- 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. (light music) 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'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, 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 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. You know, 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 the 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 length 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." 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 from 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 like 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 own 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 own
mathematical prowess and specifically, his ability
to solve the depressed cubic. On February 12, 1535, Fior challenges mathematician
Niccolo Fontana Tartaglia who has recently moved to
Fior's hometown of Venice. Niccolo 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 a 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. Take the equation x cubed
plus nine 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 nine x, you get 26. So just like with completing the square, we need to add onto the cube to increase its volume by nine x. Imagine extending three sides
of this cube out a distance y, creating a new, larger cube
with sides of length, call it z. z is just x plus y. The original cube has been padded out and we can break up the additional
volume into seven shapes. There are three rectangular prisms with dimensions of x by x by y, and another three narrower prisms with dimensions of x by y by y, plus there's a cube with a volume y cubed. Tartaglia rearranges the
six rectangular prisms into one block. One side has length three y, 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, three yz times its height, x. And Tartaglia realizes this
volume can perfectly represent the nine x term in the equation, if its base is equal to nine. So he sets three yz equals nine. Putting the cube back together, you find we're missing the
one 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. We have two equations and two unknowns. Solving the first equation for z and substituting into the second, we get y to the six plus
26 y 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 six, instead of just three. 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 one, which means y equals one,
and z equals three over y, so z is three. And since x plus y equals
z, x must be equal to two, 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, 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. 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. 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." Cardono 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 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 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 CC's a good fraction of
the mathematics community. 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. 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 four. 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 five. 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
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 the 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 own 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 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 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 Euler later
introduces the letter i to represent the square
root of 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 Schrödinger 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 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. Watch what happens when we
repeatedly multiply by i. Starting with one. One times i is i, i times i is negative one, by definition. Negative one times i is negative i, and negative i times i is one. 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 the 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 two 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, "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. 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
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Historical discussion of physical geometric-based formulation of algebraic cubics, transition to use of imaginary numbers uncoupled from real geometric space, finally leading to complex numbers as basis for early quantum mechanics formalized by Schrodinger.
Note: I didn't validate any of the historical portion of the video. Some enlightening thoughts comparing properties of negative roots. It also (helped me personally) visualize the degrees of freedom not captured by employing the traditional physicality of geometry. There are some discussion points at the end by Freeman Dyson which I would love to see another video about.
What software is used to make these animations? Specifically in 19:30
Very interesting story, but the last bit about imaginary numbers in Schrodingers equation seems exaggerated. Surely physicists must have used complex numbers long before the 1900s. I mean, Euler used complex numbers to describe vibrating strings. Arguably, he wasn't a physicist, but I cannot imagine it took 200 years before physicists caught on.
Or is there something I don't understand about how Schrodinger applied "i" to his equation?
His videos are always so good. He really has a talent for explaining complicated concepts which would normally go right over most peoples' head, but thanks to his gift, wider range of people can enjoy this stuff. So grateful for him.
Amazing video! There shoild be more content about history of math and physics
The animations of historical figures... is this the same animator that did the animations for the new Cosmos?
[removed]
Thanks for sharing! The visuals used to describe the math in his videos are so helpful.
This video is amazing.