Hello everyone, I'm Norman Wildberger. Today
we're carrying on with our sociology and pure mathematics series. We're moving into the 19th
century. We're looking at the history of geometry, and this is a really critical epoch in terms
of the sociology of pure mathematics because pure mathematics as a really sort of identifiably
separate discipline from applied mathematics raises its head, I think, for the first
time somewhere in the 19th century. Now tracking exactly when this occurred
is a very subtle and delicate business, and there's of course been aspects of pure
mathematics throughout the history. But I think it's pretty safe to say that for
the majority of the history of mathematics, the focus has been very much in the applied
direction, even though often applied mathematicians were interested in the theoretical
aspects of their investigations. But the idea that there was a completely separate kind of discipline
from the real world, the real everyday world of science and observable phenomena, that
was this world of pure mathematics, and that it was not necessarily directly in step
with science or with observational reality, that I think first emerged in the 19th century. And
I think we can trace some of its origins to this very big broadening of understanding with respect
to geometry that occurred during this time. So I want to tell you about this because it's
essential for a sociological person interested in this story to appreciate this huge flowering
of geometry that took place in the 19th century. It was a golden time for the subject, and
one of the really interesting, if somewhat disappointing, aspects of the subject
is that in the 20th century, this took a nosedive and the importance of geometry much
diminished, even though it was still around, but not in the glory that it was in the 19th century.
So trying to understand why that happened, what was responsible, what were the forces at play,
what were the reasons, this is something that there'll probably be a wide variety of opinions
in the mathematical community, and I think it's safe to say that the entire question really
needs somebody with some sociological training, some historical training perhaps, to investigate
this. It's not purely a mathematical issue. So to prepare you for that today, we're
going to talk about a number of different aspects of 19th-century mathematics,
concentrating on geometry. But of course, I have to keep in mind that you
are maybe math-phobic. Okay, you're a sociologist. You haven't had extensive training in mathematics.
You may never have even taken a first-year course at the university level in mathematics. You may
have dropped out of mathematics in high school. So I have to attempt to lay out what's going
on here in a very general bird's-eye view. Okay, so I'm going to simplify things,
I'm going to leave out lots of things, but I'm hoping that my explanation
will still be meaningful to you. Okay, so let's jump in now. Let's start. There
are lots of different geometries involved, and let's start with inversive or Mobius
geometry, okay, named after a German mathematician Mobius. And this is a geometry where circles
play the dominant role. So we're talking about the plane, and we're thinking about circles. We're
especially interested in circles, and there's a particular kind of operation that you can do with
a circle called inversion. So I have to explain what that is. So here's a circle, okay, its name
is C. I'm going to explain to you inversion, say denoted by I sub C, in circle C. What this
is is a way of starting with a point somewhere in the plane and associating to it another point, say
B, in a plane so that this is a sort of a reverse operation. If you apply this inversion to A, you
get B. If you apply the inversion to B, you get A. And this inversion has a lot of really beautiful
properties, and it's very important in lots of things. So how do I define this? Well, there's
a number of ways. I'm going to choose this way, but here's the point A. Suppose that
it's outside the circle, okay, and here is the Apollonius dual of that point, sometimes
called the polar. And I remind you that's one obtained one way to think about it is to look
at the tangents from the point A to the circle. Okay, the tangents meet the circle at these two
points, that determines this dual or polar line. Now, the point B that we're interested in is
on this polar line, and it's the meet of this polar line with the line from the center of the
circle to A. All right, so that's how you get B from A. Now, that also works the other way around.
