Introduction

Despite how “obvious” they might appear today, many discoveries in mathematics took centuries before they became widely accepted. This is particularly visible when we talk about one of the most fundamental concepts in mathematics: different types of numbers.

Humans have counted using the natural numbers for at least 20,000 years. Mathematicians in ancient Egypt also developed fractions, and students of Pythagoras in ancient Greece discovered the first irrational number: 2. They were initially terrified by this discovery, and – according to legend – tried to cover up its existence because it didn't fit into their pure, perfect model of the world.

Natural Numbers
Prehistory
20,000 BCE

Fractions
Ancient Egypt
2000 BCE

Irrational Numbers
Ancient Greece
400 BCE

Negative Numbers
China
100 BCE

Zero
Khmer
700 CE

Negative numbers and zero proved even harder. In ancient Greece, numbers were defined geometrically. Numbers that couldn’t be directly measured or represented by lines, curves, or shapes were not even considered. Over the next 1000 years, different Asian and Arabic civilisations developed number systems that included zero and negatives, but even in the 16th century, some European mathematicians called equations with negative solutions “false” or “absurd.” Continue

There don’t seem to be any obvious solutions, but Algebra had been invented just a few centuries before Cardano was born. Let’s try to write the problem as an equation:

At this stage, most mathematicians would have stopped, because you can’t take square roots of negative numbers always need to have two results need to solve the equation for x, not y.

You may have noticed that sqrt(-15) is a very strange number. For square roots of positive numbers like sqrt(9) or sqrt(5), one can simply look for a number that, when squared, yields 9 or 5. But there is no number on the number line that yields -15 when squared. If the number is positive, multiplying it by itself results in a positive negative number. If the number is negative, multiplying it by itself also results in a positive negative number. As the Hindu mathematician Bhaskara II put it: “There is no square root of a negative number, because a negative number is not a square.”

Basic and established rules about numbers made solving this equation seem impossible. But this difficulty was not limited to this one problem. The issue of taking the square root of a negative number popped up everywhere for Cardano and his contemporaries. It appeared in the complicated cubic and quartic equations they were studying, but also in simple quadratic equations like x2+1=0. What made Cardano special is that, instead of stopping and giving up, he took a different approach. He was the first to say: Maybe this only seems impossible. Maybe we don’t think we have the numbers to solve this problem, but we actually do. In other words, Cardano proceeded to solve the problem as if sqrt(-15) existed, despite all evidence to the contrary. His solution, updated only to use modern notation, is shown below:

algebra flow y-5= ± sqrt(-15) y=5 ± sqrt(-15) Picking up where we left off, we can solve the equation for y. If y=5+sqrt(-15) x+5+sqrt(-15)=10 x=5-sqrt(-15) We can then solve for x, using either logic or by substituting one of the values of y into Equation 1. By a similar logic, if y=5-sqrt(-15), x=5+sqrt(-15) This means that either way our two numbers are 5+sqrt(-15) and 5-sqrt(-15). (5+sqrt(-15))(5-sqrt(-15)) 25+5sqrt(-15)-5sqrt(-15)+(sqrt(-15))^2 To verify that our solution is correct, let’s multiply x and y and see if the product is 40. We start by expanding using the distributive property. 25-(sqrt(-15))^2 We can simplify using the fact that adding any number to its opposite gives 0. 25-(-15) To evaluate (sqrt(-15))^2, consider what sqrt(-15) means. By definition, if you square sqrt(-15), you should get -15. 40 When we simplify, we get 40, which proves that our answer does work, if we let sqrt(-15) exist!

The first and most important thing to notice about this solution is that, with one exception, Cardano treated sqrt(-15) the same way he would treat any other number. In his calculations, he still obeyed known rules of arithmetic, such as using the distributive property when multiplying. Only when considering the expression (sqrt(-15))^2 did he do something that had never been done before. Yet here his calculation also makes sense: if sqrt(-15) does, in fact, exist, it has to be a number that, when you square it, you get -15.

