotations (multiplication by complex numbers) will maintain the distance from the origin, making it easier to visualize the effects of exponentiation using De Moivre's theorem.