Why We Need Numbers That Don't Seem to Exist

 

I always have at least one student who wants to know why we have to learn about imaginary number since we can't see them.  To students, the concept of "imaginary" numbers might seem, well, imaginary. After all, how can a number not exist? Yet, these elusive entities play a crucial role in numerous fields, from physics and engineering to signal processing and even music.  

The story begins with a simple question: what is the square root of -1? In the realm of real numbers, this question has no answer. Any real number, when squared, results in a positive number. Mathematicians, ever curious, decided to invent a solution: the imaginary unit, denoted by the letter "i," where i² = -1.  At first glance, this might seem like a purely abstract concept, a mathematical curiosity with no real-world applications. However, the implications of imaginary numbers extend far beyond the realm of pure mathematics.

In electrical engineering, imaginary  numbers are fundamental to understanding alternating current (AC) circuits. They allow engineers to represent voltage and current as complex numbers, simplifying the analysis of circuits with capacitors and inductors. This is crucial for designing everything from power grids to electronic devices.   In control systems, such as those used in aircraft autopilots or industrial robots, imaginary numbers help engineers analyze and design systems that maintain stability and respond effectively to changing conditions.

Then in physics and quantum mechanics, one finds imaginary numbers playing an import part.  The wave function, a central concept in quantum mechanics, is often represented using complex numbers. These numbers describe the probability of finding a particle in a particular state, providing insights into the bizarre and counterintuitive world of the quantum realm. In addition, imaginary numbers are essential in describing electromagnetic waves, such as light and radio waves. They help represent the oscillating nature of these waves, which are crucial for various technologies, including communication systems and medical imaging. 

Another place imaginary numbers are important is in digital signal processing.  In fields like audio and image processing, imaginary numbers are used in techniques like the Fourier Transform. This powerful mathematical tool allows signals to be decomposed into their constituent frequencies, enabling tasks such as noise reduction, image compression, and audio filtering.

Beyond these fields, consider fractals.  Many beautiful and complex fractals, like the Mandelbrot set, are generated using complex numbers.These intricate patterns, with their infinite detail, have applications in fields like computer graphics and even art.  

While the term "imaginary" might suggest a lack of reality, these numbers are anything but. They are a powerful and indispensable tool with profound implications across various fields of science and engineering. From the intricate workings of our electrical grid to the fundamental nature of the universe itself, imaginary numbers play a vital role in shaping our understanding of the world around us.  Let me know what you think, I'd love to hear.  Have a great day.