- Mathematics began as a way to quantify the world and solve practical problems
- The problem of solving the cubic equation was considered impossible for 4,000 years
- Mathematicians struggled with negative numbers and finding a general solution to the cubic
- Scipione del Ferro solved the depressed cubic but kept it a secret
- Antonio Fior boasted about solving the depressed cubic, leading to a challenge by Niccolò Fontana Tartaglia
- Tartaglia solved all 30 problems in 2 hours, leading to the discovery of a method to solve the depressed cubic
- Tartaglia used three-dimensional geometry to solve the depressed cubic equation.

- Tartaglia solves the depressed cubic equation and keeps his method a secret
- Cardano persuades Tartaglia to reveal his method and then publishes it
- Cardano encounters cubic equations with square roots of negatives and struggles to find a solution
- Bombelli introduces the concept of imaginary numbers to solve these equations
- This leads to the development of complex numbers and the use of imaginary numbers in physics, specifically in the Schrodinger equation

- The function e to the ix creates a spiral by repeatedly multiplying by i as you go down the x-axis
- The real part of the spiral is a cosine wave and the imaginary part is a sine wave
- Schrodinger assumes solutions to his wave equation will look like e to the ix
- The exponential function has useful properties for solving the Schrodinger equation
- The Schrodinger equation describes the behavior of atoms and is the basis of chemistry and physics
- Nature works with complex numbers, not real numbers
- Brilliant, a website and app, teaches STEM concepts and offers a 20% discount for viewers of the video.

- Mathematics began as a way to solve practical problems and quantify the world
- The problem of solving the cubic equation was considered impossible for 4,000 years
- Mathematicians struggled with negative numbers and finding a general solution to the cubic
- Scipione del Ferro solved the depressed cubic but kept it a secret
- Antonio Fior boasted about solving the depressed cubic, leading to a challenge by Niccolò Fontana Tartaglia
- Tartaglia solved all 30 problems in 2 hours, leading to the discovery of a method to solve the depressed cubic
- This led to the development of complex numbers and their use in physics, specifically in the Schrodinger equation
- The Schrodinger equation describes the behavior of atoms and is the basis of chemistry and physics
- Nature works with complex numbers, not real numbers
- Brilliant, a website and app, teaches STEM concepts and offers a 20% discount for viewers of the video.

A general solution to the cubic equation was long considered impossible, until we gave up the requirement that math reflect reality. This video is sponsored by Brilliant. The first 200 people to sign up via https://brilliant.org/veritasium get 20% off a yearly subscription.

Thanks to Dr Amir Alexander, Dr Alexander Kontorovich, Dr Chris Ferrie, and Dr Adam Becker for the helpful advice and feedback on the earlier versions of the script. 

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References:
Some great videos about the cubic:

500 years of not teaching the cubic formula. -- https://youtu.be/N-KXStupwsc

Imaginary Numbers are Real -- https://youtu.be/T647CGsuOVU

Dunham, W. (1990). Journey through genius: The great theorems of mathematics. New York. -- https://ve42.co/Dunham90

Toscano, F. (2020). The Secret Formula. Princeton University Press. -- https://ve42.co/Toscano2020

Bochner, S. (1963). The significance of some basic mathematical conceptions for physics. Isis, 54(2), 179-205. -- https://ve42.co/Bochner63

Muroi, K. (2019). Cubic equations of Babylonian mathematics. arXiv preprint arXiv:1905.08034. -- https://ve42.co/Murio21

Branson, W. Solving the cubic with Cardano, -- https://ve42.co/Branson2014

Rothman, T. (2013). Cardano v Tartaglia: The Great Feud Goes Supernatural. arXiv preprint arXiv:1308.2181. -- https://ve42.co/Rothman

Vali Siadat, M., & Tholen, A. (2021). Omar Khayyam: Geometric Algebra and Cubic Equations. Math Horizons, 28(1), 12-15. -- https://ve42.co/Siadat21

Merino, O. (2006). A short history of complex numbers. University of Rhode Island. -- https://ve42.co/Merino2006

Cardano, G (1545), Ars magna or The Rules of Algebra, Dover (published 1993), ISBN 0-486-67811-3

Bombelli, R (1579) L’Algebra https://ve42.co/Bombelli

The Manim Community Developers. (2021). Manim – Mathematical Animation Framework (Version v0.13.1) [Computer software]. https://www.manim.community/