If this solution seems confusing, you are, once again, in good company. Cardano himself said that using -15 in this way was “as refined as it was useless.” Yet Cardano and his contemporaries could not shake -15 and other numbers like it; they kept appearing in the equations they were trying to solve. Another Italian mathematician, Rafael Bombelli, used the square roots of negative numbers to solve several important and previously unsolvable cubic and quartic equations. Every time these strange new numbers were used in the solutions of real problems, they gained a bit more plausibility.

Nevertheless, most mathematicians of the time still viewed them as meaningless, incomprehensible, or impossible. The philosopher – mathematician Rene Descartes found them so far-fetched he called them imaginary numbers, and labeled all other numbers real. Though many future mathematicians would object to this labeling, the terms ended up sticking and we still use them today.

Among imaginary numbers, there is one that is special: -1. It plays the role that 1 plays in the real number system: every imaginary number has -1as a factor multiple. To simplify the math surrounding imaginary numbers, Leonhard Euler gave -1 a special name: i, the imaginary unit. In almost every respect, i behaves like a real number, with one (important) exception: when i is multiplied by itself, it yields . This is the defining property of i.

To see the special role that i plays, consider the imaginary number -28. We could leave it as is, but we can also rewrite it as follows:

algebra flow -1 * 28 Break up -28 into a purely imaginary and a purely real part. i28 Use the fact that i=-1 to rewrite further. i4 * 7 2i7 Simplify 28 by finding the highest perfect square that goes into 28.

This may seem like an overly complicated way of writing imaginary numbers, but as we will see, it can actually greatly simplify the arithmetic surrounding imaginary numbers. For now, let’s practice writing a few imaginary numbers using the number i:

-36 = 6i -48 = 4i3 -14 = i14 -49 = 7i -50 = 5i2

Though the existence of imaginary numbers certainly seemed a bit more possible after the work of Cardano and Bombelli, it would be another 200 or so years before imaginary numbers would become fully accepted by scientists and mathematicians around the world. It is worth asking how and why this happened. How did these numbers, which throughout history and even into the late 1700s were regarded as impossible, false, and absurd, eventually come to be seen as imaginable, believable, and even convincing?

To help answer this question, it is useful to consider exactly what was so impossible about these types of numbers. By the 18th century, we no longer associated numbers strictly with counting, measurement, or geometry, but instead we viewed them as points along a continuous line with negatives to the left of zero and positives to the right. Even irrational numbers like had their place in line, although admittedly their exact position could be a bit hard to pinpoint. But imaginary numbers were nowhere to be found. They weren’t negative, nor were they positive, and they definitely weren’t zero. If they didn’t exist anywhere along the line, then where did they exist?

The answer came from three separate sources at around the same time: a Norwegian surveyor named Caspar Wessel, a Parisian bookkeeper named Jean-Robert Argand, and one of the greatest mathematicians of all time, Carl Friedrich Gauss. To see the logic behind their interpretation of complex numbers, it is helpful to take a closer look at the operation of multiplication, understood geometrically.

One way of representing multiplication is as a transformation of vectors (arrows) along the number line. Consider the following examples:

If we multiply any number by a positive number, we or the resulting vector, and the direction of the vector .

If we multiply any number times a negative number, it will still or the resulting vector, and the direction of the resulting vector when we it . Note that we can also think of the change in direction as a reflection, but as we will see, thinking of it as a rotation is more helpful when we start to consider imaginary numbers.

In this way, we can think of multiplying by -1 as a pure rotation dilation translation by .

With this geometric understanding of multiplication, we can begin to think where imaginary numbers might fit into this picture. We already know that i × i = . To understand what the imaginary number i really means, let’s also think about it as a transformation. What transformation can we apply twice, starting from 1, to arrive at –1?

