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HERBERT GROSS: Hi.
Standing here waiting for
today's lesson to begin,
I was thinking of a
story that came to mind.
And that was the story of
the foreman yelling down
into an excavation, how
many of you men down there?
And the reply came back, three.
And he said, OK, half
of you come on up.
Now with that funny
story, I would
like to launch today's lesson,
which could be, I guess,
sub called, or whatever
word that you'd
like to pick, realness is
in the eyes of the beholder,
the question being, of course,
that when you talk about taking
one half of three people, ah--
well, let's put it this way.
I was going to say,
you can't do it.
Let's say, if you could
do it, you wouldn't
like to see the answer.
On the other hand, to
take three inches and say,
let's divide that
into two equal parts,
there's a case where the answer
does happen to make sense.
Now today, you see
we're going to talk
about a new phase of our course
called the complex numbers.
The complex numbers happen
to be a delightful topic,
from the point of view
that on the one hand,
they offer a great deal of
enrichment in pure mathematics,
and on the other hand, they
contribute a great deal
to our physical
understanding of reality.
Now to launch into this, let's
get right into the topic.
I call today's lesson,
The Complex Numbers.
And going back to my
opening hilarious story,
what we're saying is if the
only numbers that we knew
were the integers and we
were given the equation
2x equals 3--
and notice that in
terms of integers,
the equation makes sense,
because 2 and 3 happen
to be integers-- we're looking
for to solve this equation.
And the question is, does
this equation have a solution?
The answer is it only
has a solution provided
that you want to invent
numbers which are not integers.
In other words, this
does not have a solution
if we insist that the
answer be an integer.
But if we're willing to invent
a new batch of numbers called
the fractions or the
rational numbers,
then this equation
will have a solution.
Do we want this
to have solutions?
Well, sometimes there are
meaningful situations in which
it's meaningful to
solve this equation,
other times when it isn't.
Again, the story of
going into the bank
and into the post office and
asking for $0.03 worth of $0.02
stamps.
And we can think of all sorts of
hilarious ways of embellishing
this story.
Going on further, though, let's
take what ultimately became
known as the "Real Numbers."
And I'm going to put
this in quotation marks.
Because even though
these are genuinely
called the Real Numbers,
they are no more real
than the integers were real.
By definition, a Real
Number is simply any number
whose square is non negative.
Assuming that to be the
definition of real numbers,
we come to the equation
x squared equals minus 1.
We say, does this
equation have a solution?
And the answer is if we
insist that the solution has
to be a real number,
the answer, obviously,
is that this does
not have a solution.
Because the definition of a
real number is that it's square
cannot be negative, and this
says that x squared is minus 1.
So either we must say that
this equation does not
have solutions.
Or if we want it
to have solutions,
we must invent an extension
of the number system.
We must extend the real
numbers in the same way
that if we wanted this
equation to have a solution,
we had to extend the integers.
The extension of
the integers that
was necessary to
solve this equation
happened to be called
the Rational Numbers.
Let's talk about, first
of all, whether we
want an extension
of the real numbers,
and secondly, if we do,
what shall we call it,
and what properties
shall it have?
To motivate why we
would like the equation,
x squared plus 1 equals
0 to have solutions.
And that, of course,
is just equivalent to x
squared equals minus 1.
Let's go back to
an example that we
used when we first
introduced exponents
in part 1 of our course.
We observed that one
of the properties
of an equation like
y equals e to the rx
where r is a constant is
that if we differentiate this
with respect to x,
the r comes down.
But the fact, the e
to the rx remains.
What this told us was that
in taking derivatives,
we would get powers of r.
But e to the rx would remain.
And we use that
technique for solving
certain types of second,
well, differential equations.
For example, looking at the
equation y double prime plus y
equals 0, one technique would
be to replace y by e to the rx.
When I differentiate twice, I
have an r squared e to the rx
here.
Here is an e to the rx.
The e to the rx cancels out.
And I wind up with a
polynomial equation for r.
And we then solve for
the values of r, such
that this would be
a solution of this.
Now back in part 1, because we
didn't know non-real numbers.
We always fixed it up so
that the resulting equation
that we got here would have
real numbers for values of r.
By the way, I have chosen
this particular equation
for two reasons.
One is, the values of r can't
be real when I get through here.
The other is, I already know
an answer to this problem.
