Invention of Imaginary Numbers

Mathematics began as a way to quantify our world, to measure land, predict the motions of planets, and keep track of commerce. Then came a problem considered impossible. The secret to solving it was to separate math from the real world, to split algebra from geometry, and to invent new numbers so fanciful, they are called imaginary. Ironically, 400 years later, these very numbers turn up in the heart of our best physical theory of the universe. Only by abandoning math’s connection to reality could we discover reality’s true nature.

In 1494, Luca Pacioli who is Leonardo da Vinci math teacher publishes “Summa de Arithmetica,” a comprehensive summary of all mathematics known in Renaissance Italy at the time. In it, there’s a section on the cubic,

any equation which today we would write as ax cubed plus bx squared plus cx plus d equals zero.

People have been trying to find a general solution to the cubic for at least 4,000 years, but each ancient civilization that encountered it, the Babylonians, Greeks, Chinese, Indians, Egyptians, and Persians, all came up empty-handed. Pacioli concluded that a solution to the cubic equation is impossible. Now, this should be at least a little surprising, since without the X cubed term, the equation is simply a quadratic.

And many ancient civilizations had solved quadratics thousands of years earlier. Today, anyone who’s passed eighth grade knows the general solution. It’s minus b plus-minus root b squared minus four ac all over two a. But most people just plug and chug into this formula completely oblivious to the geometry that ancient mathematicians used to derive it.

Back in those days, mathematics wasn’t written down in equations. It was written with words and pictures. Take, for example, the equation x squared plus 26x equals 27. Ancient mathematicians would think of the x squared term like a literal square with sides of length x. And then 26x, well, that would be a rectangle with one side of length 26 and the other side of length x, and these two areas together add to 27. So how do we figure out what x is? Well, we can take this 26x rectangle and cut it in half. So now I have two 13x rectangles and I can position them so the new shape I create is almost a square, It’s just missing this section down here. But I know the dimensions of this section. It’s just 13 by 13. So I can complete the square by adding in a 13 by 13 square. Now, since I’ve added 13-squared or 169 to the left-hand side of the equation, I also have to add 169 to the right-hand side of the equation to maintain equality. So now I have this larger square with sides of length X plus 13, and it is equal to 196. Now the square root of 196 is 14. So I know that the sides of this square have a length of 14, which means X is equal to one. Now, this is a great visual way to solve a quadratic equation, but it isn’t complete. I mean, if you look at our original equation, x equals one is a solution. But so is negative 27.

For thousands of years, mathematicians were oblivious to the negative solutions to their equations because they were dealing with things in the real world, lengths and areas, and volumes. I mean, what would it mean to have a square with sides of length negative 27? That just doesn’t make any sense. So, for those mathematicians, negative numbers didn’t exist. You could subtract, that is find the difference between two positive quantities, but you couldn’t have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic equation. Instead, there were six different versions arranged so that the coefficients were always positive. The same approach was taken with the cubic.

In the 11th century, Persian mathematician Omar Khayyam identified 19 different cubic equations, again, keeping all coefficients positive.

He found numerical solutions to some of them by considering the intersections of shapes, like hyperbolas and circles, but he fell short of his ultimate goal, a general solution to the cubic. He wrote, “Maybe one of those who will come after us will succeed in finding it.”

About 400 years later and 4,000 kilometers away, the solution begins to take shape. Scipione del Ferro is a mathematics professor at the University of Bologna. Sometime around 1510, he finds a method to reliably solve depressed cubics. These are a subset of cubic equations with no X squared term. So, what does he do after solving a problem that has stumped mathematicians for millennia? One considered impossible by Leonardo da Vinci’s math teacher? He tells no one. See, being a mathematician in the 1500s is hard. Your job is constantly under threat from other mathematicians who can show up at any time and challenge you for your position. You can think of it as a math duel. Each participant submits a set of questions to the other, and the person who solves the most questions correctly gets the job while the loser suffers public humiliation.

As far as del Ferro knows, no one else in the world can solve the depressed cubic. So, by keeping his solutions secret, he guarantees his job security. For nearly two decades, del Ferro keeps his secret. Only on his deathbed in 1526 does he let it slip to his student Antonio Fior. Fior is not as talented a mathematician as his mentor, but he is young and ambitious. And after del Ferro’s death, he boasts about his mathematical prowess and specifically, his ability to solve the depressed cubic. On February 12, 1535, he challenges mathematician Niccolo Fontana Tartaglia, who has recently moved to Fior’s hometown of Venice. Fontana is no stranger to adversity. As a kid, his face was cut open by a French soldier, leaving him with a stutter. That’s why he’s known as Tartaglia, which means stutterer in Italian. Growing up in poverty, Tartaglia is largely self-taught. He claws his way up through Italian society to become a respected mathematician. Now, all of that is at stake.