If you started with B and you took the polar line of B with respect to the circle, which is a
projective idea which makes sense, then you would intersect with that line again and you would
get A. So this is a really important notion which has remarkable properties. The most remarkable
perhaps immediately is the property that it takes circles/slash lines to circles/slash
lines. Let me try to illustrate that. So, the inversion of this point A is this point
B. Suppose that I take a little circle over here that happens to pass through A, and I look at
the inversion of a number of points on A. So, you know, I populate this with a bunch of points,
and I calculate the inverse of each one of these points. And I'm going to get, you know, like
for this one here, I have to find the polar, then I have to join the polar with the line from
the center to there, so we get some other point. It turns out that what I'm going
to get is another circle, okay? And the circle's a smaller circle, which was sort
of like this. Okay, so every point on this circle, its inverse will be a point on this circle, and
conversely, the image of that circle will be this. Now, there's another sort of case that needs to
be looked at. If you have a line, so for example, if I drew a line through a point A, okay,
some line like this, and I looked at a lot of points on the line and calculated their
inverses in this sense with respect to C, then what I'm going to get is another circle. And it's a circle that, well, it goes through
this point B, but it also goes through the origin O, okay, and so, it will
probably be something like this. So, it's not entirely true that the image of
circles are always circles because the image of this circle passing through the center is
not in fact another circle but rather a line. So, for the purposes of this Mobius geometry,
you have to think of lines as sort of being really big circles, okay? You could think of them
maybe as circles whose center is really far away, maybe at infinity in some sense. And in fact,
you actually do need to think about infinity because if you ask, well, what's the inverse
of the center itself, there is no point in the plane which plays the role of the inverse of the
center, and that means that you have to actually augment this plane with a distant point, a
new point which we call the point at infinity. So, it's a very rich geometry, and okay,
it's an example of a geometry where the main objects are not points and lines as they
usually are, but rather these circles. So, geometry of circles which was discovered
in the first half of the 19th century. Now, it turns out this inversive geometry of Mobius is
somehow intimately connected also with a sphere. And on a sphere, here's an example of a sphere in
three dimensions. There's also a notion of circle because if you intersect this sphere with a
plane, then you're going to get a circle on the surface of the sphere. In the case when the
plane is passing through the center of the sphere, you're going to get a great circle, the biggest
kind of circle which is like an equator, okay, like the equator is an example of a great
circle. So, these lines of longitude. Now, there is a relationship between the sphere and
the plane which is a kind of mapping, a way of mapping the sphere or most of the sphere
onto a plane called stereographic projection. And what that is, it's represented by this
thing here. So, here's the sphere, okay, imagine an equatorial plane, so imagine a plane
which is passing through the equator. Okay, and so, we're going to define a map from points
on the surface of the sphere to points in this equatorial plane, possibly outside, possibly
inside the sphere. So, how do we do that? Well, we start with the north pole, there it is right
up there, okay, north pole, we designate that as a special point. Now, if we have any point P on
the sphere, we join the north pole to this point P with a straight line. Okay, so imagine a straight
line passing through the north pole and the point P that comes out perhaps of the sphere, and we'll
meet this equatorial plane in a particular point, say in this case R. Okay, so there's that
line, and it sort of goes through the sphere, it comes out and meets this equatorial plane
at this point R that then gives us a map from points on the sphere to points in the plane. If
P is in the northern hemisphere like this one, then you're going to get a point outside the
sphere. However, if P is down here somewhere in the southern hemisphere, then the line
through the north pole and this point will meet this equatorial plane in a point inside the
sphere. There's only really one point that doesn't have a good image which is the north pole
itself, okay, and that's sort of, of course, if you actually looked at the image of a point as
it moves towards the north pole, the corresponding image in the equatorial plane will go further
and further away, and so it's usual to sort of think of the north pole as being represented by
a point at infinity in this equatorial plane. So that fits really nicely with what we were
just talking about with this inverse of plane and Mobius, and in fact, there's an intimate
connection between the stereographic projection and the inversive geometry of Mobius.
Okay, so you can see that by analyzing what happens when you look at not just a
point but actually a circle on the sphere. Okay, so you suppose you take some circle, so
you just, you slice the sphere with the plane and then you're getting a circle, and then you look at
the image of that circle under this stereographic projection. So, for every point or for lots of
points on the circle, you compute their images on the equatorial plane and you plot them, and lo
and behold, you find that you're getting a perfect circle. The image of a circle on the surface
here is a circle in the plane, unless the circle that you've chosen happens to go through the
north pole. In that case, the image is a line. So, stereographic projection interchanges
circles on a sphere with circles on an equatorial plane where we sort of enlarge our view
of circles in the plane to include these lines. And then it turns out that inversion,
inversion in a fixed circle in the plane, has a really nice correspondence in the terms of
the geometry of this sphere. Okay, so for example, suppose you have some circle in the plane and
you're interested in inversion in this circle, and let's say that under stereographic projection, this circle in the plane corresponds
to a little green circle on the sphere. So, this little green circle on the sphere,
okay, if you imagine a tangent to the sphere, if you draw all the tangents to the all the points
on this little circle that you've got, okay, they will all meet at a point outside here somewhere.