Remember that if you
differentiate sine twice,
you get minus sine.
If you differentiate cosine
twice, you get minus cosine.
Consequently, the
second derivative
of sine x plus sine x is 0.
Second derivative of cosine
x plus cosine x is 0.
So I know at least two
solutions to this equation,
two real solutions
to this equation,
this equation has physical
meaning because it essentially
says, if you think of the
parameter as being time here,
look at the second
derivative is equal to minus
the function itself.
That says that the
acceleration is
equal to minus the displacement.
And that's a form of
simple harmonic motion.
You see, what I'm
trying to establish here
is that this does
have a real solution.
And the problem itself has
real physical significance.
At any rate, if we now try
to solve this problem using
this technique, and
remember, again,
I can give you
problems where we won't
know the answer intuitively,
and the values of r
will still come out
to be imaginary.
I simply chose this
problem so that we
could correlate a
so-called imaginary answer
with a real answer.
The idea, again,
is I differentiate
each of the rx twice.
I get this.
So y double prime
plus y is this.
I factor out the e to
the rx, which can't be 0.
Therefore, it must be that
r squared plus 1 is 0,
or r is equal to plus or minus
the square root of minus 1.
Now you see, so far, I don't
know what this thing means.
Because there certainly
is no real number,
which has this property.
Because the square of real
numbers have to be positive.
They can't be minus 1.
Let me, for the
sake of argument,
invent a number, which is equal
to the square root of minus 1.
I'll call that number i.
So r is equal to
plus or minus i.
And if I now remember what
equation I'm solving--
it was y equals e to the rx--
I can replace r by
either i or minus i.
So the two solutions I would get
would be y equals e to the ix
or y equals e to the minus x.
But this is bothering me.
Because I really have
no real feeling for what
I means right now.
What I do know, however,
is this, that mechanically,
this is a solution
to a real problem.
y double prime
plus y equals zero
is a real problem,
not an imaginary one.
And by the way, notice
mechanically here,
taking i to be a
constant, if we still
assume that the derivative of
e to the x is re to the rx,
even when r happens
to be non-real,
notice that if I differentiate
this thing twice,
the first time I
differentiate, I bring down
an i, the second time I
differentiate, I bring down
an i.
That gives me i squared,
which is minus 1.
This is going to be
minus e to the ix.
And if I add on e to the
ix, I really do get 0.
Notice also, now, that I do
know that a real solution
of this equation is y equals
cosine x, or also y equals sine
x.
So that somehow or
other, I get the feeling
that I should exist, and more to
the point, that this expression
e to the ix should somehow be
related to sign x and cosine x.
In other words, if I had
never been lucky enough
to invent the
trigonometric functions,
and I want to solve the
problem, y double prime plus y
equals 0 and wound
up with this, then
my feeling is, I must find
a real interpretation,
a physically real interpretation
for each of the ix.
Because I sense that my
problem has a solution.
Again, I can take
the coward's way out.
I can say, if it's going
to be this much work,
I don't want the problem
to have a solution.
But once I want the
problem to have a solution,
I must extend the number system.
And the upside of
that whole thing
is that we now invent something
called the Complex Number
System.
First of all, the
complex numbers
are defined to be the
set of all symbols--
let's just call them
Symbols for the time being--
all symbols of the
form x plus iy,
where x and y are real numbers.
And just to refresh
our memories here,
i is the symbol which
numerically represents
the square root of minus 1, OK?
Notice that the X and the
y are real numbers, though,
where by Real, we mean what?
The squares are non negative.
Now you will recall, that
in any mathematical system,
the thing that
you're dealing with
is more than a set of
numbers, it's a set of numbers
or a set of objects together
with a structure, certain rules
that tell us how to work with
these elements that make up
the set.
Remember, the difference
between a set and a system
is a set that's just a
collection of objects.
The system is the
collection of objects
together with how we combine
these objects to form
similar objects, et cetera.
And what is the
structure in this case?
Given the two complex numbers,
x1 plus iy1 and x2 plus iy2,
we define equality to be sort
of component by component.
The real parts x1
and x2 must be equal.
And the imaginary parts-- and by
the way, just a word of caution
here, the imaginary
part of a complex number
is, by definition,
the coefficient of i.
Notice that the imaginary
part is itself a real number.
The coefficient of i
is the real number y1.
