Complex Numbers and Addition Formulas
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Circle geometry and related formulas in calculus are closely connected to the algebra of complex numbers. In particular the all-important rational parametrization of the unit circle has a beautiful interpretation in terms of a quadrance normalization of a square, which gives the natural association between lines through the origin and points on the circle. In this video we review the usual story of complex numbers with an eye towards extending it to the relativistic, or red, setting involving the parametrization of the hyperbola with equation x2-y2=1. In both cases we get addition formulas for the corresponding signed areas associated to segments of these curves. This topic links naturally to Chromogeometry, a remarkable three-fold symmetry in planar geometry which brings Euclidean (blue) and relativistic geometries together. The third geometry here is the green one, and it will figure prominently in the introduction to "logarithms" and the "exponential function". If you value this content, please consider becoming a supporter via Patreon at https://www.patreon.com/njwildberger. This helps us in bringing consistently high quality content to our channel, and also enables you to join the Algebraic Calculus One online course!
00:01
[Music]
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hello everyone I'm Norman wahlburger and
00:15
welcome to the algebraic calculus in
00:18
today's video we're going to be looking
00:19
at complex numbers actually of two
00:22
different kinds and addition formulas
00:25
relating both to the circle and to the
00:29
geometry of the relativistic hyperbola
00:31
so complex numbers are of course an
00:34
important algebraic domain but from a
00:38
calculus point of view the complex
00:41
numbers are really just one possible
00:44
extension of arithmetic that can be
00:47
geometrically useful so we want to put
00:51
things on a more equal footing by having
00:54
the relativistic or hyperbolic analogues
00:58
of complex numbers getting your closer
01:01
to being on the same level with complex
01:03
numbers okay so circle geometry
01:06
typically the unit circle x squared plus
01:08
y squared equals one is intimately
01:10
connected with the algebra of complex
01:14
numbers which is very closely connected
01:17
with addition formulas for circular
01:20
signed areas and notably we're gonna be
01:23
talking about this as we like to do
01:25
without any notion of angle or in fact
01:28
any transcendental quantities that's the
01:30
challenge so that where we're thinking
01:33
about things purely in terms of a poly
01:35
number or maybe quotient polynomial now
01:39
in analogy to this there's a
01:41
relativistic story where we look at the
01:44
relativistic hyperbola with equation x
01:46
squared minus y squared well either
01:48
equal to one or even minus one and then
01:51
there's a corresponding algebraic
01:54
structure which we might call the red
01:55
complex numbers and we might call these
01:58
original complex numbers the blue
02:01
complex numbers okay so it turns out
02:03
that these red complex numbers then are
02:05
intimately connected with addition
02:07
formulas for hyperbolic sine Daria's now
02:13
happening here is that these two
02:15
geometries here one Euclidean one
02:18
relativistic they're actually just two
02:20
of a triple of geometries so that's all
02:25
explained in my theory of chromo
02:27
geometry which introduces this threefold
02:30
symmetry into a planar geometry where he
02:34
seeing a facet of that here but the
02:37
point is that there's a third geometry
02:39
floating around which will also figure
02:41
in our story but dotted today's video
02:43
later on shortly in fact it will be
02:47
coming even perhaps more important in
02:49
some sense than these two so let me
02:52
remind you very quickly the usual story
02:54
about complex numbers the blue complex
02:56
numbers all centering around this sort
02:59
of new object i whose property is that I
03:03
squared equals -1 and then what happens
03:06
is that a complex number is something of
03:08
the form a plus bi where a and B are for
03:11
us rational numbers and that's naturally
03:14
identified with the point a B in the
03:16
plane and so that's may be typically a
03:19
complex number said and the way you add
03:21
them is component wise like this and the
03:25
way you multiply them is essentially by
03:27
using this rule so that a plus bi times
03:30
c plus TI is AC minus BD plus ad plus BC
03:37
times I then one introduces the complex
03:42
conjugate Z bar so this is Z then Z bar
03:45
is a minus bi and then we can introduce
03:49
the quadrants of the complex number Z
03:51
which is Z times Z bar it's a difference
03:55
of squares and we get a squared plus B
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squared so notably the modulus which is
04:01
usually talked about in classical
04:04
courses the square root of a squared
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plus B squared is gonna play no no role
04:09
at all for us okay because square roots
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are highly problematic theoretically
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okay so the quadrants is the fundamental
04:16
concept and the actual main theorem in
04:19
the subject actually goes back in some
04:21
sense to diophantus who stated it not in
04:25
terms of complex numbers but doesn't
04:27
identity here which is responsible for
04:29
this crucial equation which is that the
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quadrants of the product of two complex
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numbers at times w is the product of the
04:37
quadrants of said times the quadrants of
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W and just to reiterate the modulus or
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the so-called argument of Z which is
04:45
some kind of angle play no role at least
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initially for us and setting up this
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theory
04:50
okay so we're strictly in a rational
04:54
setting here every every number this
04:57
appear here is a rational number now
05:01
there's an important modern way of
05:03
thinking about complex numbers which
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ties it more closely to linear algebra
05:08
and that is to identify the complex
05:10
number A plus bi with the matrix the
05:14
two-by-two matrix a B minus B a so in
05:20
this way the algebraic properties of the
05:24
complex numbers are just derived from