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Special thanks to Patreon supporters: Luis Felipe, Anton Ragin, Paul Peijzel, S S, Benedikt Heinen, Diffbot, Micah Mangione, Juan Benet, Ruslan Khroma, Richard Sundvall, Lee Redden, Sam Lutfi, MJP, Gnare, Nick DiCandilo, Dave Kircher, Edward Larsen, Burt Humburg, Blake Byers, Dumky, Mike Tung, Evgeny Skvortsov, Meekay, Ismail Öncü Usta, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Ruslan Khroma, Robert Blum, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson,Ron Neal 

Executive Producer: Derek Muller
Writers: Derek Muller, Alex Kontorovich, Stephen Welch, Petr Lebedev
Animators: Fabio Albertelli, Jakub Misiek, Ivy Tello, Jesús Rascón
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Camerapeople: Derek Muller, Emily Zhang
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Producers: Derek Muller, Petr Lebedev, Emily Zhang
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Music from Epidemic Sound and Jonny Hyman

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's math teacher, publishes Summa de Arithmetica, a comprehensive summary of all mathematics known in Renaissance Italy at the time.

In it, there is a section on the cubic, any equation which today we would write as ax cubed plus bx squared plus cx plus d equals 0.

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, they all came up empty-handed.

Pacioli's conclusion is 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 8th grade knows the general solution.

It's minus b plus minus root b squared minus 4ac all over 2a.

But most people just plug and chug into this formula completely oblivious to the geometry that ancient mathematicians used to derive it.

You know, back in those days, mathematics wasn't written down in equations.

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 1 side length 26 and the other side of length x.

Well, we can take this 26x rectangle and cut it in half.

So now I have 2 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 the equality.

So now I have this larger square with sides of length x plus 13, and it is equal to 196.

So I know that the sides of this square have length 14, which means X is equal to 1.

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 1 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 2 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 6 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 1 of those who will come after us will succeed in finding it.

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?

1 considered impossible by Leonardo da Vinci's math teacher?

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.

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 1 else in the world can solve the depressed cubic.

So by keeping his solution secret, he guarantees his own job security.

For nearly 2 decades, Delfero 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 Delfero's death, he boasts about his own mathematical prowess and specifically his ability to solve the depressed cubic.

On February 12th, 1535, Fiore challenges mathematician Niccolò Fontana Tartaglia, who has recently moved to Fiore's hometown of Venice.

Niccolò 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.

As is the custom, in the challenge Tartaglia gives a varied assortment 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.

Tartaglia solves all 30 of Fior's depressed cubics in just 2 hours.

It seems Fior's boastfulness was his undoing.

Before the challenge came, Tartaglia learns of Fiore's claim 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 a great mathematician had revealed the secret to Fiore, 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 3 dimensions.

Take the equation X cubed plus 9 X equals 26.

You can think of X cubed as the volume of a cube with sides of length X.

And if you add a volume of 9 X, you get 26.

So just like with completing the square, we need to add onto the cube to increase its volume by 9x.

Imagine extending 3 sides of this cube out a distance y, creating a new larger cube with sides of length, call it z.

The original cube has been padded out and we can break up the additional volume into 7 shapes.

There are 3 rectangular prisms with dimensions of x by x by y, and another 3 narrower prisms with dimensions of x by y by y.

Plus there's a cube with a volume y cubed.

Tertalia rearranges the 6 rectangular prisms into 1 block.

1 side has length 3y, the other has a length x plus y, which is z, and the height is x.

So the volume of this shape is its base, 3yz, times its height, x.

And Tartaglia realizes this volume can perfectly represent the 9x term in the equation if its base is equal to 9.

Putting the cube back together, you find we're missing the 1 small y cubed block.

So we can complete the cube by adding y cubed to both sides of the equation.

Now we have z cubed, the complete larger cube, equals 26 plus y cubed.

Solving the first equation for z and substituting into the second, we get y to the 6 plus 26y cubed equals 27.

At first glance, it seems like we're now worse off than when we started.

The variable is now raised to the power of 6 instead of just 3.

However, If you think of y cubed as a new variable, the equation is actually a quadratic.

The same quadratic that we solved by completing the square.

So we know y cubed equals 1, which means y equals 1.