What we're saying is we define
the definition of equality
for complex numbers to be
that two complex numbers are
equal if and only if
their real parts are equal
and their imaginary
parts are equal.
Secondly, we agree to add two
complex numbers, component
by component, so to speak.
In other words, to add
two complex numbers,
we add the two real parts, and
we add the two imaginary parts,
OK?
That's all this says.
And thirdly, to multiply
a complex number
by a real number, we
multiply the real part
by the given real number
and the imaginary part
by the given real number, which
is what this third rule says.
And all I want to
see from this is
that by the very
definition of these three
structural properties, we
have made the complex numbers
a two-dimensional vector space.
In other words, it
means that we can now
visualize the complex
numbers, geometrically,
in the same way that
we could visualize
the real numbers geometrically.
Remember, geometrically-- when
we define the real numbers
to be those numbers whose
squares are non negatives,
we do not need any
picture to visualize that.
It just happens that the
number line, the x-axis,
is a very convenient
way of visualizing
real numbers pictorially.
What we're saying is that
since complex numbers seem
to indicate a two-dimensional
vector space, namely,
real and imaginary parts
which are independent,
it would appear that the
analog for visualizing
complex numbers pictorially
would be to use the plane.
And that's exactly what we do.
The diagram is called
the Argand Diagram.
The idea is this.
Given the complex number x
plus iy, which we'll call z,
we visualize z as
either the point
x comma y in the xy
plane-- in other words,
notice that the x-axis
is the real axis,
meaning it denotes the real
part of the complex number.
The y-axis is called
the imaginary axis,
because it denotes
the coefficient of i,
the imaginary part, OK?
And what we're saying is we can
visualize the complex number x
plus iy to be the
point x comma y.
Or for that matter,
the vector that
goes from the
origin to the point
z, whose components are x and
y, meaning what? x plus iy.
The components are the real
and the imaginary parts here.
We can translate this
into polar coordinates,
meaning that we can measure
the point z by its r
value and its theta value.
But one important
thing to remember
is that in polar
coordinates, it's always
assumed that r is positive.
Unlike the usual
polar coordinates,
where r could be either
positive or negative,
the idea is that we
would like to identify
the absolute value
of a complex number
with being its distance
from the origin.
Since we want absolute
value to be non-negative,
we simply say that r is
the positive square root
of x squared plus y squared.
And that names the magnitude
of the complex number.
The angle theta is called the
Argument, abbreviated Arg,
not to be confused with this
funny coincidence of Argand
diagram.
This comes from the
word Argument, argument.
The angle is called the
Argument, all right?
And if you elect to
write the complex number
x plus iy as r
comma theta, because
of the connection between polar
and Cartesian coordinates, r--
see, x is what? x
is r cosine theta.
y is r sine theta.
So the complex number
is r cosine theta
plus ir sine theta.
That's what r comma
theta denotes.
Now using this as background--
so we now have what?
We've invented, at least
abstractly, a system of numbers
called the Complex Numbers.
And by the way, let me point
out here that don't use the word
Imaginary too strongly.
There is certainly
nothing imaginary
about this geometric
interpretation.
What may be imaginary
is I may not
have a place where I want to
use these kind of numbers.
But believe me, we wouldn't have
made a whole block of material
about these numbers
if there weren't
real interpretations of the
so-called complex numbers.
This is a perfectly real
geometric interpretation.
So far, we have
established the fact
that this interpretation
gives us a vector space.
But the complex numbers have an
additional structure as well,
namely, if I saw the two complex
numbers, a plus bi times--
and c plus di, and I multiplied
them-- remember now, a, b, c,
and d are real--
I would like to believe that
the rules of ordinary algebra
still apply here.
To multiply this, I would
want this times this,
this times this, this times
this, and this times this.
Notice that a times c as ac.
bi times di should be
bd times i squared,
if the ordinary rules of
arithmetic are to apply.
Since i squared is minus 1,
this term becomes minus bd.
Since a, c, b, and
d are real numbers,
this is a real number, right?
Correspondingly, the
coefficient of-- the multiplier
of i, the coefficient of
i is what? bc plus ad,
which is also a real number.
So if we make up this
definition of multiplication
for complex numbers,
we have made sure
that the rules of arithmetic
for complex numbers
parallel the rules
for real numbers.
And by the way,
it's very important
here to make the aside that
this is a crucial result.
