Problem 31 - Complex numbers from A to...Z

Question

Find the geometric images of the complex numbers \(z\) for which $$ z^{n} \cdot \operatorname{Re}(z)=\bar{z}^{n} \cdot \operatorname{Im}(z) $$ where \(n\) is an integer.

Step-by-Step Solution

Answer

Answer: The geometric images of the complex numbers \(z\) satisfying the given condition are the origin in the Argand plane, represented as a point (0,0), and the lines in the Argand plane that have an angle of \(\frac{\theta}{n}\) with the positive real axis (measured counterclockwise) for integer values of n.

Step 1: Write the complex number in polar form

Let \(z = a + bi\), where \(a\) and \(b\) are real numbers. Let's write \(z\) in polar form: \(z = r(\cos{\theta} + i\sin{\theta})\), where \(r\) and \(\theta\) are the magnitude and argument of \(z\), respectively. Similarly, we can find the polar form of the complex conjugate \(\bar{z} = a - bi = r(\cos{\theta} - i\sin{\theta})\).

Step 2: Express the given condition using the polar form

Now let's express the given condition \(z^n\operatorname{Re}(z) = \bar{z}^n\operatorname{Im}(z)\) in terms of the polar form. Substitute \(z = r(\cos{\theta} + i\sin{\theta})\) and \(\bar{z} = r(\cos{\theta} - i\sin{\theta})\): $$ z^n\operatorname{Re}(z) = (r^n(\cos{n\theta} + i\sin{n\theta}))(a) = \bar{z}^n\operatorname{Im}(z)= (r^n(\cos{n\theta} - i\sin{n\theta}))(b) $$ This gives us: $$ r^n(a\cos{n\theta} - b\sin{n\theta}) + r^n i(a\sin{n\theta} + b\cos{n\theta}) = r^n(a\cos{n\theta} + b\sin{n\theta}) - r^n i(a\sin{n\theta} - b\cos{n\theta}) $$

Step 3: Equate the real and imaginary parts

As two complex numbers are equal if and only if their real and imaginary parts are equal, we can equate the real and imaginary parts of the above expression: \begin{align*} r^n(a\cos{n\theta} - b\sin{n\theta}) &= r^n(a\cos{n\theta} + b\sin{n\theta})\\ r^n(a\sin{n\theta} + b\cos{n\theta}) &= r^n(a\sin{n\theta} - b\cos{n\theta}) \end{align*} From the first equation, we get the following cases: 1. \(r=0\), which implies \(z = 0\). 2. \(r \neq 0\): In this case, we can divide both sides by \(r^n\): $$ a\cos{n\theta} - b\sin{n\theta} = b\sin{n\theta} + a\cos{n\theta} $$ Simplifying further, we get \(a\cos{n\theta} = b\sin{n\theta}\). This implies that \(\tan{n\theta} = \frac{a}{b}\).

Step 4: Describe the geometric images

Using the results from Step 3, we can find the geometric images of the complex numbers \(z\) satisfying the given condition: 1. When \(r=0\), we have the origin as a geometric image, represented as a point (0,0) in the Argand plane. 2. When \(r>0\), the geometric images correspond to the lines in the Argand plane that have an angle of \(\frac{\theta}{n}\) with the positive real axis (measured counterclockwise) and n is an integer.

Key Concepts

Geometric Interpretation

Understanding the geometric interpretation of complex numbers is vital in visualizing their behavior. When we deal with equations involving complex numbers, like the one given, these numbers can be visualized as points or vectors in the Argand plane. This exercise has us discovering the geometric images (or sets of points) that represent complex numbers satisfying a certain condition. Think of it as plotting a path or a shape made up by the solutions to the equation.

• The origin is one possible solution, appearing when the magnitude, or the 'length' of the vector, is zero.
• For non-zero magnitudes, the solutions lie on lines at specific angles.

This helps us see not only where the solutions lie but also their relationship in terms of angles and positions.

Polar Form

The polar form of complex numbers expresses them in terms of a magnitude and angle. This provides a powerful alternative to the standard rectangular form, which just has real and imaginary parts. For a complex number given as \[z = a + bi\] it can be transformed into its polar form:\[z = r(\cos{\theta} + i\sin{\theta})\]where \(r\) is the magnitude, given by \(r = \sqrt{a^2 + b^2}\), and \(\theta\) is the argument or angle with the positive real axis. This form is particularly helpful when multiplying or raising complex numbers to powers, as it simplifies operations through rotational geometry.

Complex Conjugate

The complex conjugate of a complex number is another concept that helps us in many mathematical operations. For a complex number \(z = a + bi\), the complex conjugate, denoted \(\bar{z}\), is \(a - bi\). The idea is to reflect the number across the real axis in the Argand plane.

• This reflection can be useful in solving equations or simplifying expressions.
• In this exercise, both \(z\) and \(\bar{z}\) are explored in polar form, showcasing how they relate.

Utilizing complex conjugates helps in equating real and imaginary parts, as well as understanding the behavior of complex functions more deeply.

Argand Plane

The Argand plane is a vital tool when it comes to visualizing complex numbers. Much like a coordinate system, each point on the plane represents a complex number, with the x-axis showing real parts and the y-axis showing imaginary parts. In this context, imagining the Argand plane allows us to see complex numbers not as abstract entities but as real points and vectors on a 2D plane. This visualization facilitates the understanding of geometric interpretations and interactions between complex numbers.

• The origin corresponds to the complex number 0, where both real and imaginary parts are zero.
• Lines or curves can represent sets of solutions or specific magnitudes and arguments.

Therefore, the Argand plane is critical for illustrating solutions and understanding their spatial significance.

Equating Real and Imaginary Parts

A crucial technique in complex number equations is to equate their real and imaginary parts. This means breaking down the equation of two complex numbers into separate equations of real and imaginary components.For the equation given, the complex expressions were broken down:\[\begin{align*}r^n(a\cos{n\theta} - b\sin{n\theta}) &= r^n(a\cos{n\theta} + b\sin{n\theta})\r^n(a\sin{n\theta} + b\cos{n\theta}) &= r^n(a\sin{n\theta} - b\cos{n\theta})\end{align*}\]The equality of each part leads to specific conditions that the real numbers \(a\) and \(b\) must satisfy.

• This technique simplifies complex equations, breaking them down into familiar algebraic terms.
• It also helps to find conditions for solutions, such as specific angles or positions in the Argand plane.