As is the custom, in the challenge Tartaglia gives a very discernment of 30 problems to Fior. Fior gives 30 problems to Tartaglia, all of which are depressed cubics. Each mathematician has 40 days to solve the 30 problems they’ve been given. Fior can’t solve a single problem Tartaglia solves all 30 of Fior’s depressed cubics in just two hours. It seems Fior’s boastfulness was his undoing. Before the challenge came, Tartaglia learns that Fior’s claimed to have solved the depressed cubic, but he’s skeptical. “I did not deem him capable of finding such a rule on his own,” Tartaglia writes. But word was that great mathematician had revealed the secret to Fior, which seems more plausible. So with the knowledge that a solution to the cubic is possible, and with his livelihood on the line, Tartaglia sets about solving the depressed cubic himself. To do it, he extends the idea of completing the square into three dimensions. Tartaglia becomes the second human on the planet to solve the depressed cubic. To save himself the work of going through the geometry for each new cubic he encounters, Tartaglia summarizes his method in an algorithm, a set of instructions. He writes this down not as a set of equations like we would today. The modern algebraic notation wouldn’t exist for another hundred years, but instead, as a poem. Tartaglia’s victory makes him something of a celebrity.

Mathematicians are desperate to learn how he solved the cubic, especially Gerolamo Cardeno, a polymath based in Milan. As you can guess, Tartaglia will have none of it. He refuses to reveal even a single question from the competition. But Cardano is persistent. He writes a series of letters that alternate between flattery and aggressive attacks. Eventually, with the promise of an introduction to his wealthy benefactor, Cardono manages to lure Tartaglia to Milan. And there on March 25, 1539, Tartaglia reveals his method, but only after forcing Cardono to swear a solemn oath not to tell anyone the method, not to publish it, and to write it only in cipher. Quote, “So that after my death, no one shall be able to understand it.” Cardano is delighted and immediately starts playing around with Tartaglia’s algorithm. But he has a loftier goal in mind, a solution to the full cubic equation, including the x squared term. And amazingly, he discovers it. If you substitute for x, x minus b over three a, then all the X squared terms cancel out. This is the way to turn any general cubic equation into a depressed cubic, which can then be solved by Tartaglia’s formula.

Cardano is so excited to have solved the problem that stumped the best mathematicians for thousands of years, he wants to publish it. Unlike his peers, Cardano does not need to keep the solution a secret. He makes his living not as a mathematician, but as a physician and famous intellectual. For him, the credit is more valuable than the secret. The only problem is the oath he swore to Tartaglia who won’t let him break it. And you might think this would be the end of it. But in 1542, Cardano travels to Bologna and there he visits a mathematician who just happens to be the son-in-law of one Scipione del Ferro, the man who on his deathbed, gave the solution to the depressed cubic to Antonio Fior. Cardano finds the solution in del Ferro’s old notebook, which is shared with him during the visit. This solution predates Tartaglia’s by decades. So now, as Cardano sees it, he can publish the full solution to the cubic without violating his oath to Tartaglia.

Cardano is so excited to have solved the problem that stumped the best mathematicians for thousands of years, he wants to publish it. Unlike his peers, Cardano does not need to keep the solution a secret. He makes his living not as a mathematician, but as a physician and famous intellectual. For him, the credit is more valuable than the secret. The only problem is the oath he swore to Tartaglia who won’t let him break it. And you might think this would be the end of it. But in 1542, Cardano travels to Bologna and there he visits a mathematician who just happens to be the son-in-law of one Scipione del Ferro, the man who on his deathbed, gave the solution to the depressed cubic to Antonio Fior. Cardano finds the solution in del Ferro’s old notebook, which is shared with him during the visit. This solution predates Tartaglia’s by decades. So now, as Cardano sees it, he can publish the full solution to the cubic without violating his oath to Tartaglia.