Notice that among
the complex numbers,
the real numbers are
included, namely,
thinking of it geometrically,
if you look at the plane,
the real numbers
are those points
in the plane which
lie along the x-axis.
It's the same as saying that
when you extended the integers
to form the rational numbers,
the rational numbers included
the integers.
In other words, if I
look at the number 3,
I can think of it
as the integer 3.
I could also think of it as the
ratio 3 divided by 1, you see?
And therefore, I
would like whatever
rules I have for
the complex numbers
to remain valid if the complex
numbers happen to be real.
It would be terrible to
have two different rules
for multiplication,
one for real numbers
and one for complex
numbers, and then
given two real numbers,
when I multiply them,
I get one answer if I think
of them as being real,
and another answer if I think
of them as being complex.
At any rate, the important point
is with this as motivation,
I have a way of defining the
product of two complex numbers
to be a complex number.
I don't mean that these things
are equal to r, so they simply
stand for Real.
This is real plus
i times a real,
which is a complex number.
A very special case occurs when
you take the complex number
a plus bi and multiply it by a
minus bi, the same number only
changing the plus to a minus.
Notice that we would get what?
The sum and difference
here gives us the sum
of difference of two squares.
It's going to be what? a squared
minus b squared i squared.
But since i squared is minus
1, minus b squared i squared
is minus b squared times minus
1, which is plus b squared.
And notice, therefore, if
you multiply a plus bi, which
is a complex number, by this
other complex number, a minus
bi, you get a squared
plus b squared,
which is a non-negative
real number.
In particular, if you think back
to the geometric interpretation
of this, the point a plus bi
in the plane is a comma b.
And the distance of a
comma b from the origin
is the square root of a
squared plus b squared.
So this product is
actually the square
of the magnitude, the
absolute value of a plus bi.
This little gizmo
is so important
that it's given a special name.
By definition, given
any complex number z,
written in the form x plus iy,
the complex conjugate of z,
called z bar--
not to be confused with a z with
a bar underneath that we were
using to denote vectors--
but the complex conjugate
z bar is what you get by just
changing the plus sign here
to a minus.
In other words, the complex
conjugate of x plus iy
is x minus iy.
And geometrically,
all this means
is, you see, when you change
the sign of the imaginary part,
remember, the imaginary
part is the y-axis.
You're just reflecting
the number symmetrically
with respect to the x-axis.
You're leaving the
x-coordinate alone.
And you're changing the
y-coordinate to minus y.
In polar coordinates,
what you're saying
is a complex number
and its conjugate
are the same distance
from the origin.
But in one case,
the angle is theta.
And in the other case,
the angle is minus theta.
One of the many applications
of complex conjugates--
I'll give you some of
these other applications
in the exercises.
But for now, what I
think is important
is a simple one that
essentially shows us
how we find the quotient
of two complex numbers.
Quite simply, sparing
you the details,
if I have c plus di
divided by a plus bi,
I simply multiply
numerator and denominator
by the complex conjugate
of the denominator, OK?
What will that do for me?
Well, when I multiply the
two factors in the numerator,
I'm going to get a real
part, namely, ac plus bd.
In other words, it's
minus bd i squared.
But i squared is minus 1.
So I'm going to get ac plus bd.
I'm going to get an imaginary
part, which is ad minus bc.
But the denominator will just
be a squared plus b squared,
which is a real number.
In other words, what
this tells me is
I can now write the quotient of
two complex numbers in the form
what--
a real number plus i
times a real number.
In fact, the only
time I can't do this--
I have to be careful.
a or b must be unequal to 0.
Because if both
and b are 0, this
is a 0 denominator, which
I don't allow myself
to divide by.
And the only way
both a and b can be 0
is if the complex number is 0
plus 0i, which is the number 0.
So I'm saying, again, just
like in the real case,
I still can't divide by 0.
Anyway, to give you a more
practical illustration to see
this with numbers-- not
practical, but, at least,
concrete--
3 plus 2i divided by 4 plus
i, I simply multiply numerator
and denominator by 4 minus i.
You see, what that
does is downstairs,
I get 4 squared, which is
16, minus i squared, which
is plus 1, 16 plus 1 is 17.
The real part is
going to be what?
Here's 12.
2i times minus i
is plus 2 is 14.
8i minus 3i is 5i.
