Engineers work in the real world, civil engineers designed
buildings and bridges. But it turns out that some
of the computations that engineers do can make
use of complex numbers. We'll see that in
the differential equations course, that to solve certain equations, to find the real solution, the easiest path is to go through
the complex number space. So, in this video I want
to teach you the basics of complex numbers that you'll need say in a differential
equations course. What is a imaginary number
or the imaginary unit? The imaginary unit we call that i is the number that
when you square it, you get minus one. So, in fact there are two numbers
that when you square it, you get minus one. That's i and minus i. In a real expression, when you use complex numbers, typically, you'll end
up with solutions that contain both i and minus i. Okay. So, symbolically, we just write I is equal to the square
root of minus one. That just means that I squared
is equal to minus one. And then we represent
a complex number z as a real number i plus i
times a real number y. So, x and y are real and i
is our imaginary unit here. There are functions associated with complex numbers that we haven't
seen before with real numbers. One of them is called
the complex conjugate. We write that as z bar
that's equal to x minus iy, that's called the complex conjugate. All you do is in an expression
with real numbers and i. You form the complex conjugate
by changing i to minus i. These two other functions, there's the real part
of z that's equal to x. The part of z that
doesn't have i in it. So, x is the real part of z. You see you can form
x using z and z bar. You can add z and z bar
and the iy cancels, and you end up with 2x. So, this is a linear combination
of z and z bar is equal to z plus z bar divided by two. Okay and that's real. That's the real part of z. The other function then is
called the imaginary part of z. We write that as IM for
imaginary part of z. That's equal to y a piece
that multiplies i. Then how do you get the imaginary
part of z from z and z bar? You can say z minus z bar
so the x is eliminated. So, that's z minus z bar. Then you get 2iy. So, you have to divide by 2i. Okay. z and z bar are useful in
one more way if you multiply them. So, if you take z times z bar, that would be x plus
iy times x minus iy. So, the cross term cancels. So, you would get x
squared minus iy squared. But that's simplifies. So, when you have
an expression and you have an I squared in the expression, I squared becomes minus one. So, here we have an I squared in the expression and that
simplifies to x squared plus y squared which is the real part of z squared plus the imaginary
part of z squared. Okay, you can add complex numbers
and get a complex number. You can subtract
complex numbers you can multiply complex numbers
and get a complex number. In that case, you have to take care that i squared equals minus one. I cubed then would be minus i, i to the fourth would be one. So, you just use the principle
that i squared equals minus one. You can also divide complex numbers. Division of complex numbers
requires a trick. So, let's say you have a
two complex numbers z and w. If you wanted to divide z by w. The trick you use is that you want the denominator then to
become a real number. So, you multiply by w bar in
the numerator and the denominator. Okay. I think there is one
more point here that I should make. So, say we have two complex numbers
z and w and they're equal z or z can be a function that is a function that has
complex expressions in it. W can be another function
with complex expressions. If you have these two objects
z and w that are equal, this is a complex expression, but very nicely this is equivalent
to two real expressions. So, z equals w is equivalent to the real part of z equals
the real part of w and the imaginary part of z equal to the imaginary
part of w. So, we'll see that when
we study differential equations that we may have a differential equation that we write as a complex
differential equation, a differential equation
for a complex function. But, that's actually equivalent to two differential equations
for two real functions. The real functions would
be the real part of the complex function and the imaginary part of
the complex function. Okay. These are the real fundamentals of complex numbers in
terms of their algebra. We have to think about one
function though that's very important in the differential
equations course. That's the exponential function. So, let's look at e_i Theta. Okay. So, what is this function? So, theta is some real angle but here i is the square root
of negative one. So, we have to define this
exponential function in some way. The proper way to define this
is through the Taylor series. The Taylor series you can
show in a math course. The Taylor series is convergent
for all arguments here. So, we don't have to worry in
this course for engineers about convergence but we can divine
that through the Taylor series. So, if you remember from
your calculus course the Taylor series for the
exponential function would be one plus i Theta plus i theta
squared over two factorial, plus i Theta, cubed
over three factorial plus i Theta_fourth over
four factorial etc. So, that's basically
the definition of e_i Theta through the Taylor series. Then we want to write
