Well, I'm going to bring you back to the sixteenth century first.
The Italians, especially, were interested in cubic polynomials
x cubed plus, oh, five x plus seven, something of that sort
Now I'll draw you the graph of any of these cubic polynomials
That is to say I'll draw it qualitatively. Unless you're particularly perverse
in your choice of cubic polynomial it's gonna look sort of like that
Brady: this kind of distinctive wiggle
Barry Mazur: there's a wiggle there. There's a ... it looks like it's getting to a maximum
and then it changes its mind, then goes to a dip there, and then it thinks 'oh gosh this is a valley' and then
especially if the coefficient of the X cube is positive it'll go thphsssss
off to plus infinity
otherwise it will be reversed I'll draw you a few of them
now you might say why did I call this a few of them rather than just one
the reason is I haven't told you where the x-axis is and I can change my
cubic polynomial by adding a constant
so raising its graph by adding a constant or lowering its graph by subtracting a constant
so maybe the x-axis of this one is there
the x-axis of this one is there
and the x-axis of this one is there now once I have told you when the x-axis is
I can tell you when this polynomial is zero
at least graphically because the 0s of this cubic polynomial will just be
where its graph intersects the x-axis
so this one has only a single 0 there
this one has only a single 0 here
and this one has 3 of em
these would be the solutions to the equation a
for example this one, we will have to figure out which of these three it is
but doesn't matter
we're looking for solutions to equations of this sort
so as I say unless you're really perverse you either have one or three solutions
and it's the 16th century that people began to get formulas and equations
for the solutions of cubic polynomials
that if they for the solutions ... when a cubic polynomial,
of a problem, when a cubic polynomial is 0
now, something very funny happened
you think, this guy has three solutions so it must be easier to find them because they are 3 of em
rather than these two which have only one but it's quite the reverse
be equations that lead to you to a solution for either of these two, are, well
they're a bit of algebra, but a bit of algebra that's perfectly happily in the, sort of, mental theatre
of the 16th century people. This one was a weirdy, it was weird.
and in order to solve this, these are real numbers so all you have to do is work in real numbers
this one too, you work in real numbers and you get be solution
for this one something very funny happened
I need another sheet of paper Brady: you can have as many as you ...
Barry Mazur: well this is the toughie and now for this one of course the equation for this is real numbers
the solutions, there are three of them, are real
and in order to pass from the cubic equation to the solution you have to do various things
and what seemed a little perturbing, in fact very perturbing, to the 16th century people was this passage
you'd have to pass through a region
which seem to be (how do I wanna put it) out of the familiar mental theatre
of the mathematicians of the time. Let us say imaginary numbers or complex numbers I'll say imaginary.
Mathematicians were perturbed but not that perturbed. And the first recorded instance
of clear perturbation that I know is around 1545, Cardano, who had to use a certain imaginary number
in fact he had to use the numbers five plus the square root of minus 15
and five minus the square root of minus 15
in order to solve his equation or to deal with this issue that he writes: "dismissing mental tortures"
and multiplying (5 plus the square root of minus 15)
times (5 minus the square root of -15)
you get, and he tells you how to do it, well the 5 times 5 is 25
the square root of minus 15 times the five cancels the minus square root of minus 15 times the 5
The 4th term in these crossed terms that you have to consider is the square root of minus fifteen times
minus the square root of minus 15
which if you work it out is 15 and so you get 25 plus fifteen
equals 40 and he writes "multiplying five plus the square root of minus 15 by five
minus the square root of minus 15 we obtain
25 minus minus
15 and therefore the product is 40 and thus far does
arithmetical subtlety go of which
this the extreme
is as I have said, so subtle
that it is useless"
he's saying that he hasn't yet
gotten into his mathematical repertoire
this strange creature the square root of minus 15
or complex numbers in general and yet
certainly has an intense understanding
that we had better
begin to deal with it despite the fact that it may be subtle
and useless
it's the first instance where someone complains bitterly
and yet uses such a number Brady: does that still happen now do you think or
Barry Mazur: do we complain bitterly about things that we're on the verge of understanding but not quite?
You know, I'm about to say
yes it doesn't matter what blob
they are as long as they're similar
councillor says 'yes' 'what do you call that?' says the king
'oh I call that the blah Pythagorean theorem'