05:27
the usual algebraic properties of
05:30
matrices and crucially then the the
05:33
associative and distributive and
05:35
commutative laws of the addition and
05:38
multiplication are just consequences of
05:40
those laws for matrices so we save
05:43
ourselves a lot of work and then in
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addition this quadrants of a plus bi
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which was a squared plus B squared is
05:51
just the determinant of this thing and
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that makes the the theorem that I I
05:56
mentioned almost obvious because the
05:59
determinant of a product of two matrices
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is the product of the determinants so
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that's very natural so we have one
06:06
identification of complex numbers with
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two by two matrices then we have the
06:10
more traditional geometrical connection
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with simply pairs of numbers forming
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points which we can think of as being a
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fine points so this is powerful because
06:22
the complex numbers then connect with
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geometry intimately with Euclidean
06:27
geometry because the quadrants a plus bi
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equals one condition really amounts to a
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squared plus B squared equals one so
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we're really talking about Euclidean
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geometry where the unit circle figures
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prominent
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okay so this is a nice extra way of
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thinking about complex numbers that
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clarifies a lot of things for us and
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actually makes the transition to the
06:49
relativistic setting easier so a lot of
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the content of a first course in
06:56
calculus revolves around the geometry of
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the circle and also the geometry of a
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hyperbola and there's a particular
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diagram here that really ought to be
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standard material for undergraduate
07:12
students but is generally not really
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taught so it's very important it harks
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back to the rational parameterization of
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the unit circle which I'm always talking
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about crucial kind of understanding but
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it's a little bit more elaborate here so
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I'm gonna give you a little bit more
07:27
detail how to connect this rational
07:30
parameterization also with complex
07:33
number multiplication so this makes
07:35
clear the intimate connection between
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complex number algebra and this geometry
07:40
send it around the circle okay so here's
07:43
our unit circle x squared plus y squared
07:45
equals one and we're thinking about
07:48
parameterizing it and I guess you know
07:51
that the way we do this is we join the
07:54
point minus one to a point with
07:56
coordinates in the Y direction now in
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this case we're thinking about
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everything being in the complex plane so
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this actually is the number I T because
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we're thinking about this as being the
08:06
complex number I okay and then you join
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these two points and then you get this
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second point on the circle and that
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turns out to be 1 minus T squared over 1
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plus T squared plus I times 2t over 1
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plus T squared if we express it using
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complex number notation and let's call
08:26
this e of T so there is e of T right
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there now algebraically there's another
08:34
way of sort of understanding what's
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going on here if we take this line here
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and we look at a line parallel to it
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through the origin then have a look at
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where that line meets the line x equals
08:48
1 ok so then this triangle here will be
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congruent to this triangle here they're
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exactly the same so this is 1 and this
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is 1 this is T or I T so this will also
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be T or I T so this point here as a
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complex number is 1 plus I T that
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describes the points lying on this
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vertical line now if we take Z and we
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square it
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okay square it in the sense of complex
09:18
numbers what are we going to get we're
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gonna get 1 squared minus T squared plus
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I times 2t and when we normalize that
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point of actually geometrically that's
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that's going to be up here somewhere in
