Question

Bestimmen Sie die 4 . Wurzeln der imaginären Einheit \(i\).

Short answer

The 4th roots of \(i\) are \(e^{i\frac{\pi}{8}}, e^{i\frac{5\pi}{8}}, e^{i\frac{9\pi}{8}}, e^{i\frac{13\pi}{8}}\).

Step-by-step solution

Express Imaginary Unit in Polar Form

The imaginary unit is represented as \(i\), which can be expressed in polar form. Knowing that \(i = 0 + 1i\), it can be represented as having a modulus of 1 (distance from origin) and an angle of \(\frac{\pi}{2}\) radians (since it lies on the positive imaginary axis). Hence, its polar form is \(1 \cdot e^{i\frac{\pi}{2}}\).

Use De Moivre's Theorem

We know from De Moivre's Theorem that if a complex number is written in polar form as \(r \cdot e^{i\theta}\), then its n-th root is \(r^{1/n} \cdot e^{i(\theta + 2k\pi)/n}\) for \(k = 0, 1, 2, \ldots, n-1\). For \(i\), this becomes the 4th root expression as \(1^{1/4} \cdot e^{i(\frac{\pi}{2} + 2k\pi)/4}\).

Calculate Each Root

Calculate the actual roots by setting \(k = 0\), 1, 2, and 3:- For \(k = 0\), the root is \(e^{i\frac{\pi}{8}}\).- For \(k = 1\), the root is \(e^{i\frac{5\pi}{8}}\).- For \(k = 2\), the root is \(e^{i\frac{9\pi}{8}}\).- For \(k = 3\), the root is \(e^{i\frac{13\pi}{8}}\).

Convert Polar Roots to Rectangular Form

Convert each polar root to rectangular (Cartesian) form using \(e^{i\theta} = \cos\theta + i\sin\theta\):- \(e^{i\frac{\pi}{8}} = \cos\frac{\pi}{8} + i\sin\frac{\pi}{8}\).- \(e^{i\frac{5\pi}{8}} = \cos\frac{5\pi}{8} + i\sin\frac{5\pi}{8}\).- \(e^{i\frac{9\pi}{8}} = \cos\frac{9\pi}{8} + i\sin\frac{9\pi}{8}\).- \(e^{i\frac{13\pi}{8}} = \cos\frac{13\pi}{8} + i\sin\frac{13\pi}{8}\).

Key concepts

Polar Form

Complex numbers can be expressed in multiple forms, and one of the most insightful is the polar form. It provides a powerful way to understand complex numbers through their magnitude and direction. A complex number, say \(z = x + yi\), can be represented in polar form as \(r \cdot e^{i\theta}\). Here, \(r\) is the modulus, calculated as \( \sqrt{x^2 + y^2}\), and \(\theta\) is the argument, or the angle made with the positive x-axis.
When you convert a complex number like \( i \) into polar form, you can simplify multiplication and division, and find powers and roots with ease. In our problem, the imaginary unit \( i \) is located at \((0, 1)\) in the complex plane, making its modulus 1, with an angle of \( \frac{\pi}{2} \) radians from the positive x-axis. Its polar form is thus \(1 \cdot e^{i\frac{\pi}{2}}\).

• Modulus \(r\) gives the distance from the origin.
• Argument \(\theta\) measures the angle in radians.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number theory. It allows us to find powers and roots of complex numbers when they are expressed in polar form. The theorem states that if \(z = r \cdot e^{i\theta}\), then \(z^n = r^n \cdot e^{i n \theta}\). This can be particularly useful for calculating powers, as the exponential term simplifies the process greatly.
To find roots, we slightly modify the expression. The nth roots of \(z\) are given by \(r^{1/n} \cdot e^{i(\theta + 2k\pi)/n}\) for \(k = 0, 1, 2, \ldots, n-1\). This formula generates all distinct nth roots by simply varying \(k\).

• Efficiently handles powers and roots.
• Encapsulates complex arithmetic using angles.

De Moivre's theorem simplifies our task by turning the problem of finding the fourth roots of \( i \) into calculating \(1^{1/4} \cdot e^{i(\frac{\pi}{2} + 2k\pi)/4}\) for \(k\) values from 0 to 3.

Nth Roots

Finding nth roots of a complex number involves a set of procedures specific to handling both the modulus and the angle of the number in polar form. Here, the nth root of a complex number \(r \cdot e^{i\theta}\) is determined by evaluating \(r^{1/n} \cdot e^{i(\theta + 2k\pi)/n}\) where \(k\) ranges from 0 to \(n-1\).
This guarantees all possible roots are executed, which can be visualized as evenly distributing these roots around a circle in the complex plane.
For instance, calculating the fourth roots of \(i\) involves splitting the angle \(\frac{\pi}{2}\) into four parts. Therefore:

• Select \(k = 0, 1, 2, 3\).
• Compute the angle for each root using \((\theta + 2k\pi)/4\).

This provides the solutions \(e^{i\frac{\pi}{8}}\), \(e^{i\frac{5\pi}{8}}\), \(e^{i\frac{9\pi}{8}}\), and \(e^{i\frac{13\pi}{8}}\).

Imaginary Unit

The imaginary unit, denoted as \(i\), serves as the basis for complex numbers. It is defined by the fundamental property \(i^2 = -1\). This property allows for the extension of real numbers to complex numbers, accounting for solutions to equations that involve the square root of negative numbers.
In the complex plane, \(i\) is positioned at the point \((0, 1)\), laying on the imaginary axis. As such, \(i\) is purely imaginary, and its presence allows for rich mathematics in complex number theory, including the ability to find roots of negative numbers.
When described in polar form, \(i\) is represented as having a modulus of 1 and an angle of \(\frac{\pi}{2}\), symbolized by \(e^{i\frac{\pi}{2}}\).

• Enables expression of complex numbers.
• Integral to polar and exponential forms.

Understanding \(i\) is essential when exploring more advanced concepts such as roots and polar representations, acting as a stepping stone into the broader world of complex arithmetic.