91Ó°ÊÓ

Complex numbers are numbers that have both a real part and an imaginary part. The standard form of a complex number is written as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary part is denoted by \( i \), where \( i^2 = -1 \). This unique characteristic allows complex numbers to extend the real numbers and solve equations that have no real solutions.
For example, the complex number from the exercise we are dealing with is expressed as \(-\frac{1}{2} - \frac{1}{2} \sqrt{3} i\). This means its real part is \(-\frac{1}{2}\) and its imaginary part is \(-\frac{1}{2} \sqrt{3} i\). These numbers provide a way to deal with two-dimensional quantities using one-dimensional notation.

• Real Part (\(a\)): Corresponds to the x-axis on the complex plane.
• Imaginary Part (\(bi\)): Corresponds to the y-axis on the complex plane.

Understanding complex numbers is essential for solving problems that require more than one dimension of solutions.

Polar form is a way of expressing complex numbers in terms of a magnitude (or modulus) and an angle (or argument). Instead of using rectangular coordinates with real and imaginary parts, a complex number is described by how far it is from the origin and the angle it makes with the positive real axis.
The polar form is given as \( r\text{cis}\theta \), where:

• \( r = \sqrt{a^2 + b^2} \) is the modulus, representing the distance from the origin to the point in the complex plane.
• \( \theta = \tan^{-1}(\frac{b}{a}) \) is the argument, representing the angle to the positive x-axis.

Using this form is particularly useful for multiplication and division of complex numbers. It simplifies the operations by transforming them into operations of moduli and angles.
In the given exercise, the complex number \(-\frac{1}{2} - \frac{1}{2} \sqrt{3} i\) was converted into polar form as \(1\text{cis}\frac{4\pi}{3}\), making further calculations straightforward and revealing its rotation around the origin.

Finding the square roots of a complex number involves determining two numbers, which when squared, return the original complex number. In the context of complex numbers, this is efficiently done using DeMoivre's theorem.
DeMoivre's theorem states that for a complex number in polar form \( r\text{cis}\theta \), any \( n \)-th root is given by \(\[ w_k = r^{1/n} \text{cis} \left(\frac{\theta + 2k\pi}{n}\right) \]\) for integer values of \( k \). This theorem helps find not only the square roots but all roots of any degree. In polar coordinates, rotating and rescaling are simplified operations.
Using this method in the exercise, we calculated the two square roots of the polar form \( 1\text{cis}\frac{4\pi}{3} \). The roots are:

• \( \text{cis}\left(\frac{2\pi}{3}\right) \), which converts to \(-\frac{1}{2} + i\frac{\sqrt{3}}{2} \) in rectangular form.
• \( \text{cis}\left(\frac{7\pi}{3}\right) \), which simplifies to \(\frac{1}{2} + i\frac{\sqrt{3}}{2} \) in rectangular form.

This method illustrates the geometrical nature of complex numbers, with roots appearing as evenly spaced points on the complex plane's circle.