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this situation so there's said square
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it's another point on the plane if we
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normalize that dividing by quadrants of
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Z the quadrants of Zed is 1 plus T
09:45
squared so if we divide that then it
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turns out that we get exactly the point
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that we started with over here so that's
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another way of describing the rational
09:54
parameterization of the circle
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it's obtained by taking a point on this
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line in this case Z equals 1 plus I T
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and squaring it and then adjusting by
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the quadrants so crucially usually to
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get on the unit circle we we divide
10:12
something by its modules in in
10:15
traditional courses but the modulus is a
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transcendental object it's it's
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something that you can't get hold of
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because it involves the square root okay
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so that's why we're doing this we're
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squaring the thing first and then we can
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just divide by the quadrants to get
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something that's on the unit circle
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okay now geometrically one way of
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thinking what what that's happening
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there is that you're taking the point
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one and you're reflecting it in this
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this line and getting that point there
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this shows also by the way that actually
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it's really only this line that matters
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you don't actually have to take this
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point on the line if you applied the
10:50
same formula with any multiple of Z well
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then this ratio of Z squared over
10:55
quadrants of Z is not going to change so
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this point is really associated to this
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line so what happens is that if we as we
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modify the line the core
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responding point moves around twice as
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fast
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okay now the rational parametrization of
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the unit circle that's what we have here
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and now we can observe that if we have
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two points on this vertical line say 1
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plus I T and 1 plus I you if we multiply
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them will get 1 minus 2 u plus I T Plus
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u and then if we think about that what
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that tells us because of this formula
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here is that a of T times e of U will be
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e of this thing ok but it's really this
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thing normalized so it has the same form
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as this so in order to to normalize it
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so that it's on this line here we have
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to divide by this first term so if we
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scale this by dividing by 1 minus tu we
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get 1 plus PI T plus u over 1 minus tu
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that's telling us that the
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multiplication of points on the circle
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can be described in these coordinates as
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Y of T times e of U equals e of T plus u
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over 1 minus tu this is the group
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structure on the circle well there's a
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little bit of a problem with minus 1
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because minus 1 has a special role in
12:37
this story so it's kind of excluded from
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this story to include that we would have
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to go to a projective setting but we're
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not going to do that here but anyway
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this is an important formula telling us
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what the group operation is and it's
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purely algebraic and rational and
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crucially it does not depend on any
12:58
angles ok so this is an alternative to
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the usual annular picture for describing
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the unit circle and what's happening on
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it so this multiplication on the unit
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circle is capturing what's usually
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described in terms of angles but it
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allow