So this quotient, 3
plus 2i times 4 plus i,
is 14/17 plus 5/17 i.
And by the way, as a
trivial check, and you
can do this if you
wish for an exercise,
simply take the answer
that we got here,
multiply it by the
denominator, 4 plus i
here, and actually check
that that product comes out
to be 3 plus 2i.
At any rate, this use
of complex conjugates
shows us that when you
divide two complex numbers,
the result will be
a complex number,
except for division by 0.
You see, that's what went
wrong with our integer problem,
2x equals 3.
The reason that we couldn't
solve it is that to solve 2x
equals 3 required that
the process of taking
the quotient of two integers,
and the quotient of two
integers did not have
to be an integer.
Notice on the other
hand, that the quotient
of two real numbers
is a real number,
as long as the
denominator is not 0.
And the quotient of
two complex numbers
is a complex number,
except for division by 0.
While we're talking
about multiplication,
a very enlightening
thing happens
if we think of multiplication
in terms of polar coordinates,
namely, take the two numbers,
which in polar coordinates,
are r1 comma theta 1
and r2 comma theta 2.
Now again, remember
from our Argand diagram
what it means to say that
the complex number has
polar coordinates
r1 comma theta 1.
Remember, the complex number is
written in the form x plus iy.
The x-coordinate is
r1 cosine theta 1.
The y-coordinate
is r1 sine theta 1.
At any rate, this product, in
terms of the standard notation,
the x plus iy form, says we're
multiplying these two numbers
together.
And using our rule
for multiplication,
multiplying term by
term, and remembering
that i squared is minus
1, observe that if we
collect the terms here--
and I'll leave the details for
you to verify at your leisure--
we wind up with r1 r2
being a common factor.
The real part is r1 r2
times cosine theta 1
cosine theta 2 minus sine
theta 1 sine theta 2.
The minus coming, because
i squared is minus 1.
The imaginary part is r1 r2
sine theta 1 cosine theta 2 sine
theta 2 cosine theta 1.
Remember, theta
is a real number.
If we recall our geometric--
trigonometric definitions,
cosine theta 1
cosine theta 2 minus
sine theta 1 sine theta 2
is cosine theta 1 plus theta 2.
This expression this sine
theta 1 plus theta 2.
And remembering now that in
polar form, what we're saying
is what?
This is the complex
number whose magnitude
is r1 r2 and whose argument
is theta 1 plus theta 2.
And putting these two steps
together, what it says
is that to multiply two complex
numbers, the same definition
of multiplication
that we were using
before, if we interpret this
in terms of polar coordinate,
it says, look at, to multiply
two complex numbers [INAUDIBLE]
lengths, you simply do what?
Multiply the two lengths
to get the resulting
length of the product and add
the two arguments, the two
angles.
That's a rather
interesting result.
You see, for example,
if we now want
to think of the complex
numbers as being vectors,
this gives us a
third vector product
that we had never
talked about before
and which I'll reinforce
in the learning exercises.
The idea is that
using this as a model,
and here's another
real application,
why not define a new product
of two vectors obtained
by multiplying the two lengths
and adding the two angles?
And the point was, that
in our physical examples,
there was really no motivation
to invent this vector
definition, in the
same way that we
could define the dot product
and the cross product
for complex numbers.
Because after all, they are
viewed as vectors in the plane.
But again, we have no great
practical application for this.
So we don't bother doing it.
By the way, just
for kicks here, I
thought you might
enjoy the aside
that this little
interesting result explains
why the product of two negative
numbers happen to be positive.
Remember a negative
number, if r has to-- see,
a negative number
lies on the real axis.
And if r has to
be positive, that
means you're measuring
in the direction
of the negative x-axis, which
means that the polar angle is
180 degrees.
If you multiply two
numbers whose angles
are 180 degrees, if you add
the two angles, you get what?
360 degrees.
You see what I'm
driving at here?
In other words, if I multiply
two numbers in polar form, each
of whose angles is
180, the product
will have the angle
equal to 360, you see?
And that puts you back on
the positive real axis.
And so you have such
real interpretations
as y in terms of
complex numbers,
we can explain very nicely what
it means for the product of two
negatives to be a positive.
By the way, we can
carry this result
further if we had n factors
written in polar form.
Then to multiply
these n factors,
we would simply, by
induction, so to speak,
multiply the n magnitudes
together and add the n angles.
