Part I: Complex Variables, Lec 1: The Complex Numbers Video Lecture - Engineering Mathematics

the following content is provided under
a Creative Commons license your support
will help MIT OpenCourseWare continue to
offer high quality educational resources
for free
to make a donation or view additional
materials from hundreds of MIT courses
visit MIT opencourseware at ocw.mit.edu
hi standing here waiting for today's
lesson to begin I was thinking of a
story that came to mind and it 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 okay 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 would 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 well
let's put it this way I was going to
saying 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 & 3 happened 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
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 nesting
for three cents worth of 2 cent stamps
and we can think of all sorts of
hilarious ways of embellishing the 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
and we say does this equation have a
solution and the answer is if we insist
that the solution have 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 a
square can not 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 wanted 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 property 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 a
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 that e to the
rx remains
what this told us was that in taking
derivatives we would get powers of our
but easy rx would remain and we use that
technique for solving certain types of
second while differential equations for
example looking at the equation y double
prime plus y equals zero one technique
would be to replace Y by e to the rx
when I differentiate twice I have an I
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 our
such that this would be a solution of
this now back in part one 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
because for two reasons one is the
values of our 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 zero second derivative of
cosine cosine X plus cosine X is zero so
I know at least two solutions to this
equation two real solutions in 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 e to 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 -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 our by either I
or minus I so the two solutions that I
would get would be y equals e to the i-x
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 0 was 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 a
derivative of e to the rx is our e to
the rx even when our happens to be non
real notice that if I differentiate this
thing twice the first time I
differentiate I bring down an eye the
second time I differentiate I bring down
an eye that gives me I squared which is
minus 1 this is going to be minus e to
the i-x and if I add on e to the i-x I
really do get zero 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 i-x should somehow be related to
sine 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
wound up with this then my feeling is I
must find a real interpretation
physically real interpretation for e to
the i-x 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
upshot 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 such as column 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 -1 ok
notice that the X and the y are real
numbers though whereby real we mean what
they're 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 is just a collection of
objects the system is the collection of
objects together with how we combine
these objects to form similar objects
etc and what is the structure in this
case given the two complex numbers X 1
plus iy 1 and X 2 plus iy 2 we define
equality to be sort of component by
component the real parts X 1 and X 2
must be equal and the imaginary parts
and by the way just a word of caution
here the imaginary part of the 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
y 1 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 they're 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 okay 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 you 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
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 okay
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
whose components are X and Y meaning
what X plus iy the components are the
real and imaginary parts here we can
translate this into polar coordinates
meaning that we can measure the point Z
by its our value and its data 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 with the split where I
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 argon diagram this comes
from the word argument are you meant 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 our cosine theta Y is our sine
theta so the complex number is R cosine
theta plus I R sine theta that's what
our 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 as
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 ABC and D
are real I would like to believe that
the rules of ordinary algebra is still
applied here to like that to multiply
this I would want this times this this
times this this times this and this
times this notice that a times C is 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 ACB and D
are real numbers this is a real number
right correspondingly the coefficients
about the multiplier revive 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 plain 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
a complex number I don't mean that these
things that equal ours here 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 summin difference of
two squares it's going to be what a
squared minus B squared I squared but
since I squared is minus one minus B
squared I squared is minus B squared
times minus one 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 that means
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
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 this 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 okay 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 at 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 have to be
careful A or B must be unequal to 0
because if both a 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 0 I which is the number 0 so I'm
saying again just like in the real case
I still can't divide by 0 any rate 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 and 17 the real part is going
to be what his 12 to I times minus I is
plus 2 is 14 8 I minus 3 is 5 I so this
quotient 3 plus 2i times 4 plus I is 14
17
Plus 5:17 sigh 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 not the denominator 4 plus I here
and actually check that that trailer
comes out to be 3 plus 2 I 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 zero you see
that's what went wrong with our integer
problem two x equals three the reason
that we couldn't solve it is that the