And since x plus y equals z, x must be equal to 2.

Which is, indeed, a solution to the original equation.

And with that, 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, Tertullius summarizes his method in an algorithm, a set of instructions.

He writes this down not as a set of equations like we would today, 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 Cardano, 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.

He writes a series of letters that alternate between flattery and aggressive attacks.

Eventually, with the promise of an introduction to his wealthy benefactor, Cardano manages to lure Tartaglia to Milan.

And there, on March 25th, 1539, Tartaglia reveals his method.

But only after forcing Cardano 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 1 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.

If you substitute for x, x minus b over 3a, 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 has no 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 1 Scipione del Ferro, the man who, on his deathbed, gave the solution to the depressed cubic to Antonio Fiore.

Cardano finds this 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.

3 years later, Cardano publishes Ars Magna, the Great Art, an updated compendium of mathematics, written in 5 years, may it last for 500.

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, Delfero, and Fior, Tartaglia is displeased to say the least.

He writes insulting letters to Cardano and CCs a good fraction of the mathematics community.

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, literally.

While Cardano is writing Ars Magna, he comes across some cubic equations that can't easily be solved in the usual way, like x cubed equals 15x plus 4.

Plugging this into the algorithm yields a solution that contains the square roots of negative numbers.

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 similar 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 5.

Since the full square has an area of 25, to complete the square, Cardano has to somehow add negative area.

That is where the square roots of negatives come from, the idea of negative area.

Now this isn't the first time square roots of negatives show up in mathematics.

In fact, earlier in Ars Magna is this problem.

Find 2 numbers that add to 10 and multiply to 40.

You can combine these equations into the quadratic x squared plus 40 equals 10x.

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 2 real numbers which add to 10 and multiply to 40.

So mathematicians understood square roots of negative numbers were math's way of telling you there is no solution.

With a little guessing and checking you can find that x equals 4 is a solution.

So Why doesn't the approach that works for all other cubics find the perfectly reasonable solution to this 1?

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 Raffaello 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 own new type of number.

Bombelli assumes the 2 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 negative 1.

And this way, Bombelli figures out that the 2 cube roots in Cardano's equation are equivalent to 2 plus or minus the square root of negative 1.

So when he takes the final step and adds them together, the square roots cancel out, leaving the correct answer 4.

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, François Viette 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 Euler later introduces the letter I to represent the square root of negative 1.

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 1 of the most important and famous equations in all of physics, the Schrodinger equation.

And featured prominently within it is I, the square root of negative 1.

While mathematicians have grown accustomed to imaginary numbers, Physicists have not and are uncomfortable seeing it show up in such a fundamental theory.

Schrodinger 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.

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.

Watch what happens when we repeatedly multiply by I.

Negative 1 times I is negative I, and negative I times I is 1.

We've come back to where we started and if we keep multiplying by I, the point will keep rotating around.

So when you're multiplying by I, what you're really doing is rotating by 90 degrees in a complex plane.

Now there is a function that repeatedly multiplies by I as you go down the x-axis, and that is e to the ix.

It creates a spiral by essentially spreading out these rotations all along the x-axis.

If you look at the real part of the spiral it's a cosine wave and if you look at the imaginary part, it's a sine wave.

The 2 quintessential functions that describe waves are both contained in e to the ix.

So when schrodinger goes to write down a wave equation, he naturally assumes that the solutions to his equation will look something like e to the ix, specifically e to the ikx minus omega t.

You might wonder why he would use that formulation and not just a simple sine wave, but the exponential has some useful properties.

If you take the derivative with respect to position or time, that derivative is proportional to the original function itself.

And that's not true if you use the sine function, whose derivative is cosine.

Plus, since the Schrodinger equation is linear, you can add together an arbitrary number of solutions of this form, creating any sort of wave shape you like, and it too will be a solution to Schrodinger's equation.

The physicist Freeman Dyson later writes, Schrodinger put the square root of minus 1 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 1 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 math's connection to reality could it guide us to a deeper truth about the way the universe works.

Not gonna lie, I learned a ton while making this video because I really had to engage with some ideas that I was already familiar with.

And that is exactly what happens with Brilliant, the sponsor of this video.

Brilliant is a website and app that teaches you all kinds of STEM concepts, from geometry to quantum mechanics, and you learn by doing.

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How Imaginary Numbers Were Invented