Up to now, mathematicians had always thought of the number line as 1-dimensional: a simple line that goes from –infinity to +infinity. And we already knew that none of these numbers could ever satisfy i² = –1.

Around 1797, mathematicians like Gauss and Argand realised that to find these new “imaginary” numbers, we had to look in an entirely new dimension. They discovered that numbers form a 2-dimensional plane, not just a line.

This is an amazing discovery! It means that instead of existing along the number line, imaginary numbers like i exist perpendicular to it. For any real number, you can turn it into an imaginary number by multiplying it by i, the imaginary unit. Visually, this will rotate it [{90°]] onto what we now know as the imaginary number line. Similarly, for any imaginary number, you can multiply it by i again, and it will rotate again, and return to a place along the real number line.

When we combine the real and imaginary number lines, we end up with a complex plane. In addition to locating purely real and purely imaginary numbers, we can locate combinations of real and imaginary numbers, which we call complex numbers, at different points in the plane. You can write any complex number in the form a+bi, where a is referred to as the real part and bi is referred to as the imaginary part. Sometimes, the real part is written as Re(z ), and the imaginary part as Im (z), where z=a+bi. For example, the complex number -2+3i could be written as Re(z)= and Im(z)=3. For each number given below, place it in its proper position on the complex plane.

As you saw with some of these examples, you can write even purely real numbers (like sqrt(5)) or purely imaginary numbers (like -3i) as complex numbers. In this sense, the complex numbers contain all possible real numbers, imaginary numbers, and combinations thereof.

It is hard to overestimate the importance of these discoveries. Numbers that had once seemed like empty, meaningless symbols were shown to have a meaningful and concrete interpretation that both fit in with and expanded known rules of geometry and arithmetic. As Tobias Danzig memorably put it, the discovery of the complex plane “gave these phantom beings . . . flesh and blood. It took the imaginary out of the complex and replaced it with an image.” Just as we had extended our concept of number in the past to make room for fractions, decimals, irrationals, zero, and negative numbers, we now had to extend it once more to make room for imaginary and complex numbers.

With an entirely new dimension of numbers discovered, new worlds opened up. Complex numbers were invented simply to solve certain polynomial equations, but they actually have important applications in many areas of engineering and technology:

Electrical engineering Electrical engineers use complex numbers when working with alternating currents, voltages, and electromagnetic fields. // https://en.wikipedia.org/wiki/Electrical_engineering#/media/File:Silego_clock_generator.JPG

Signal processing Complex numbers are often used in technologies that analyze, edit and even produce sounds, images, and other information. // https://entertainment.howstuffworks.com/sound-editing4.htm

Civil and mechanical engineering Civil and mechanical engineers use complex numbers to model the vibrations of buildings, bridges, roads, and even airplanes.

Quantum Physics // https://phys.org/news/2019-06-quantum-physics-heisenberg-uncertainty.html Complex numbers (and some even crazier numbers) are used by physicists that try to explain the nature of the fundamental particles that make up the building blocks of matter and the forces between them.

Complex numbers played a role in some of the most important scientific breakthroughs of the 19th and 20th centuries. Just as important, complex numbers revolutionized mathematics. They opened the door to new types of geometry that challenged and expanded our understanding of space, and they paved the way for even more innovative answers to the question: what are numbers and what can they be used for?

In many ways, the idea of complex numbers was the point of departure for a thousand other innovations, questions, and discoveries. But in at least one important way, it was the end of a chapter in the history of mathematics. As we will soon see, for certain types of equations, we could finally say that we had all of the numbers we needed to solve any equation we wanted to. Better yet, we knew for sure exactly how many solutions each of these equations would have. In this sense, it is fitting that the journey to find the solutions to these equations, a journey that went through so many twists and turns, so many fits and starts, ended with numbers that we call imaginary. It is a testament to the role that imagination, intuition, and conjecture play in mathematical discovery. It is a reminder that even the numbers that we now call real had to be imagined first.