solve 2x equals three require 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
zero and the quotient of two complex
numbers is a complex number
except for division by zero 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 R 1 comma
theta 1 and R 2 comma theta 2 now again
remember from our argon diagram what it
means to say that the complex number has
polar coordinates R 1 comma theta 1
remember the complex number is written
in the form X plus iy the x coordinate
is R 1 cosine theta 1 the y coordinate
is R 1 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
you Lygia we wind up with R 1 R 2 being
a common factor the real part is r 1 r 2
times cosine theta one cosine theta 2
minus sine theta 1 sine theta 2 the -
coming because I squared is minus 1 the
imaginary part is R 1 R 2 sine theta 1
cosine theta 2 sine theta 2 cosine
theta one if we now remember failure is
a real number if we recall our geometric
trigonometric definitions cosine theta
one cosine theta two minus sine theta
one sine theta two is cosine theta one
plus theta two this expression is sine
theta one plus theta two and remembering
now that in polar form what we're saying
is what this is the complex number whose
magnitude is R 1 R 2 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 says
look at two multiply two complex numbers
viewed as 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 our has to
be positive that means you're measuring
in the direction of the negative x-axis
which means that the pole
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 than to multiply
these n factors we would simply by
induction so to speak multiply the N
magnitudes together and add the end
goals and a very interesting special
case is if all 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 nth power a very
interesting thing is is that our comma
theta to the nth 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
nth power is that complex number in
polar form 1 comma and theta and by the
way again remember what this thing means
to say at 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 pole
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 D moths
theorem in my high school was called de
Moira's theorem but it's the mob'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 I 2 sine theta cosine theta by the
mob sturm 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 m to be 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
five theta plus I sine 5 theta and wind
up with real identities in fact for what
sine and theta and cosine n theta for
any whole number value event see I'm
trying to hammer home the fact that
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 got too
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 leads to a very
interesting way of extracting roots of
complex numbers for example suppose I
want to find the sixth root of AI in
other words what complex number raised
to the sixth power gives I in fact is
there such a complex number you know
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 -1 there was no real number whose
square was -1 the question now is 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 call it X plus
iy which when raised to the sixth power
equals I in other words the sixth root
of 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 I Y to the sixth power
on the other hand I is written as zero
plus one I 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 -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 6 X to the fifth iy plus
15 X to the fourth I Y squared plus 20x
cubed iyq
gube plus 15 x squared I Y to the fourth
plus six X I Y to the fifth plus I Y 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 eyes to the best of
my ability
she's 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 of the
fourth is one I have the six is I fourth
times I squared which is I squared so
this just comes out as minus 1 etc 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 real 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 cut of polar
coordinates you see in polar coordinates
how would I write I I is what its
magnitude is 1 see it's the point 0
comma 1 and the argon diagram its
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 not only PI over 2 it could be 5 PI
over 2 9 PI over 2 etc these are 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
sixth root of I let's assume that the
answer can be written in polar
coordinates if I write in polar
coordinates the answer is r comma theta
oliver 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
and this is our karma theta to the sixth
power but the beauty of multiplication
and polar coordinates is that our karma
theta to the sixth power is R to the
sixth comma six data the magnitude is
this in other words you multiply the
magnitudes and you add the angles
therefore what this tells me is remember
our must be real there is only one real
number which when raised to the sixth
power is one and that's one itself see
the enth root see the nth root of any
positive number has exactly one real
solution that's why we can always find
the R in this case I picked a simple
example where R turns out to be 1 C
minus 1 also raised to the 6th power is
1 but that doesn't count because we
agree that our had to be positive I was
measuring the magnitude of the complex
number so our must be 1 by virtue of the
fact that our 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
etc the important point being that we
now take 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 1/2 pi K
over 6 PI over 3 K 62 like essentially
in terms of angles tack-on 60 degree
increments here to make a long story
short are 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 6 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 one unit from the origin
that means it's on the circle of radius
one 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 that one 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 you to raise this
to the sixth power we must raise one to
the sixth power that will still be one
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 are
supposed to be in other words to do this
thing backwards so to speak I know
that's that six 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 eight times I want 90 so
the angle would have had to been 90 over
eight and again I leave this for you as
exercise as 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 six times
notice that I come back what 75 times
six is 450 it means I go all
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 nine PI over twelve is
three PI over four 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 sign 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 the vision 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 will not close 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 they have
to do to 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 public
of this video was provided by the
Gabriella and Paul Rosenbaum foundation
help ocw continue to provide free and
open access to MIT courses by making a
donation at ocw.mit.edu/donate