Okay, so another way of thinking about that is if you kind of imagine like an ice cream cone
with an apex outside and enveloping the sphere tangent to the sphere, then that associates to
the original circle a point, and it's that point that's out here somewhere that is responsible for
the inversion at the level of the sphere. How so? We take this point, I haven't drawn it, it would
be out here somewhere in this example, and what we do is we use that point to interchange a point
on the sphere with the other point on the sphere which is obtained by taking the line from this
distinguished point, drawing it through the point on sphere, and out the other side. Okay, so once
we have a fixed point outside the sphere, then by drawing lines through that fixed point, that gives
us a way of interchanging points on the sphere, and because the circle that we started
with, are the points on it are obtained by looking at tangents, those points
are going to be fixed under this sort of reflection, and so the net effect is that
this sort of reflection in this external point ends up being the spherical version of inversion
in the corresponding circle in the plane. Okay, that's a whole mouthful, a
whole theory there which I've just sort of given you in a few minutes. I don't
want you necessarily to understand all of it. I just want you to appreciate that there is some
really interesting deep connection between circles on a sphere and circles in the plane
given by stereographic projection, and that this correspondence allows us to think
about inversion in the plane in another way using sort of our three-dimensional space and
points external to the sphere, sort of reflecting points in the sphere with respect to some external
point. So, it's a rich kind of theory which is quite different from Euclidean geometry, but
it's a rich consistent geometry all on its own. The next important topic that I want
to talk a little bit about is the re-emergence of projective geometry in the 19th
century. So, projective geometry in modern times has its origin with the work of Desarg, which I've
already told you about, and Pascal also made an important contribution, but the subject was more
or less forgotten about for a couple of hundred years. And then in the 19th century, a number
of people started thinking about this again, and they had a crucial insight into the nature,
the essential nature of projective geometry, and this also Mobius was involved
with this also Plucker, and in fact, there were a whole bunch of 19th-century
mathematicians very interested in projective geometry. So, I want to explain that a little
bit because it's also quite interesting in that a notion of infinity arises
here which is quite interesting, so kind of a geometrical infinity, not a
numerical infinity exactly, but a geometrical one. So here is an ordinary line with some
usual kind of affine coordinates, okay, and the idea here is that we're
going to look at this one-dimensional object from a two-dimensional point of
view, okay, so this turns out to allow us a bigger picture, and the way we're going
to set this up is we're going to think about embedding this picture in a two-dimensional plane
whose origin is right here, okay, so I'm going to make, so this is like the real x-axis, this is
the real y-axis, okay, and here is the origin of this two-dimensional plane. Now, I want
you to observe that points on this line here are in correspondence with lines through the
origin, okay, so there's a point on our original number line and to this point, I can associate
the line through the origin, and that point. Okay, so every point on this line gives
us a unique line through the origin. Okay, so choose one over here, there's another point
that will have this line associated to it. So different points get associated to different
lines. So do we get every line through the origin this way? What happens, for example, if we start
going in this direction and we start looking at the lines through the origin that are determined
by some variable point moving this direction? Well, the corresponding lines are going to
become sort of more flatter and flatter as we go further and further out, and after we're
a million miles out, the line joining, you know, the point there will be to us indistinguishable
almost from this horizontal line with the x-axis. However, we don't actually ever get
exactly to the x-axis line because this line is parallel to that one, so they don't
actually meet. But any little adjustment to it and we do get a line which meets the given line.
That means that this x-axis itself is somehow a distinguished line in this picture, okay? That
line there, okay, has a sort of a different role and we might end up calling it like infinity.
So the projective geometry is realized that you could make a model, okay, a model of
projective geometry by looking at lines in a space of one higher dimension. So the
projective line can be realized as essentially one-dimensional subspaces or lines
to the origin in this two-dimensional space. And when we do that, then the point at infinity
ends up becoming completely unmysterious. Previously had been mysterious. What
do we mean by points at infinity? From this point of view, the point at infinity in
the projective line is just that particular line through the origin which happens to be horizontal.