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us to connect with some important
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formulas called addition formulas
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relating to areas of sectors of the
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circle so here is our unit circle here
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is a quotient polynomial of it the
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standard one
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so gamma equals these two functions 1
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minus alpha squared over 1 plus alpha
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squared + 2 alpha over 1 plus alpha
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squared and then the corresponding
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addition formula relates these 3 sign
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areas one at e of T so T is some
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parameter and a of T is this point here
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and so the first one is from 0 up to T
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so it's that signed area that's sector
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the next one is from 0 to another value
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say u so from there the first one is the
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red one and the second one is the blue
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one and the sum of these two is then the
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sign area of a third sector from 0 to
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say V where this V is exactly that
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expression we've just seen t plus u over
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1 minus T u that point over there in
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this situation here so in terms of
14:31
angles this would be your adding that
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angle to this angle to get this angle
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but we don't have to use angles and we
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don't want to use angles because their
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transcendental things they require
14:42
infinite processes to deal with and to
14:44
calculate with okay so this is a
14:46
rational kind of version of this story
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but geometrically the content is the
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same that this area here plus this area
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here is the total area determined from
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here to here ok that big area there now
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considerable care is needed here so
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first of all the thing is in quotes
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because I remind you we actually have
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not properly defined what we mean by the
15:13
signed area of such a sector of the
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circle in fact I'm trying to convince
15:17
you that it's almost impossible to get
15:20
an exact definition of that quantity
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it's somehow illusory no matter what we
15:25
do but nevertheless from an engineering
15:28
point of view we might want you know to
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know what the approximate relations are
15:32
between the
15:33
these kinds of things so this is perhaps
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interpreted as an approximate relation
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okay so this considerable care needed
15:40
just for that approximate aspect but
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also because it turns out that the
15:46
formula only works sort of in certain
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ranges so that's because our
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parameterization has this difficult
15:55
point that I've already mentioned okay
15:57
and minus one the circle is not actually
16:00
parametrized so it's safest if we
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restrict ourselves to save values T and
16:06
u between minus 1 and 1 so that these
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two points that we're starting with are
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on the right-hand side of the circle
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then we don't run into any problems of
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sort of going across this this boundary
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here okay so we're touching base with
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some more sophisticated things
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we're just taking an applied point of
16:27
view right now we are looking at these
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formulas and say okay an engineer could
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use that to to study areas on circle in
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particular we have some control of not
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just the total area of the circle but
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also sectors of a circle okay so our
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theory is is quite a lot more powerful
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than the old story of calculating just
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areas of circles by approximating
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polygons okay so now let's introduce the
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red complex numbers that's the same kind
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of story except that we're introducing a
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new object called J such that J squared
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equals 1 J is not 1 or minus 1 J is some
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new object ok so then to a pair of
