A Mathematical History: “Imaginary” Numbers. Part 1: what’s so imaginary?
By: Arya Vohra
Tags: Mathematics, Imagination

Algebra, functions, and imaginary numbers—the key paradigm shifts I felt when learning math. In this article, I’ll introduce you to the commonly known (yet misunderstood) idea of imaginary numbers, the newest paradigm shift I went through.

one of the few times Hobbes is wrong.

To begin, we need to ask: what are numbers, and what is the purpose of a number? This seems somewhat ridiculous, but bear with me.

The number system, and by extension mathematics, is a tool that humans use to add order and understanding into our lives. The use of numbers for counting brings us back to humble beginnings, yet our number system today is founded upon the same basic set of arithmetic operations: addition, subtraction, multiplication and division. These all apply to imaginary numbers, just as they apply to all other numbers.

Our number system has evolved just as our societies and knowledge have. The ancient Egyptians were among the first to use fractions — a type of number that was “new” to all of us at some point—as a solution to the need for more granularity in our arithmetic system, likely used in some sort of agricultural economy.

We see that fractions were invented with a clear purpose in mind: representing a part of a whole number. This was a revolutionary step for our number system; fractions allowed the Egyptians to accomplish new, previously unexplored things.

Note that this utility isn’t exclusively practical: even though there is practical value in representing the ratios between integers, we use fractions for the sake of mathematical exploration, and that’s okay! The invention of mathematical tools for the sake of exploration will become important later.

Let’s get back to history. A few millennia later, the ancient Greeks found irrational numbers during their exploration of right-angled triangles. The word irrational means illogical / unreasonable, but irrational in mathematis denotes numbers that are not expressible in terms of a ratio, hence, ir-ratio-nal (check this out to learn more about the source of the word). Irrational numbers contradicted the understanding of the greatest mathematicians of the time, yet they extended our number system nevertheless.

Practical solutions to human problems (the Egyptians and their agriculture) arise from a more versatile number system. However, the Greeks (with their right-triangles) show us that great mathematical discoveries can be made in the pursuit of the beauty of mathematics in itself, rather than for some external practical purpose. These Greeks found a hole in our number system that needed filling, and they filled it!

Now, curious math students like yourself may find a pattern emerge. “Imaginary” numbers are just another class of number, exactly like the two “new” classes of numbers we’ve seen so far. Let’s see why and how imaginary numbers came about.

Rafael Bombelli!

Picture Bologna, Rome, back in the 16th century. Mathematician Rafael Bombelli begun pivotal work in his extension of the quadratic formula to cubics. Bombelli explored the del Ferro-Tartaglia-Cardano (quite the mouthful) formula, shown below:

Consider a cubic of the above form.

See here for derivation

This formula, however, was a bit troubling. Bombelli noticed that the solutions given often contained the square root of negative numbers. In fact, this happened so often that this issue is sometimes referred to as Cardano’s problem, named after the inventor of the formula, Jerome Cardano.

Bombelli persevered, though, and found a solution to Cardan’s problem. This is detailed below.

Since there exists no positive number whose square is negative, and no negative number whose square is negative, Bombelli realised that a new class of numbers had to be created that would have a negative square.

This declaration seems simple, but it broke the minds of the greatest mathematicians of the time. This is for the same reason that the ancient Greeks were baffled by the discovery of irrational numbers: all humans tend to be a little uncomfortable when we go through paradigm shifts. Mathematicians are humans too!

As a result of Bombelli’s declaration, the solutions given by the del Ferro-Tartaglia-Cardano formula can be shown to hold. Take the example of x³=15x+4.

Using the above del Ferro-Tartaglia-Cardano formula, we arrive at:

The solution above can be broken down into two components, and these can be manipulated to give the integral root of the cubic x³=15x+4:

Solutions for a & b in full can be found in Welch Labs’ video

Using this system of equations, we can find values for a and b:

Unfortunately, this is where a little bit of thoughtful playing about with integers and inspection is necessary. I haven’t found any rigorous methods for finding these roots (please comment if you can help!). We need find initial values for a and b by inspection, and can use long division to find new roots given these values.

Go ahead, try to find the integer values for a and b by inspection. Don’t worry, I’ll be here when you’re done…

Hopefully, you arrived at the solution a=2 and b=1. ∴ By the above solution for x:

Wonderful! We can now solve previously unsolvable cubics with a rigorous and repeatable method. One needs a bit of processing time and a few double-takes at the algebra to see the magnitude of the discovery here. I strongly recommend you try to follow the derivation of the del Ferro-Tartaglia-Cardano formula, it’s a fun algebraic challenge!

You might still be wondering about what i actually means. Is it simply an algebraic hack? Will it change my conception of the reality? Well, I’ll be following up on this article with a more comprehensive introduction to the actual uses of i — where you might encounter i, how i might even revolutionise your mental image of the number line!

Stay tuned for part 2 of this piece, thanks for reading!