There's nothing magical about it at all. And moreover, this gives us a
way of describing things. So this line here, okay, it's determined by that
point, and that point has coordinates. What? Well, maybe this point has coordinates. Maybe
that's a little bit more than a half. Um, maybe it's a little more than half. Maybe 11
fifths. That's a little bit more than half, right? It's a little bit more than one and a
half. Also, one and half, okay? Maybe 11 and seven's. How's that? That's a bit more
than that's a bit more than one and a half. Okay, so maybe that's the coordinates 11 over seven, and
the y-coordinate will be zero. I will be one. The y-coordinate will be one now. So this line here
is determined by this point that lies on it, but in fact, it's really determined up to scale
by the proportion between these two things. So what we can end up doing, and this is what
the 19th-century geometers did, is they said, "Okay, what's really involved in describing
this line is the proportion 11 over 7 to 1." It's the proportion. And a pleasant thing about
proportions is that they're unchanged if you scale the two entries by the same factor. So in
particular, we could clear denominators and say, "Well, this is the same as
the proportion 11 to 7," moving us from fractional arithmetic to integer arithmetic.
So instead of fractional arithmetic with single numbers, we have sort of an integer arithmetic
of these proportions involving a pair of numbers. And the beautiful thing about this is that
this new line here, this horizontal line which corresponds to some point at infinity
which is hard to express in the original system, can be expressed in this
homogeneous coordinate system by looking at the point on here. Okay, it's going to be so that
that point there is in the point one comma zero. Okay, and so this line is represented
by the proportion one to zero. And that then, so that's really like
infinity in this projective geometry. So this is a way of completely demystifying
infinity in this geometrical sense. Instead of replacing it with something that's arbitrarily big
or something like that, we just replace it with this really simple proportion, the proportion
one to zero, which arguably is just as simple, maybe simpler than this one. So any
proportion between two non-zero things, as long as they're both nonzero, represents a
line, and that's called homogeneous coordinates. So this was a great advance because
it allowed geometers to start understanding projective geometry arithmetically,
indeed algebraically. And this became such a powerful tool that the 19th-century geometers
were able to start incorporating more and more of classical geometry into this somewhat
bigger framework where you allow yourself points at infinity. This is just
the one-dimensional situation. It gets more interesting when you go to higher
dimensions. In two dimensions, you have to sort of add a whole circle of points at infinity,
but it's all the same. It's just done with these homogeneous coordinates very simply,
and so algebraically becomes actually quite easy to manage and enlarges the subject
dramatically because it turns out that these points at infinity actually seem to have
often important geometrical significance. Okay, so this subject with this new point
of view, okay, initiated by Möbius also and Plücker and others, this new point of view was
so powerful that, in the middle of the century, Arthur Cayley, famous British mathematician,
claimed that all geometry is really descriptive or projective geometry. That's sort of another name
for it. That's probably a bit of an overstatement, but you know, there was definitely this view
that, okay, this is like the biggest geometry, that this geometry includes all others. It's a
very interesting, um, powerful point of view. The next very important chapter in the story
of 19th-century geometry is the emergence of non-Euclidean geometry, and this is a very rich
story. I do suggest that you have a look at my history of math lecture. I have an entire one-hour
lecture on this where I go into a lot more detail, and I think that's a pretty decent lecture.
So I'm just giving you a bird's-eye view, but you have to appreciate that this
was really a monumental advance and it really shook the foundation of mathematicians
thinking with regard to geometry. Suddenly, they're loomed in front of them or off to the
side this alternate universe which was somehow parallel to the one they were familiar with from
Euclid. Everybody was firmly entrenched in the logical structure and development of Euclid, but
now suddenly there was this new kind of geometry which on the face of it was very unfamiliar but
still obviously very rich, so it was kind of an alternative to the standard genre. Now there
was the question, okay, well, we have these different geometries around, which is the actual
true one? Does that question even make sense? So the story here is always associated
with Lobachevsky, Bolyai, and Gauss—three mathematicians, Russian, Hungarian, Gauss
German, of course. And Lobachevsky actually published this first and Bolyai independently, and
then after the fact, Gauss pointed out that he had made a lot of investigations on this much earlier,
okay, but had left off publishing it because he didn't want to shock people too much. So very
roughly, okay, in a couple of minutes here is Euclidean geometry, and in Euclid's development,
there are these five postulates, sometimes called axioms, these are sort of things that Euclid
starts his work with, things that we're going to assume because they're kind of obvious, okay, and
four out of the five really are kind of obvious, sort of at least if you're drawing pictures, but
the fifth one is more challenging. The fifth one says that if you have a line and you have a
point which is not on the line, then there is exactly one, there's one and only one line through
that point which is parallel to the given line. And what does that mean? It means that it doesn't
meet, no matter how far you go, these lines never meet. So there's exactly one such parallel
line through a point outside a given line. So for a long time, this was more complicated than
the other postulates, and mathematicians tried to prove it independently, and eventually these
three gentlemen started to realize that this did not follow logically from the others. In fact,
you could create an alternate geometry in which it was not true, and that's hyperbolic geometry. The
starting point that they had that you can imagine a geometry in which there's here's a line, here's
a point on the line where there's two different lines, maybe even more, but at least two different
lines which are passing through this point which are parallel to this line in the sense that they
don't meet that line even if you extend the lines. So, this is really a non-intuitive, but from my
point of view, okay, this whole story is, in some sense, a manifestation of the over-reliance on
Euclid. People were just too fixated on Euclid that they couldn't see the forest for the trees.