And a very interesting special
case is if all of the factors
happen to be equal--
in other words,
if we want to raise the complex
number written in polar form
as r comma theta to the n-th
power, a very interesting thing
is that r comma theta to
the n-th power is what?
You multiply the magnitude.
So the magnitude of the
product will be r to the n.
You add the angles.
So the angle of the product, the
argument will be n times theta.
A special case of the special
case is if r equals 1.
And if r equals 1, that says
that the complex number whose
polar form is 1 comma theta,
when raised to the n-th power,
is that complex number in
polar form 1 comma n theta.
And by the way, again,
remember what this thing means?
To say that the complex number
is 1 comma theta means what?
That the distance from the
origin is 1 and the angle
is theta.
See, that was what we mean
by 1 comma theta over here.
And notice that
in Cartesian form,
that makes this length cosine
theta and this length sine
theta.
So i comma theta is cosine
theta plus i sine theta.
At any rate, translating both
of these into Cartesian form,
we wind up with a
very famous result,
called De Moivre's theorem.
In my high school, it was
called "De Moyvre's" theorem.
But it's De Moivre's theorem.
And it simply says that
cosine theta plus i sine
theta to the n is cosine n
theta plus i sine n theta.
That result may not seem
that remarkable to you.
But let me give you another
example using this result that
shows how we can get
real results using
imaginary numbers.
Let me take the special
case n equals 2, just
for the sake of argument.
If I take n equals
2, this gives me
cosine theta plus i sine
theta squared equals cosine
2 theta plus i sine 2 theta.
This says what?
The real part of this
is cosine squared theta
minus sine squared theta.
Again, it's i squared
sine squared theta, which
is minus sine squared theta.
The imaginary part is 2
sine theta cosine theta.
So if I square
this, I get cosine
squared theta minus sine squared
theta plus i2 sine theta cosine
theta.
By De Moivre's theorem, that
equals cosine 2 theta plus i
sine 2 theta.
We saw that the only way
that two complex numbers can
be equal is if the
real parts are equal
and if the imaginary
parts are equal.
Comparing the real parts and
comparing the imaginary parts,
we get that sine 2 theta equals
2 sine theta cosine theta.
Cosine 2 theta is cosine squared
theta minus sine squared theta.
Notice, by the way, that
these are real results.
And also, notice, by the
way, that even though these
may look like old
hat to you, I could
just as easily, for
example, I could have picked
n 5 here or 6.
And in fact, I will do that
in the learning exercises.
The point being that,
what, I can raise this
to the fifth power, compare
this with cosine 5 theta plus i
sine 5 theta, and wind
up with real identities,
in fact, for what?
Sine n theta and cosine n
theta for any whole number
value of n.
See, I'm trying to
hammer home the fact
that as we're doing the
complex number arithmetic,
don't forget that this stuff
does have real applications,
and we haven't even started
to scratch the surface yet.
This is just our baby lecture
introducing the arithmetic
of complex numbers.
We haven't even
gotten to anything
like algebra or calculus yet.
Wait till that happens,
and you're really going
to see some nice applications.
At any rate, by the way, the
polar form of multiplication
that leads to a very
interesting way of extracting
roots of complex numbers.
For example, suppose I want
to find the sixth root of i.
In other words,
what complex number,
raised to the sixth
power, gives i?
In fact, is there
such a complex number?
After all, we could raise
real numbers to powers.
But one of the
reasons that we had
to invent the complex numbers
is that we couldn't extract
the square root of minus 1.
There was no real number
whose square was minus 1.
The question now is
there a complex number,
which when raised to the
sixth power, equals i?
One way of doing
this is to say, OK,
let's assume there
is a complex number.
We'll call it x plus iy, which
when raised to the sixth power,
equals i.
In other words, the sixth
root the i is x plus iy.
And let's see if we
can solve for x and y.
One way of doing this is
to raise both sides here
to the sixth power,
in which case
we see that i has to be x
plus iy to the sixth power.
On the other hand, i is
written as 0 plus 1i.
If I raise this to
the sixth power,
I don't know if
you've noticed this,
every time I raise i to an even
power, I get a real number.
Why?
Because i squared is minus 1.
Therefore, i to the fourth
is i squared squared, which
is minus 1 squared, which is 1.
i to the sixth is i
to the fourth times
i squared, which is 1 times
i squared, which is minus 1.
