Illustrating Imaginary Numbers
Advanced students of math and science are familiar with the concepts of complex and imaginary numbers due to the various problems they encounter. When you have two different quantities, or when you have orthogonal entities that are paired together, or if you are dealing with vectors — all great ways to understand imaginary and complex numbers. However, how does it apply to everyday problems? I found one way to introduce students to this concept. Any corrections are welcome.
The misnomer imaginary numbers immediately discourages students of math from attempting to visualize it. Imaginary numbers evaded me as a math student myself. Now, decades later, when I tried to teach my daughter the same, I was looking for some visualization or teaching aid to help her understand what imaginary numbers are in the real world.
The Royal Land Grant Problem
Consider the following scenario from ancient times
A king wishes to grant a rectangular plot of land to a favored noble. The king, having a particular fondness for certain numbers, decrees the following:
The perimeter of the rectangular plot must be exactly 20 units (perhaps 20 lengths of a standard measuring rope).
The area of this plot must be precisely 28 square units. The royal surveyor is tasked with determining the length and width of this plot.
To break it down further, if the size of the rectangular plot is L x W, where L represents the length and W represents the width of the land, we need to find two values of L and W such that
2L + 2W = 20, or
L + W = 10.
Attempting to solve this using intuition immediately tells us that there is no ‘real’ solution possible. No two numbers that add up to 10 can multiply to make 28 units. Next, let us try to solve it algebraically
W = 10 - L, which helps us rewrite L x W = 28 as
L ( 10 - L ) = 28, or
10L - L² = 28, or
L² - 10L + 28 = 0
Solve using the quadratic formula,
We have a complex root, as expected. If we simplify it, L turns out to be
5 ± √(-3) =
5 ± √3 i, or 5 ± √3 i
Get eggcigar’s stories in your inbox
Join Medium for free to get updates from this writer.
What does this mean in the real world? What is the royal surveyor to do?
The Number Line and Complex Numbers
“Mathematics is not only real, but it is the only reality.” — Albert Einstein
As children, we all have learned about the number line. The number line represents the numbers we know on a single horizontal line of infinite length, thereby depicting that there are infinite numbers. Centered with zero, numbers are evenly spaced, and the numbers progressively increase as we go right and progressively decrease as we go left.
Some students were taught that imaginary numbers exist on a scale perpendicular to the number line. This imaginary line axis is also a number line, orthogonal to the original one, and shares the same zero. This plane, where both real and imaginary numbers exist, is also known as the complex plane.
The complex plane is a concise and precise definition and representation of the complex number system. The number would increase in value by going right or up. When they go up, the imaginary value of the number increases, and when they go down, the value decreases.
Every point where the grid lines meet is a complex number with a whole number for the real part and a whole number for the imaginary part.
The above picture depicts a few complex numbers: the orange one is 4+2i, the cyan one is at -3+4i, and the purple one is at -2-3i.
Armed with this knowledge, let us return to the original Royal Land Grand Problem.
The Solution
The solution we obtained was 5 ± √3i, which means each side of the plot is either 5 + √3i or 5 - √3i. This means the sides of the plot are 5 units (feet, let us say) wide and are elevated off the ground (or dug into the ground) at √3 units!
Is this correct? Let us verify. The side of the rectangle is 5 units wide and √3 units high. Applying Pythagoras' Theorem, we realize the side is √(5² + √3²) = √(25 + 3) = √28. This means it is a square of √28 length! It is elevated off the ground, giving us the extra 3 square units.
Summary
Not all mathematical concepts are intuitive. Many branches of abstract algebra, linear algebra, and mathematical logic are not easily accessible to our natural senses. It takes years of study, mentorship, and guidance to master the concepts such as higher dimensions, abstract structures, and probabilistic reasoning, which is what we instinctively experience.
However, many concepts, especially those that are misunderstood, misnamed, or made to appear esoteric, are in fact very accessible and understandable at a young age. Imaginary numbers and complex numbers are one such concept — students can be made intuitively aware of these concepts. I sincerely hope this blog post inspires someone to become enamored with mathematics and fall in love with it.
Addendum: Background of Imaginary Numbers
Consider the operation of squaring a number. Consider the square number, 4: 2 x 2 = 4. For example, a sheet of paper 2 inches wide and 2 inches long will have an area of 4 square inches. In other words, the piece is comprised of four pieces, each one inch long and one inch wide. Now, consider -2 x -2 = 4. It is possible to imagine this scenario. Let’s say you are playing a game where a curse causes you to lose 2 hearts (-2). However, you get another curse with it, which reverse the last effect twice. The reverse is negative, and because it does reverse it twice, you end up gaining 4 hearts. However, imaginary numbers was not understood intuitively.
Making the concept of imaginary numbers public knowledge is attributed to a 16th-century mathematician, Cardano, as part of solving cubic equations. Backed by a fascinating story and with accounts calling him many things, from mathematical genius to formula thief, Cardano published the technique in Ars Magna (The Rules of Algebra). While solving cubic equations, he encountered the problem of finding square roots of negative numbers. What two numbers, when multiplied, create -4? Note that the concept of adding the (-) sign before negative numbers is from Rafael Bombelli L’Algebra. The square root of -4 cannot be 2, or -2, because both yield +4. Rene Descartes termed them imaginary numbers, due to skepticism of their existence, in his La Géométrie. It was Euler, in the 18th century, who introduced i in his works such as Introductio in analysin infinitorum. Seeing that 4 is a square number, but -4 is not, a shorthand for √-1 (symbol i) works perfectly. Then, it was easy to solve √-4, now √4 x √-1 = 2i. The rest is, really, history.