In fact, okay, the reality is that non-Euclidean geometry had been well studied for centuries, if
not millennia before this, okay, and that was in the context of spherical geometry, in the context
of astronomy, at the heavens, and in particular, the Islamic mathematicians' work on the geometry
of the sphere. Also, Indian mathematicians made contributions there too, and in fact, so did
the ancient Greeks, okay, so the geometry of the sphere also is a non-Euclidean geometry,
but here you have to think of the lines as being these great circles that I was telling you about,
the things that you get when you slice a sphere with a plane through the center. Now, if
you have some point which is on the sphere, not on this plane, there are zero lines that pass
through this point and are parallel to this given line. You take any plane, which, any great
circle which passes through the center and this point here, there's a whole family of
them, but any one of them will give you a line. In other words, a great circle, which will
necessarily meet the equator that you're talking about. So, this is a non-Euclidean geometry, but
for sort of a different reason than this one is. Here we have more than one line which is parallel,
here we have no lines which are parallel, but nevertheless, this is a really valid, it ought
to be just as valid a non-Euclidean geometry as this one. In fact, ultimately, it was seen that
these two geometries are sort of on different sides of Euclidean geometry, sort of somehow fits
in between them, so from my point of view, there was a lot of confusion in the 19th century, so if
they could listen to this lecture, you know, like 200 years ago, it would
have been hugely influential because they just didn't think about
things in the right way. They couldn't see that this spherical or elliptic geometry that
was really completely in front of their eyes was actually a valid example of non-Euclidean
geometry, and it turned out that, in fact, these two geometries, the theorems in them are
very parallel, but that required the emergence of another figure, another key figure in
the story, so this was Eugenio Beltrami, Italian mathematician, who quite some time
after Lobachevsky and Bolyai did their work, roughly in the 1830s, so he came up with these
models of hyperbolic geometry, maybe 1870s, 1880s, and he was really the first person to logically
put this geometry on the map in the sense of providing actually a completely clear model for
what these geometries were really about. So, in short, somewhat simplified, here is what
he realized that you could think about both the elliptic geometry or spherical, almost
the same thing, and the hyperbolic geometry in very parallel ways by going to three
dimensions. So, we're talking about two-dimensional geometry, but we're thinking
about it being embedded in three dimensions. We've already seen this is a key sort of idea to
try to understand things in certain dimensions by looking at them from a larger point of view.
We saw that already with inversive geometry, something similar here. So, the elliptic
geometry, which is the geometry of the surface of the sphere, is naturally studied by thinking
of the sphere as embedded in three-dimensional linear space, given by an equation x squared
plus y squared plus z squared equals one. That's the equation of a sphere in three dimensions.
