The relevance and application of Imaginary numbers

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What do you think about when I say “i”? Most people know “i” to be the square root of -1, and some may even remember it from the quadratic formula. Most people struggle when they are asked how it can be useful in terms of the real world.
In the past, great mathematicians used to have competitions to see who was better at math, and this was one of the main ways they would make money. In order to prepare for this, Gerolamo Cardano wanted to have a formula to obtain the roots of the equation x^3 = cx +d. He discovered that for particular cases in his formula, you would need to take the square root of a negative number, but he knew that since he was dealing with a cubic equation, it needed to cross the x-axis somehow and a real root must exist somewhere. It was Rafael Bombelli, Gerolamo Cardano’s student, who made more progress was made to this idea. Through his substitutions and manipulation of the formula, he found the real answer to the problem, but he did not seem to appreciate it as he just thought it was ‘broken’ math. For example, the area of a square is often expressed as the length squared, but what does it mean to have a negative area as represented by i^2? Since people did not understand the importance of these numbers, they were called imaginary.
Progress was continuing and the complex plane was developed, where the vertical axis would represent the imaginary numbers and the horizontal axis would represent real numbers. This plane may be used for vector calculations and displacement problems. It is also useful when learning about polar graphs and their properties. A complex number is made up of a real part and an imaginary part. The imaginary part represents how far up or down you need to go on the imaginary axis, and the real part represents how far left or right you need to go on the real axis. Unlike the x and y plane, the dimensions in the complex plane are connected. One example of this is when you multiply 1 by i, you get i, which is a point 90 degrees counterclockwise from the point you started on the complex plane. You can keep multiplying the result by i and you will notice that the point will continue to shift 90 degrees counterclockwise from the last.
But how are imaginary numbers useful? Imaginary numbers have a variety of applications in math, physics, and other sciences. For example, they help us model electric circuits, electromagnetic waves, fluid flow, and understand the physics of the quantum world.
One of the most famous formulas involving imaginary numbers in Euler’s formula. This states that e^ix=cosx+i*sinx. This can be derived from the Maclaurin series (a series which expresses e^x, sinx, and cosx), and can be used to derive the fact that e^pi*i=-1. This formula can be used to add two trigonometric functions of different phases, which may be applied when dealing with AC circuits with alternating current. The math to represent this is all complex numbers. Additionally, there are some peculiar things you can prove with this formula like how cos(i)=1.54 and i^i is roughly .208.
Another popular idea is the Fourier Transform, which states that any function can be made up of a lot of sine and cosine functions added up. This is particularly used in messy signals since they could be broken up into the frequencies that they are composed of (the sine and cosine functions that they are made of), which can then be analyzed and manipulated. In the real world, this is helpful for audio processing, speech recognition, radar, etc. The equation for Fourier Transform involves calculus, but it is based on imaginary numbers, and a term like e^ix is present in the equation.
Imaginary numbers have prevalence in other areas as well. One of the ideas in quantum mechanics state that you cannot be certain of the position and the velocity of a particle at the same time. When studying this alongside other ideas, you can see the prevalence of imaginary numbers in the math. Additionally, when dealing with control theory in engineering, graphs are used to portray the stability of systems and how certain parameters change with respect to individual components of the control system. These types of systems are found in rockets, fighter jets, robots, autonomous vehicles, and more.
It is important to understand that you cannot necessarily represent imaginary numbers in the real world. For instance, you cannot just have 5i pounds of cheese, or 3i grams of sugar. Overall, imaginary numbers help simplify the math for dealing with the real world and has a variety of useful applications.