Three years later, Cardano publishes “Ars Magna,” The Great Art, an updated compendium of mathematics. “Written in five years, may it last for five hundred.” Cardano writes a chapter with a unique geometric proof for each of the 13 arrangements of the cubic equation. Although he acknowledges the contributions of Tartaglia, del Ferro, and Fior, Tartaglia is displeased, to say the least. He writes insulting letters to Cardano. And he has a point. To this day, the general solution to the cubic is often called Cardano’s method. But “Ars Magna” is a phenomenal achievement. It pushes geometrical reasoning to its very breaking point. While Cardano is writing “Ars Magna “ he comes across some cubic equations that can’t easily be solved in the usual way. Cardano asks Tartaglia about the case, but he evades and implies Cardano is just not clever enough to use his formula properly. The reality is Tartaglia has no idea what to do either. Cardano walks back through the geometric derivation of a problem to see exactly what goes wrong. While the 3D cube slicing and rearrangement works just fine, the final quadratic completing the square step leads to a geometric paradox.

Cardano finds part of a square that must have an area of 30, but also sides of length five. Since the full square has an area of 25, to complete the square, Cardano has to somehow add a negative area. That is where the square roots of negatives come from, the idea of the negative area. Now, this isn’t the first time square roots of negatives show up in mathematics. Earlier in “Ars Magnae” is this problem. Find two numbers that add to 10 and multiply to 40. You can combine these equations into the quadratic X squared plus 40 equals 10 X. But if you plug this into the quadratic formula, the solutions contain the square roots of negatives. The obvious conclusion is that a solution doesn’t exist, which you can verify by looking at the original problem. There are no two real numbers, which add to 10 and multiply to 40. So, mathematicians understood square roots of negative numbers were Maths way of telling you there is no solution. But this cubic equation is different. With a little guessing and checking, you can find that x equals four is a solution. So why doesn’t the approach that works for all other cubics find the perfectly reasonable solution to this one? Unable to see a way forward, Cardano avoids this case in “Ars Magna” saying the idea of the square root of negatives “is as subtle as it is useless”.

But around 10 years later, the Italian engineer Rafael Bombelli picks up where Cardano left off. Undeterred by the square roots of negatives and the impossible geometry they imply, he wants to find a way through the mess to the solution. Observing that the square root of a negative “cannot be called either positive or negative”, he lets it be its new type of number. Bombelli assumes the two terms in Cardano’s solution can be represented as some combination of an ordinary number and this new type of number, which involves the square root of the negative one. And this way, Bombelli figures out that the two cube roots in Cardano’s equation are equivalent to two plus or minus the square root of the negative one. So, when he takes the final step and adds them together, the square roots cancel out, leaving the correct answer, four. This feels nothing short of miraculous. Cardano’s method does work, but you have to abandon the geometric proof that generated it in the first place. Negative areas, which make no sense, in reality, must exist as an intermediate step on the way to the solution.

Over the next hundred years, modern mathematics takes shape. In the 1600s, Francois Viete introduces the modern symbolic notation for algebra, ending the millennia-long tradition of math problems as drawings and wordy descriptions. Geometry is no longer the source of truth. Rene Descartes makes heavy use of the square roots of negatives, popularizing them as a result. And while he recognizes their utility, he calls them imaginary numbers, a name that sticks, which is why Leonhard Euler later introduces the letter i to represent the square root of the negative one. When combined with regular numbers, they form complex numbers. The cubic led to the invention of these new numbers and liberated algebra from geometry. By letting go of what seems like the best description of reality, the geometry you can see and touch, you get a much more powerful and complete mathematics that can solve real problems. And it turns out the cubic is just the beginning.

In 1925, Erwin Schrodinger is searching for a wave equation that governs the behavior of quantum particles building on de Broglie’s insight that matter consists of waves. He comes up with one of the most important and famous equations in all of physics, the Schrödinger equation.

And featured prominently within it is i, the square root of the negative one. While mathematicians have grown accustomed to imaginary numbers, physicists have not and are uncomfortable seeing it show up in such a fundamental theory. Schrödinger himself writes “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. The wave function Psi is surely fundamentally a real function.” This seems like a fair objection, so why does an imaginary number that first appeared in the solution to the cubic turn up in fundamental physics? Well, it’s because of some unique properties of imaginary numbers. Imaginary numbers exist on a dimension perpendicular to the real number line. Together, they form the complex plane.

The physicist Freeman Dyson writes, “Schrödinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation. And Schrodinger found to his delight that the equation has solutions corresponding to the quantized orbits in the Bohr model of the atom. It turns out that the Schrodinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers. This discovery came as a complete surprise, to Schrodinger as well as to everybody else.” So, imaginary numbers discovered as a quirky, intermediate step on the way to solving the cubic turn out to be fundamental to our description of reality. Only by giving up Maths connection to reality could it guide us to a deeper truth about the way the universe works.

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