Okay, and he realized that correspondingly, the hyperbolic geometry can also be viewed in a very
parallel way, just by making a single change of sign by replacing this plus sign with a minus sign
and looking at the what we call the quadratic form x squared plus y squared minus z squared. It's
a little bit more complicated when you set this equal to zero over here, you just get the origin,
but over here you get a cone, more complicated, and when you set this equal to one, you get
what's called a hyperboloid of revolution, a surface which has one branch which is like
a hyperbola sort of rotated around and here's the second branch on the other side rotated
around, and he realized that what Lobachevski, Bolyai, and Gauss were really doing could
be described as geometry on the surface of one of these branches of this hyperboloid,
and what you do is you do really the same thing as you do over here for the sphere to get lines,
straight lines, what you do is you choose some plane that passes through the origin, and then
you intersect the passes through the origin, and then you intersect the sphere with that plane, and
you get a great circle. Let me draw it like this. Okay, so that's an example of like a straight
line in the spherical or elliptic geometry. And over here you do very much the same thing: you
take a plane that passes through the origin, okay, and you intersect it with this hyperboloid, okay,
and you're getting okay, something will, this thing carries on, so it goes up like this, you
get essentially what looks to us as a hyperbola. And when you set this thing up in this
way, then the parallels between elliptic geometry and hyperbolic geometry completely
demystifies the subject to a certain extent, okay, and a lot of things become much
clearer. Unfortunately, this is not even taught in undergraduate courses very much, it's
very unfortunate, and in fact, the whole story is, you know, sort of extended massively by
what I call universal hyperbolic geometry, which is my own take on things, which says that
actually what you really should be doing here, okay, is looking at this whole thing, but
not just the portion inside this cone, but also the portion outside the cone, which maybe
I can try to draw it here. The portion outside the cone is also a hyperboloid of revolution but
looks something like this. So here's a typical um, sort of orbit when you place the one say with a
minus one you get a surface that looks like this which is found in architecture quite a lot,
okay. And in fact, so when you're studying universal hyperbolic geometry you learn that
the same formulas once you set them right in sort of the rational trigonometry format,
um, work, um, both for the geometry of this two-sheeted hyperbola as well as the geometry
of the one-sheeted hyperblend and there's a unification that goes on, okay, that you don't
really just want to be studying one part of it. So Beltrami's understanding is a key point
in helping us move to this much larger, more beautiful, and more general point of
view which is universal hyperbolic geometry. Okay, but as you can see that we're expanding
our our vistas, uh, considerably here. I just want you to get an overview of this, uh,
this sense that from a sociological point of view the 19th century people were carrying this
big burden of Euclid on their back and that somehow um, constrained their thinking to
go in certain directions and not in others. So there's definitely a very
interesting sociological aspect of this. So I'm just part of the way
through the 19th century story here but I'm going to finish the lecture with this, uh, other
development that's really important which is the introduction of complex numbers into geometry
which transformed a lot of the subjects. So complex geometry we can think of this as
sort of getting two-dimensional arithmetic for the price of one dimension, okay,
so it's a really interesting thing that we're enlarging the arithmetic, okay, we're
enlarging the arithmetic and that ends up enlarging the geometry that's built up from that
arithmetic from a from a Cartesian point of view. So we're following Descartes and Fermat
from looking at things in the Cartesian point of view in terms of coordinates then
we realize okay so if we extend our number system from ordinary numbers to so-called complex
numbers then we can extend a lot of geometrical objects which are constructed from these from
these coordinates also. So here is the usual number line and it's best that you think about the
rational number line so 0, 1, 2 and then you can also you know subdivide so there's 17 over 11,
there's minus seven-thirds, there's one-half so rational number is more or less represented here,
okay, and over here is the decimal number line which is a number system used by engineers and
physicists and you know sort of ordinary people which involves subdividing by 10. So there's
zero, there's one, then we subdivide units of 10 and then we make further subdivisions, subdivide
each one of these into 10 and so on, so we end up expressing um, numbers by decimals and
the more decimals there are sort of the more accuracy we're talking about but there's an
approximate aspect of this so a scientist thinks of 0.4 as not being an exact point but
actually sort of a range of values, okay, something is measured and it's 0.4 it means
that you know there's there's some indeterminacy it means that the next decimal point is not
entirely clear that there's some variability there. Well this number here is representing
a greater degree of accuracy, okay, but the engineer never has complete accuracy so you
know infinite decimal is not part of this story. So I want you to appreciate this. Of course, the
real number system is some kind of attempt to, uh, to take this thing, sort of drag it into the exact
world, so to pretend that we have an arithmetic of infinite decimals that extends this one but sort
of, you know, has the exactness of the arithmetic involved here. And as I've pointed out many
times, that's a very, very much wishful thinking. Okay, so what's the complex number story? So
the complex number story is the realization that these pictures can be extended from their
one-dimensional aspects to a two-dimensional aspect by introducing a new number called i whose
square is -1. And here is the usual story. So now here, there's the usual number line much like
this one in terms of fractions and now here's a new axis which notably has i on it instead of
one, and that allows us then to describe points in this plane in terms of an ordinary component,
a fraction in the x direction or the horizontal direction, and a multiple of i in this case one
half i, in the i direction. Okay, so that's what a complex number is and often denoted by z.