And in the same way, if I
take i cubed, that's what?
i squared times i,
which is minus i.
In other words, the even
powers of i are real,
the odd powers of i give
me back plus or minus i.
So if I raise this to the
sixth power and collect terms,
I'll get a certain
number of real terms
and a certain number of
purely imaginary terms.
In fact, using the
binomial theorem
and raising this
to the sixth power
and separating the terms
for you in advance,
I wind up with what?
x to the sixth plus 6x
to the fifth iy plus 15x
to the fourth iy squared plus
20x cubed iy cubed plus 15x
squared iy to the
fourth plus 6x iy
to the fifth plus
iy to the sixth.
I went through that rapidly.
It's just using the
binomial theorem,
noticing that all of these
terms will turn out to be real.
All of these terms will
be purely imaginary.
In other words,
getting rid of the i's
to the best of my ability.
See, squaring over here, this
is a minus y squared term.
This is just y to the fourth.
This is i to the sixth, which
is the same as i squared.
Because i to the fourth is 1.
i to the sixth is i fourth times
i squared, which is i squared.
So this just comes out
as minus 1, et cetera.
And making these
translations, we wind up
with the complicated algebraic
system that to find x and y,
we must be able to solve
this system of equations.
In other words, the
root part must be 0,
the imaginary part
must be 1, all right?
Now at this stage of
the game, not only
may it seem difficult
to solve this,
but it may be that there are
no real values of x and y
which solve this.
And if I can't find
x and y, if there
are no values for x and
y, it means that x plus iy
doesn't exist.
Well, here's where I wanted to
show you the tremendous power
of polar coordinates.
You see, in polar coordinates,
how would I write i?
I is what?
It's magnitude is 1.
See, it's the 0.0 comma
1 in the Argand diagram.
It's magnitude is 1.
And it's argument is pi over 2.
By the way, again, I'm in this
trouble with multiple angles.
You see, it's only pi over 2.
It could be 5 pi over 2,
9 pi over 2, et cetera.
Every time I go through 2 pi,
I come back to the same point.
I'm going to mention
that in a moment.
But the idea is, look
at, this is what I want.
This is i.
And I want the
[INAUDIBLE] divide.
Let's assume that
the answer can be
written in polar coordinates.
If I write it in
polar coordinates,
the answer is r comma theta.
All I've got to do now
is solve for r and theta.
And I claim, believe it or
not, that that's trivial,
namely, given this, I raise
both sides to the sixth power.
If I do that, I wind up
here with i, which just,
to get this all on one
line, I just repeat this.
This is 1 comma pi over
2 in polar coordinates.
And this is r comma
theta to the sixth power.
But the beauty of multiplication
in polar coordinates
is that r comma theta
to the sixth power
is r to the sixth comma 6 theta.
The magnitude is this--
so in other words, you
multiply the magnitudes,
and you add the angles.
Therefore, what this tells me
is, remember, r must be real.
There is only one
real number, which
when raised to the
sixth power, is 1.
And that's 1 itself.
See, the n-th root of
any positive number
has exactly one real solution.
That's why we can
always find the r.
In this case, I picked
the simple example
where r turns out to be 1.
See, minus 1 also raised
to the sixth power is 1.
But that doesn't count.
Because we agreed that
r had to be positive.
r was measuring the magnitude
of the complex number,
so r must be 1 by
virtue of the fact
that r is greater
than or equal to 0.
That eliminates minus 1.
Now 6 theta can either
be pi over 2 or 5 pi
over 2, et cetera.
The important point being that
we now tack on the 2 pi k.
Because notice that as this
changes by 360 degrees,
theta only changes
by 60 degrees.
Because 6 theta is
changing by 360.
Therefore, you see, we're going
to get a whole bunch of theta
values that work this way.
And again, I'll explain
this in more detail
as we go through the
exercises on this unit.
But the idea is what?
If I keep tacking on multiples
of 2 pi, what I'm saying
is r must be 1,
and theta is what?
It's pi over 12 plus what?
2 pi k over 6 pi over
3k, 60 [INAUDIBLE]..
I essentially, in
terms of angles,
tack on 60 degree
increments here.
To make a long story
short, r must be 1.
But theta could either be
pi over 12, 5 pi over 12,
9 pi over 12, 13 pi over 12.