that as a complex number. So, that means we
want to write that as a real part plus i times
an imaginary part. So, what is the real part here? One is real. i Theta, Theta is part of
the imaginary part then. i Theta squared over two factorial, that has an i squared, i squared is minus one. So, we can because
that third term then becomes minus Theta squared
over two factorial. The fourth term has an i cubed, i cubed is i squared
times i is minus i, as part of the imaginary term, and then the fourth term
has an i to the fourth. So, i to the fourth is minus one
times minus one is plus one. So, we have a plus
Theta to the fourth over four factorial, et cetera. So, this series is the real
part of e to the i Theta, plus i, times the imaginary
part of e to the i Theta. So, we have an i times Theta. So, the first term here is Theta. We have an i cubed, which is i squared
times i is minus i. Then, we have a Theta cubed
over three factorial. So, we have a Theta minus Theta
cubed over three factorial. Then, the next term here
will be Theta to the fifth over five factorial, et cetera. Okay. Now, you have to think
back to your calculus course. This is a Taylor series
expansion for cosine Theta, and the second expression here, series here is a Taylor series
expansion for sine Theta. So, we have the very
important formula here, that e to the i Theta is equal to
cosine Theta plus i sine Theta. A connection between
the exponential function and the trigonometric functions. Extremely useful, and particularly in
our differential equations course, we'll be using this all the time. There's a famous formula
that comes from this. It's called Euler's identity. If you put Theta equal to Pi, then we have cosine Pi equals one, sine Pi equals zero. So, cosine Pi equals minus one, sine Pi equals zero. So, again, e to the i Pi
equals minus one, and the famous expression
which maybe you'll see on tee-shirts is I'll
write it down here, e to the i Pi plus one equals zero. Okay. Euler wrote down this expression for King and
this is how the story goes, and said that this proves
the existence of God. Why is this such a
beautiful expression? In one expression, we have one, two, three, four,
five key numbers of mathematics,. The zero, the one, and the i, and then we have e and Pi, the two most important, what we say are
transcendental numbers. Then, we have one addition, we have one multiplication, i times Pi, and one exponentiation, e to the i Pi. A beautiful expression. Okay. One more point I want to
make about these complex numbers, we can put them in what's
called the complex plane, we can draw a graph. Let me put it here. We can say the x-axis
is the real part of z and the y axis is the
imaginary part of z, and we can put a complex number z. So, let's put it here. So, that's our complex number z, and I can write that as x plus iy, and draw a vector from the origin
to that complex number. Okay. If you remember
your polar coordinates, then we have an angle here Theta, we have a length of
this vector r. Okay. What are the polar coordinates
of this complex number? We have x equals r cosine Theta, and we have y equals r sine Theta. Putting it together, then
we have z equals x plus iy. We factor out an r, we have cosine Theta
plus i sine Theta. Then, we have
our wonderful expression, e to the i Theta, is cosine Theta plus i sine Theta. So, this becomes re to the i Theta. That's called the polar
representation of z. So, z we can write as x plus iy, or we can write it as
re to the i Theta, r is this length, Theta is the angle it
makes with the x-axis. Also, very useful in computations. So, let me review. We're introducing
the imaginary unit here, the number that when you square it, you get minus one. Symbolically, everyone writes i equals the square root of minus one. We have a complex number z equals a real number plus i times
the imaginary number, and then I define a function which is called
the complex conjugate of z, z bar is x minus iy. You just change i to minus i. We have the real part of z is x, the imaginary part of z equals y. We can get the real part of z from a linear combination of z and z bar, and we can get the
imaginary part of z from a different linear combination. Remember, the real part of
z and the imaginary part of z are both real numbers. If we multiply z and z bar, we also get a real number, which is x squared plus y squared, which is actually r, the length of this complex number
in the complex plane. Addition, multiplication,
subtraction are straightforward, if we use i squared equals minus one. Division has a trick to do, we make the denominator real by multiplying by its complex conjugate. Often, we have a complicated
complex expression, z equals w. But don't despair, these are just two real equations. The real part of z equals
the real part of w, and the imaginary part of z equals the imaginary part of w. Finally, we've defined what
it means to be e to the i Theta through
its Taylor series. When we do that, we see that e to the i Theta is cosine
Theta plus i sine Theta, a very useful expression. Our first use is here, where we say that
a complex number x plus iy can also be written as
r times e to the i Theta. I'm Jeff Chasnov,
thanks for watching, and don't be afraid
of complex numbers, they are very useful.