Okay, so that's a complex number four thirds plus one half i, and to do arithmetic with these
things, well, basically you use this crucial effect when you add them and you sort of add them
in the obvious way, but when you multiply them, you have to use that the fact that i squared is
always equal to minus 1. Now this is sort of an exact system, so I might call this the complex
rational numbers, but there's also a kind of sort of engineering or scientific version where we
work with approximate decimal quantities. So this point here, which is no longer quite as precise
as over here, minus 0.9 plus 0.4i, representing some kind of, you know, rough position, we could
call these the complex decimal numbers. Okay, so it turns out somewhat miraculously that the
arithmetic of these complex numbers is really, in first of all, very interesting
mathematically but also, uh, ends up having huge implications for 20th-century physics. So when
people start doing quantum mechanics in the 1920s, they start to realize that the essential aspect
of the workings of quantum mechanics somehow almost requires us to frame things in terms of
complex numbers. This is a deep mystery that I think we still don't really understand why is
this, you know, okay, it's pretty interesting but it's telling us that these complex numbers
are not just a figment of the mathematician's imagination, they do somehow connect very
directly with the real world or as much of that as represented by quantum mechanics; it
has a little bit of an unreal aspect sometimes. All right, so now that we have the possibility
of this two-dimensional arithmetic with complex numbers, we can look at complex geometries which
are roughly obtained by complexifying ordinary rational geometries. So what we do is we take
some classical geometry like the kinds I've been telling you about and we think about replacing
the underlying numbers which are involved with complex numbers; it's called complexification, and
we get a rich theory which is unfortunately much harder to visualize, so we lose contact a little
bit with direct physical visualization. So it is challenging and it's sort of beyond our
ordinary experience in lots of ways. So this is challenging even for undergraduates;
a lot of undergraduates will only sort of dimly be aware of some of these things, so these
are quite advanced ideas in some sense but I just want to give you a flavor of
it. Okay, so let's look at an example of complexification by taking a very simple
kind of geometrical object: the circle. Okay, the unit circle with the equation x squared plus
y squared equals one in the usual x-y plane, there it is there, and here is a point on that circle
four-fifths comma three-fifths. If you square this, you square this and you add them, you're
gonna get one, that means this point actually is on the circle, and it's four-fifths along here
and three-fifths along here, right? So we can, we can represent this point really
by two points on the axis, that point on the axis which is four-fifths, and this
point on this axis which is three-fifths. Now, more generally, if you wanted to
describe more general points on the circle, here is a really nice way of doing it. So, this is
an algebraic parametrization of the unit circle. We have these two expressions
involving some parameter t, and if you plug in any value of t in here,
then you're going to get a point on the circle, and all points on the circle are of
this form except for this one here. You might, as an exercise, try to figure out what
the t value is that gives you this point. As a clue, I'll tell you what it is geometrically:
if you join this point here and look at that point there, its coordinate is t. Okay, you might
like to see if you can figure out what t is. Okay, so now let's think about the question: how could
we complexify this story? So, it turns out that there's a number of different ways potentially
of doing this, but the simplest is probably this: we just take this equation x squared plus y
squared equals one involving sort of ordinary numbers. Ordinary for us means rational
numbers; that's sort of the simplest kind, and we replace the rational numbers with complex
rational numbers. So, instead of x and y, let's use z and w, so we get z squared plus w
squared equals one, but now z is a complex number, and there's a copy of the complex plane. It's
really the complex line, but to us it looks like a plane. This is a direct analog in the
complex world of this x-axis over here, and here is the w plane, which is a direct analog of this
y-axis. Now, you might say, "Well, why don't you make it sort of perpendicular?" I could try to do
that, but then it becomes harder, in some sense, to visualize because it's hard for us to try
to understand four dimensions. Okay, we can't: two dimensions this way, two dimensions this way.
What does that look like? So, I'm going to keep them separate. It's a little bit clearer, and I'm
going to ask: okay, so what does this equation represent in this complex world?
It's a pretty interesting question. If you're an undergraduate, you might like
to think about this. So, as an example, let's see if we can find one point on this sort
of complex circle. So, how are we gonna do that? Well, so we want some analog of four-fifths comma
three-fifths. So, what we could do is we could say, "Why don't we just use this same formula?