See, I'm adding on 4 pi over
12 each time, pi over 3,
60 degrees, in degree measure.
17 pi over 12, 21 pi over 12.
The next one would
be 25 pi over 12.
But I hope you can
see that that's
the same as, position-wise,
1 comma pi over 12
gives me the same thing
as 1 comma 25 pi over 12.
But all six of these angles
give me different positions.
Just by way of
illustration, pi over 12
turns out to be 15 degrees.
What this says is that
one of the six roots of i,
geometrically, is what?
It's that complex number, which
is 1 unit from the origin.
That means it's on the
circle of radius 1,
centered at the origin.
And the angle must
be 15 degrees.
The argument is 15 degrees.
And by the way,
just to check this
out that this really is
at 1 comma pi over 12
really is an answer here.
How do you raise a
complex number to a power
if we view it as a length?
To raise this to
the sixth power,
we must raise 1 to
the sixth power.
That will still be 1.
So I'm still going
to be on the circle.
When I multiply, I add angles.
So when I raise this
to the sixth power,
I'm taking, what, 15 degrees
six times is 90 degrees.
And that puts me right up
where i is supposed to be.
In other words, to this
thing backwards, so to speak,
I know that 6 times the angle
I'm looking for must be 90.
So the angle itself must be 15.
For example, if I was looking
for the eighth root of i,
I would do what?
I would know that when I add
the angle to itself 8 times,
I want 90.
So the angle would have
had to been 90 over 8.
And again, I'll leave
this for you as exercises.
The point is that geometrically,
the sixth roots of i
are all equally spaced points.
The first one is the
point 1 comma pi over 12.
And the rest are spaced equally
along the circle at 60 degree
intervals.
You see, it breaks the circle
up into six equal parts.
I come back to here.
Notice that, for
example, if I take
this one, which is 75 degrees,
if I take 75 degrees 6 times,
notice that I come back,
what, 75 times 6 is 450.
It means I go all the way
around and come back to i when I
raise this to the sixth power.
To help you see this
geometrically, I'll pick--
of these six, one
of these happens
to be very easy,
at least, to me.
9 pi over 12 is 3
pi over 4, which
happens to be 135 degrees.
And i-- 1 comma 3 pi over 4
is cosine 3 pi over 4 plus i
sine 3 pi over 4.
The cosine of 3 pi over 4 is
minus 1 over square root of 2.
The sine is plus 1
over square root of 2.
So in typical x plus iy
form, one of the roots
is 1 over the square root
of 2 times minus 1 plus i.
In other words, the real part of
this complex number is minus 1
over the square root of 2.
The imaginary part is 1
over the square root of 2.
And I leave it, again, as a
voluntary exercise for you
to do to actually raise
this to the sixth power
and find, amazingly enough, that
you do get 1 for an answer--
i for an answer.
Now you see, I just picked
one particular example.
But this would have
worked for any roots
that I wanted to extract.
And this is very important from
a mathematical point of view.
The complex numbers are
closed, with respect
to extracting roots.
And let me summarize that
for a part of today's lesson
over here.
The idea is that
one of the reasons
that we had to invent the
fractions after we knew
the integers was the fact that
the integers were not closed
with respect to division, that
the quotient of two integers
didn't have to be an integer.
One of the reasons that we had
to invent the complex numbers
after we had the
real numbers was
that the real numbers were
not closed with respect
to extracting roots.
What I've just shown you in
terms of a particular example
is that the complex numbers
are closed with respect
to extracting roots.
This means that, in particular,
the basic operations that
are involved in solving
polynomial equations--
in other words,
what you have to do
is solve a polynomial
equation, nothing
more than the basic operations
of adding, subtracting,
multiplying, dividing,
raising the powers,
and extracting roots.
All of these operations
are closed with respect
to the complex numbers.
And what this means
is if you wanted
to write a polynomial
equation which
had complex numbers
as coefficients,
you would not have to invent
any more complex numbers.
You would not have to
invent a new number
system to solve this
equation, namely,
any polynomial with complex
coefficients has complex roots.
And I think that's
enough for today.
I want you to drill
now on the exercises.
Next time, we will
talk about functions
using complex numbers.
At any rate, until
next time, goodbye.
Funding for the
publication of this video
was provided by the Gabriella
and Paul Rosenbaum Foundation.
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