Problem 30 - Advanced Engineering Mathematics

Question

Use (10) to compute all roots. Sketch these roots on an appropriate circle centered at the origin. $$ (-1+i)^{1 / 3} $$

Step-by-Step Solution

Answer

The cube roots are \(2^{1/6} e^{i\pi/4}\), \(2^{1/6} e^{i11\pi/12}\), \(2^{1/6} e^{i19\pi/12}\). Plot these on the Argand diagram as vertices of an equilateral triangle.

Step 1: Convert to Polar Form

The complex number \((-1+i)\) can be expressed in polar form as \(re^{i\theta}\). To find \(r\), compute the modulus: \(r = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\). For the angle \(\theta\), note that the number is in the second quadrant with \(\tan^{-1}(1)\), so \(\theta = \frac{3\pi}{4}\). Thus, \((-1+i) = \sqrt{2}e^{i3\pi/4}\).

Step 2: Apply De Moivre’s Theorem

To find the cube roots, apply De Moivre’s Theorem. Raise the polar form to the power \(1/3\): \[\left(\sqrt{2}\right)^{1/3} e^{i(3\pi/4)/3} = 2^{1/6} e^{i\pi/4}\].

Step 3: Compute All Roots

To find all cube roots, note that the roots are given by \[z_k = 2^{1/6} e^{i(\pi/4 + 2k\pi/3)}\]for \(k = 0, 1, 2\). Calculate each:- For \(k=0\): \[z_0 = 2^{1/6} e^{i\pi/4}\]- For \(k=1\): \[z_1 = 2^{1/6} e^{i(\pi/4 + 2\pi/3)} = 2^{1/6} e^{i11\pi/12}\]- For \(k=2\): \[z_2 = 2^{1/6} e^{i(\pi/4 + 4\pi/3)} = 2^{1/6} e^{i19\pi/12}\].

Step 4: Plot the Roots on the Argand Diagram

The roots are plotted on a circle centered at the origin with radius \(2^{1/6}\) in the Argand plane. This circle illustrates the magnitude of the roots. Each root is plotted at angles \(\pi/4\), \(11\pi/12\), and \(19\pi/12\) respectively, measured from the positive real axis. Ensure each point is evenly distributed since these angles are separated by \(2\pi/3\), representing an equilateral triangle.

Key Concepts

Polar Form

In Mathematics, complex numbers can be represented in different forms. One of these is the polar form, which expresses a complex number in terms of a radius and an angle. For any complex number, like \[a + bi\]its polar form is given by \[re^{i\theta}\]where:

• r is the modulus or absolute value of the complex number. It's calculated as \[r = \sqrt{a^2 + b^2}\]. It represents the distance from the origin to the point on the complex plane.
• \(\theta\) is the argument (or angle) in radians. It's found using the tangent inverse, \[\theta = \tan^{-1}(b/a)\], and it describes the direction of the point from the positive real axis.

Understanding the polar form is essential as it simplifies the multiplication, division, and finding roots of complex numbers, all of which involve manipulation of the angle \(\theta\) and the modulus \(r\). Using the polar form, we're able to harness the power of complex exponentials to ease these calculations.

De Moivre’s Theorem

De Moivre’s Theorem is a fundamental principle in complex number theory. It relates complex numbers and trigonometry, providing a formula for raising complex numbers to integer powers. This is particularly useful in finding roots of complex numbers, such as cube roots. The theorem states:\[(r \cdot e^{i\theta})^n = r^n \cdot e^{in\theta}\]where

• r is the modulus.
• \(\theta\) is the argument of the complex number.
• n is the exponent.

When we apply De Moivre’s Theorem to finding roots, we look at raising the complex number to fractional powers. For instance, if we're looking for the cube roots, we use \(n = 1/3\). By doing this, the roots are distributed evenly on the complex plane, with each root separated by an angle of \(2\pi/n\). This not only provides a systematic way to approach complex roots but also ensures each solution's magnitude and angle align perfectly in the Argand plane.

Argand Diagram

An Argand Diagram is a graphical representation of complex numbers on a plane. Named after the French mathematician Jean-Robert Argand, this diagram allows complex numbers to be plotted with the horizontal axis representing the real part and the vertical axis representing the imaginary part.

• Each complex number, say \(a + bi\), is represented as the point \((a, b)\).
• Magnitude or modulus of a complex number is represented by its distance from the origin.
• The argument or angle of the complex number is the angle from the positive real axis to the line segment joining the origin and the point.

When plotting roots on an Argand Diagram, such as cube roots, these points lie on a circle centered at the origin. The radius of this circle corresponds to the modulus of the roots. The symmetry and spacing reflect the uniform distribution around the origin.
For cube roots, you should see an equilateral triangle formation as the roots are separated by \(120^\circ\)or \(2\pi/3\)radians. This visual representation helps in understanding the positioning and nature of complex roots.

Cube Roots

Finding cube roots of a complex number involves technique and understanding of complex arithmetic. Generally, a complex number will have three cube roots due to the nature of cube functions, and these can be determined using the polar form combined with De Moivre's Theorem.

• Firstly, convert the complex number into its polar form \(re^{i\theta}\).
• Apply the formula to find the cube roots \[z_k = r^{1/3} e^{i(\theta/n + 2k\pi/3)}\]for \(k = 0, 1, 2\), where each \z_k\ is a unique root.

When plotted on the Argand Diagram, the cube roots are evenly spaced on a circle forming vertices of an equilateral triangle. This graphical representation reveals the geometric harmony in the distribution of roots on the complex plane. By understanding these unique root positions, one gains deeper insight into the symmetry and balance inherent in complex numbers.