91Ó°ÊÓ

The polar form is a way to express complex numbers using a magnitude (or length) and an angle. This is especially useful when dealing with complex numbers as it simplifies many mathematical operations like multiplication and finding roots.
- A complex number is usually written as \( z = r \angle \theta \) in polar form, where \( r \) is the magnitude and \( \theta \) is the angle with the positive real axis.
- The angle, \( \theta \), is measured in radians, and \( r \) is always non-negative.
- In our exercise, the number 16 can be represented as \( 16 \times e^{i \cdot 0} \). Here, \( r = 16 \) and \( \theta = 0 \), meaning the number is purely real and non-negative, lying on the positive real axis.
Polar form is particularly helpful in finding roots because it distributes angles evenly, making calculations systematic and structured.

Rectangular form expresses a complex number as a sum of its real and imaginary components. It closely resembles the way we use coordinates to pinpoint locations on a plane.
- In rectangular form, a complex number \( z \) is written as \( z = a + bi \) where \( a \) is the real part and \( b \) is the imaginary part.
- Conversion from polar to rectangular uses two key relations: \( a = r \cos(\theta) \) and \( b = r \sin(\theta) \). This converts the polar angle \( \theta \) into a coordinate-based format.
- From the exercise, converting the fourth root \( 2 \times e^{i \cdot 0} \) gives \( 2 + 0i \), and similarly for others: \( 0 + 2i \), \(-2 + 0i \), and \(0 - 2i \).
Each conversion uses the unit circle's symmetrical properties, transforming a rotation perspective into a straightforward coordinate, making operations tangible and intuitive.

Finding the fourth roots of a complex number retrieves four distinct numbers (or roots), equally spaced around the origin in the complex plane. This is due to the fundamental property of complex roots having symmetric distribution.
- The formula to find an \( n \)th root, \( z^{1/n} = r^{1/n} \times e^{i (\theta + 2k\pi)/n} \) where \( k \) runs from 0 to \( n-1 \).
- The solution uses \( n = 4 \) (since we want fourth roots), so \( k \) takes values 0, 1, 2, and 3 to give each root. Each root angle is found by altering \( k \), yielding symmetry across the circle.
- For our example, \( r^{1/4} = 16^{1/4} = 2 \). The roots derived give distinctly spread out points: \( 2 \angle 0 \), \( 2 \angle \frac{\pi}{2} \), \( 2 \angle \pi \), and \( 2 \angle \frac{3\pi}{2} \).
These computations demonstrate a balance, positioned perfectly around the unit circle, visually illustrating rotations and solving elegantly.