This formula gives us points on this circle, but we have to put in rational numbers t over
here." Okay, why don't we use the same formula and just plug in a complex number for t, and
hopefully we'll get z's and w's? So, I'm going to illustrate some complex number arithmetic for you.
You can check this if you have some familiarity with it; if you don't, don't worry, it's just
relatively simple, sort of high school kind of fiddling around with some complex numbers,
basically always using i squared equals minus 1. Okay, so what I'm proposing to do is to take this
formula, call it e of t, and to plug in 1 plus i. Okay, so I'm going to get two numbers
which are going to be complex numbers. Okay, so what I have to do... well, I
take 1 minus 1 plus i squared over 1 plus 1 plus i squared and over here, 2 times
1 plus i over 1 plus one plus i squared. Now, when you square something like
this, like you square one plus i, you get the square of the first term which is one
plus the square of the second term. The square of i is minus one, so you get one plus minus one plus
twice the product, and twice the product is 2i. So now the 1 and the minus 1 cancel, so all you
get when you square this is 2i. So this becomes: the numerator becomes 1 minus 2i, and the
denominator becomes 1 plus 2i. Over here, we have 2 times 1 plus i, that hasn't changed
in the numerator, and this is again 1 plus 2i. Okay, so how do we simplify that? So, in
complex number world, what you do is you say, "Oh, it's a quotient: one complex number divided
by the divisor." So, we have to multiply top and bottom by the complex conjugate of the bottom, the
denominator. So, the complex conjugate of 1 plus 2i is 1 minus 2i. When you multiply 1 plus 2i by
1 minus 2i, you get the sum of the squares: 1 plus 4, which is 5. And when you multiply this by
1 minus 2i, well, it's like squaring this; you're going to get the square of the first
term, which is 1, the square of the second term, which is four, i squared, or minus four, plus twice the product, which is minus four i. So,
all together, you can get minus three-fifths minus four-fifths i. And you do the same kind of thing
over here; you get six-fifths minus two-fifths i. So, we're getting two complex numbers.
Let's picture them. Okay, so here's this first one. Is that... so it's sitting somewhere
in the z plane. Okay, there's one minus one i, minus i. So, where is minus three-fifths? So we
have to kind of go minus three-fifths over there and minus four-fifths i down here somewhere,
right? So, it will be... okay, there. Okay, that point represents that number. And what
about this one? Well, this is in the w plane, so we have to draw it up here somewhere.
The first coordinate is six-fifths, so that's about over there somewhere, minus
two-fifths around there. So, the number is there. So, these two things here that I've drawn
sort of correspond to these two things on the corresponding axes. Okay, so the
point that we're getting in this complex circle is the point which is... well, this is
it. This is the point. And we can visualize it as a pair of complex numbers, the pair
consisting of that point and that point. So, that's one point on this complex circle.
Okay, what does the rest of it look like? Well, that's an interesting question. And if you had
some mathematical experience and you don't already know, you might like to play around with this and
try to get a sense of what this circle looks like. But one thing that's certainly going to be
clear is that this is going to be essentially, from our point of view, a two-dimensional object.
The circle is one-dimensional, that's true. But here we're talking about something which is given
by a one-dimensional parameter, but the parameter is allowed to range over the complex numbers,
which for us is a two-dimensional kind of range. So when we look at this, somehow we're seeing some
kind of surface in this four-dimensional space. And that sounds complicated, but
it's also interesting because technically it should be like a circle. So
we might ask, like, of all those hundreds of theorems of circle geometry that are known, do
any of them apply to such a circle? What happens? And once you start thinking in this way, then
all kinds of new things open up because you can take all these classical curves that people
have been studying for hundreds of years, and you can start looking at them
from a complex point of view. So in the 19th century, we had these
sort of two great, you know, and large number of great enlargements, but in
particular in terms of the study of curves, we had this point of view of looking at the
projective aspects, including points at infinity, and we had this idea of extending our number
system to complex numbers. Those two things really created all kinds of new developments in geometry,
projective geometry, algebraic geometry. Okay, so I've covered a huge amount of territory here. I
hope you don't worry if you haven't understood it all. I want you to get a sense of it, okay? And
there's more to come in our next video because we're not finished with the 19th century.
I'm Norman Wildberger. Thanks for listening.
