Kind: captions
Language: en
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 maths
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 c x
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 pachioli'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 past eighth
grade knows the general solution it's
minus b plus minus root b squared minus
4 ac 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 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 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 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 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 khayam 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
shipping del faro 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
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 well 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 solution secret he
guarantees his own job security
for nearly two decades del faro keeps
his secret
only on his death bed in 1526 does he
let it slip to his student antonio fiore
fjor 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 fiore challenges
mathematician nicolo fontana tartalia
who has recently moved to fiora's
hometown of venice
nicolo 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
tartalia which means stutterer in
italian
growing up in poverty tartalia 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
tartalia gives a varied assortment of 30
problems to fiore
fiore gives 30 problems to tartalia 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
tartalia solves all 30 of fiora's
depressed cubics in just two hours
it seems fiora's boastfulness was his
undoing before the challenge came
tartalia learns of fiora'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 tartalia
writes but word was that a great
mathematician had revealed the secret to
fjor which seems more plausible so with
the knowledge that a solution to the
cubic is possible and with his
livelihood on the line tartalia 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 9x 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 9x you get
26. so just like with completing the
square we need to add on to the cube to
increase its volume by 9x
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
tertalia 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
3yz times its height x and tartalia
realizes this volume can perfectly
represent the 9x term in the equation if
its base is equal to 9. so he sets 3yz
equals 9.
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 sixth plus
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 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 tartalia 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 totalia
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
tertalia'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 tartalia 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
cardano manages to lure tartalia to
milan
and there on march 25th 1539 tartalia
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
one shall be able to understand it
cardano is delighted and immediately
starts playing around with tartalia'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 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 tartalia'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
tartalia 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 shipyone del phero 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 tartalia's by
decades so now as cardano sees it he can
publish the full solution to the cubic
without violating his oath to tartalia
three years later cardano publishes ars
magna the great art an updated
compendium of mathematics written in
five 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 tartalia del ferro and
fiore tartalia 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 ours magnet he
comes across some cubic equations that
can't easily be solved in the usual way
like x cubed equals 15 x plus four
plugging this into the algorithm yields
a solution that contains the square
roots of negative numbers
cardano asks tartalia about the case but
he evades and implies cardano is just
not clever enough to use his formula
properly the reality is tartalia 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 rs magna
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 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 two real numbers
which add to 10 and multiply to 40.
so mathematicians understood square
roots of negative numbers where math's
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 4 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 rs 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 raphael 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 mass 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 these 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 viet 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 schrodinger is searching
for a wave equation that governs the
behavior of quantum particles building
on de bruy's insight that matter
consists of waves he comes up with one
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 one
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
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 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 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 i kx 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 -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
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already familiar with and that is
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