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numbers forming an affine point we can
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associate this red complex number a + b
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J let's call it said then we can define
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addition of two of these things
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component wise in the usual way but more
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interestingly we can define
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multiplication of two of them using this
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alternate rule where the J squared is
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equal to
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so a plus B J times c plus D J is AC
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plus BD because the J times J is 1 and
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the coefficient of J is ad plus BC as it
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is in the ice story so we might call
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these the red complex numbers maybe
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denote it by something like that or if
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you're working traditionally C maybe
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with a small R underneath it okay what
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about the rest of the theory well you
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can define a complex conjugate a minus B
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J then you can define the quadrants of Z
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to be Z times Z bar and that's also a
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difference of squares and but now we get
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a squared minus B squared because the J
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times J is 1 and then it turns out that
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the quadrants of Z times W equals the
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quadrants of ZN & quadrants of w and the
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easiest way to appreciate that is to go
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back to this object and to recognize
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that you can identify that with another
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two-by-two matrix a b b:a so these red
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complex numbers also sit inside the
18:59
algebra of two-by-two matrices but in a
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different way from the blue complex
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numbers not better not worse just
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different and then this thing here this
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quadrants is just the determinant a
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squared minus b squared of that and so
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this is just the usual product of
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determined rule but crucially this
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algebra that we're getting is not a
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field because there's other things
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besides 0 which are not invertible in
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particular the quadrants of 1 + J or the
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quadrants of 1 minus J is 0 so these
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objects are not invertible in the in the
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in their algebra so that means that
19:43
algebraically it plays a little bit of a
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different role and historically complex
19:50
numbers have been much much much more
19:52
study than these ones but probably the
19:58
playing field ought to be leveled
20:00
the case could be made that these are in
20:02
fact just as important as complex
20:05
numbers okay
20:06
especially since we live in a
20:08
relativistic world you know that's the
20:10
way modern physics explains the world
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and so relativistic kinds of algebra are
20:17
you know just as potentially important
20:20
as Euclidean ones in fact maybe even
20:22
more so
20:23
okay so do not minimize the potential
20:27
importance of understanding this red
20:29
alternative to the usual complex numbers
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story I want you to think about them yes
20:35
just being different alternate algebraic
20:39
domains in which we can do arithmetic so
20:43
for an important application of red
20:45
complex numbers let's study the
20:47
relativistic hyperbola x squared minus y
20:50
squared equals 1 from a point of view
20:52
that wants to put that parallel to the
20:56
usual unit circle so here is the curve
20:59
we're talking about and we're interested
21:03
in a rational parameterization of this
21:05
and we can proceed pretty well exactly
21:08
as we did for the unit circle we can
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start with this point minus 1 we can
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choose a point J times T J is our
21:16
special number so just T times that
21:18
somewhere on this y axis we can join
21:22
minus 1 to that and see where the other
21:25
point of intersection is with the curve
21:28
so that turns out to be the point with
21:32
coordinates 1 plus T squared over 1
21:34
minus T squared and 2t over 1 minus T
21:37
squared and in this red complex number
21:41
world we can represent that by this
21:42
linear combination involving our new
21:45
symbol J and we can call this F of T
21:50
notice it's very similar to the blue
21:54
parameterization except here we have a
21:57
plus sign here and a minus sign there
21:59
and the other way around it's with a
22:01
plus sign and a and the denominator here
22:03
the minus sign there now there's another
22:06
way of thinking about this which is
22:07
entirely analogous to the picture I
22:10
showed you for the unit circle if we
22:12
look at the line x equals y
22:14
just as before and choose the point 1
22:16
plus JT on it so that's what you get
22:19
