Question

Decide whether the number is in the Mandelbrot Set. Explain your reasoning. $$c=-i$$

Short answer

The complex number \(c=-i\) is part of the Mandelbrot Set. After iteratively applying the function \(f_{c}(z)=z^2+c\) from \(z=0\), we found that the sequence created by the function doesn't tend to infinity, thus satisfying the condition to be in the Mandelbrot set.

Step-by-step solution

Understanding the Function

We first need to understand the function \(f_{c}(z)=z^2+c\) which forms the basis of the Mandelbrot Set. In particular, we need to note that this function is recursively applied, starting with \(z=0\). This means that we start with \(z_{0}=0\) and then calculate \(z_{1}, z_{2}, z_{3}\), and so on, where each \(z_{n+1}=z_{n}^2+c\) with \(c=-i\).

Calculating the First Few Iterations

Start to calculate the first few iterations of the function from \(z_{0}=0\). For \(n=0\), we get \(z_{1}=0^2 - i=-i\). Then for \(n=1\), we get \(z_{2}=(-i)^2 -i= -1 -i\). We continue this process for few more iterations.

Assessing Divergence

We continue to calculate further iterations. However, we can stop as soon as we find that the absolute value of our sequence (\(|z_n|\)) is becoming larger than 2, because once it gets over 2, the sequence will surely escape to infinity. However, if after many iterations the sequence is still bounded, it means the value of \(c\) belongs to the Mandelbrot set.

Key concepts

Complex Numbers

Complex numbers are fascinating entities in mathematics. Unlike regular numbers, complex numbers consist of two parts: a real part and an imaginary part. A complex number is written in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with \(i\) being the imaginary unit that satisfies \(i^2 = -1\).
Understanding complex numbers is crucial to exploring the Mandelbrot Set since they are used in the iterative functions that define it.
When dealing with complex numbers, calculations like addition and multiplication involve both the real and imaginary components. For example:

• To add two complex numbers \((a + bi)\) and \((c + di)\), simply add the real parts and the imaginary parts separately: \((a+c) + (b+d)i\).
• To multiply \((a + bi)\) and \((c + di)\), use the distributive property, and remember that \(i^2 = -1\). This yields: \((ac - bd) + (ad + bc)i\).

Complex numbers allow us to navigate and analyze expressions that involve the square roots of negative numbers, offering a wide array of possibilities in mathematics and beyond.

Iterative Functions

Iterative functions play a crucial role in many mathematical contexts, especially within the Mandelbrot Set. They are functions repeatedly applied in a sequence, with each iteration using the output of the previous one as its input. For the Mandelbrot Set, the iterative function is defined as \(f_{c}(z)=z^2 + c\), where \(c\) is a complex number.

The process starts with an initial value, often \(z_0 = 0\), and each subsequent term is generated by squaring the current term and adding the constant \(c\). This creates a sequence: \(z_1, z_2, z_3, \ldots\) such that \(z_{n+1} = z_n^2 + c\).
The beauty of iterative functions lies in their ability to produce complex patterns and behaviors from simple rules. For example, small changes in \(c\) can dramatically alter the behavior of the sequence, leading it to either grow without bound or remain confined within a certain range. This makes exploring iterative functions a captivating subject, particularly when visualized through fractals like the Mandelbrot Set.

Divergence

The concept of divergence is key to understanding whether a sequence will become unbounded or remain limited. In the context of the Mandelbrot Set, divergence refers to the point at which the values of the sequence \(|z_n|\) grow beyond a specific threshold (commonly 2) and continue to increase indefinitely.

To determine if a complex number \(c\) is part of the Mandelbrot Set, we iterate using the function \(f_{c}(z) = z^2 + c\) starting from \(z_0 = 0\). If through iterations, the magnitude \(|z_n|\) never exceeds 2, the sequence does not diverge, suggesting \(c\) is part of the set. However, if the magnitude surpasses 2 at any iteration, the sequence diverges, and \(c\) is not within the Mandelbrot Set.
The notion of divergence is not only critical for the Mandelbrot Set but also for numerous applications across different fields, including physics and engineering, where it helps distinguish between stable and unstable systems.