when you take this line and move it so
22:22
it's through the origin now but still
22:24
parallel ok so we're going over 1 and up
22:27
J or JT and then you take that point
22:31
zette and you think of it as a red
22:34
complex number and you square it ok
22:36
you're gonna get some point maybe up
22:37
there and then you divide by the
22:40
quadrants of that thing just the same as
22:44
we did in the other situation and you
22:46
get this point that we've already
22:48
identified as being obtained through
22:51
this construction so this is another way
22:53
of thinking about the parameterization
22:54
we take some line through the origin we
22:57
take some point on that line we square
23:00
it and divide by the quadrants and we
23:01
get a point on the conic and in fact
23:06
that gives us an identification of lines
23:08
through the origin with points on the
23:10
conic including also this branch here ok
23:15
so since if we have two points on here
23:17
say 1 + JT + 1 + J u then the way they
23:21
multiply in the red complex number story
23:24
we get 1 + T u + J times T Plus u and
23:28
that gives us this formula that f of T
23:33
times f of u so the product of two
23:35
points on the Peconic will be f of t + u
23:41
divided by 1 plus t u the appropriate
23:46
scaling of this thing so that we get on
23:48
to a point on this line so this gives a
23:53
group structure on x squared minus y
23:55
squared equals 1 again with this point
23:58
here somehow being excluded and this is
24:03
entirely analogous and for relativistic
24:07
studies like if you're a physicist and
24:09
you're interested in Lorentz
24:10
transformations and understanding
24:12
Einstein's special theory of relativity
24:13
this is this is exactly what you need
24:17
okay this is exactly the kind of algebra
24:20
that you need this kind of group
24:22
structure and the Associated
24:24
transformations of multiplying by
24:27
unit unit vector is there that's exactly
24:31
what the Lorentz transformations are the
24:34
the boosts of special relativity amount
24:37
to so this is a very important picture
24:40
alternative to the blue story especially
24:44
for modern physics but actually also for
24:47
calculus modern calculus okay so now
24:52
let's introduce an important definition
24:55
that we've used already a number of
24:58
times but now we're gonna put a name to
25:00
it so for a curve gamma say P Q usually
25:06
with P and Q being poly numbers but now
25:09
we're also thinking a little bit more
25:10
generally sometimes P and Q might be
25:13
quotient polynomial functions okay so
25:18
we're going to define the signed area
25:21
differential of this curve gamma to be
25:24
we're gonna call it D of gamma equals P
25:28
DQ minus QD P so this basic expression
25:33
that appears in the fundamental theorem
25:35
and so way to give it a name we'd call
25:38
it the area differential of the curve so
25:42
for some examples if gamma is the usual
25:46
power function y equals x to the N or
25:49
described in poly number form as alpha
25:52
alpha to the n then d of gamma is n
25:56
minus 1 times alpha to the n that
25:58
actually works for a general integer n
26:02
more generally if we have a curve given
26:05
by alpha to the M and alpha to the n
26:07
then the differential D of gamma is n
26:09
minus m times alpha to the M plus n
26:13
minus 1 it's a generalization what's
26:17
happening up here now for the circle so
26:20
now we're extending our understanding
26:23
here to having quotient polynomial
26:28
parameterization of the circle and we
26:30
made this computation that basically
26:32
amounts to that the differential of this
26:35
gamma is 2 over 1 plus alpha squared
26:40
and now we're in position to extend that
26:44
calculation to the rational
26:47
parameterization of the relativistic
26:48
hyperbola so if gamma is equal to one
26:52
plus alpha squared over one minus alpha
26:54
squared and two alpha over one minus
26:57
alpha squared that's exactly that
26:59
rational parameterization of the
27:01
hyperbola x squared minus y squared
27:03
equals one that we've been talking about
27:05
then a computation just very much like
27:08
this one is that the differential of
27:11
gamma equals two divided by one minus
27:14
alpha squared that's a good exercise to
27:18
check that so these are very close
27:22
algebraically very similar kinds of
27:25
calculations but geometrically there's
27:30
quite a big difference between these two
27:32
these two things especially if you graph
27:35
them as say as functions of alpha
27:38
they will look very different and so now
27:41
finally I want to show you the
27:43
corresponding addition formula for the
27:45
relativistic hyperbola an analogue of
27:47
the ones that we've seen for the circle
27:49
here's our hyperbola again and we can
27:53
think of it as parameterised here 1 plus
27:55
alpha squared over 1 minus alpha squared
27:58
and 2 alpha over 1 minus alpha squared
28:00
and here are some points on the
28:03
hyperbola say F of T f of U and their
28:09
product f of t plus u over 1 plus T U
28:13
and the addition formula is that the
28:17
signed area from 0 to T of gamma so this
28:21
red portion there plus the signed area
28:24
from 0 to U of gamma that's the blue
28:27
portion up there equals sign area from 0
28:30
to W of gamma where W is this number T
28:35
Plus u over 1 plus T U so in other words
28:39
the sum of these two is this bigger
28:42
signed area swept out there and
28:48
again we have a formula in quotes so we
28:52
don't have exact definitions even of
28:54
what the assigned areas are and perhaps
28:57
exact definitions are forever elusive
29:00
but nevertheless there's an approximate
29:03
aspect to this but still considerable
29:06
care is needed in particular you need to
29:08
think about which values of T and U
29:10
you're going to choose to ensure that
29:12
you get a nice formula so for example we
29:16
could restrict ourselves to T in you
29:18
being from minus 1 to 1 that effectively
29:21
means that we're talking about a T and
29:23
you being on that branch there ok
29:27
so we've done quite a lot of things
29:32
today we've seen a lot of material
29:33
connecting with geometry making that
29:37
transition from the Euclidean story to
29:39
the relativistic story and you seeing
29:42
that calculus bridges these two things
29:45
right calculus is not a Euclidean theory
29:48
it's an affine theory which can be
29:50
applied to the Euclidian case can be
29:52
applied to the relativistic case I'm
29:55
going to see the importance of the
29:57
relativistic side even more in the
29:59
subsequent lectures so there's much to
30:02
learn I'm going wild Berger thanks for
30:05
listening
30:08
[Music]