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Complex numbers are simply expressions of the form z = x + iy,

where x and y are simply real numbers.

For example, z = 3 + i * 5

would be a complex number.

Often that is just written as 3 + 5i, so

it doesn't really matter if the i comes before or after.

x, this first number right here,

is called the real part because there's no i attached to it, whereas y,

the second number with the i in front of it, is called the imaginary part.

Or, in short, x is Re of z, the real part of z, and y is Im of z,

the imaginary part of z.

The set of all complex numbers is denoted by a C, for complex plane.

This is like a C with a double bar in it.

It's a regular C with an extra bar in it.

That's the mathematical symbol for the complex plane.

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The real numbers are considered a subset of the complex numbers,

those for which that second parameter, that y, is simply equal to 0.

So a number of the form z = x + 0i is considered a real number.

And so the real numbers are kind of subsets of the complex numbers.

So our regular numbers, 3, 6, 7, they're all included in the complex numbers.

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And with that convention, we can then draw

a complex number z which is of the form x + iy,

where the coordinates are here the x coordinate,

the real part of z, and the y coordinate,

that's the imaginary part of z.

And thereby, the real numbers are actually sitting on that x-axis.

The numbers on the y-axis are called purely imaginary numbers.

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Suppose we have a number z which is of the form x + iy, and

we have a second number w of the form u + iv.

How do we add them?

So here we're defining the addition of complex numbers.

We're defining them by simply adding the two numbers that don't have any i's

attached to them, so the x and the u.

We add those two numbers, and so those two real numbers.

And then we add the y and the v and put the i in front.

And so our new complex number, our resulting complex number has a real part,

which is the sum of the two real parts, and has an imaginary part,

which is the sum of the two imaginary parts.

In other words, the real part of z + w is simply the real part

of z plus the real part of w, and the imaginary part of z + w is

simply the imaginary part of z plus the imaginary part of w.

Let me give you an example.

Suppose we were adding

the numbers (3 + 5i)

+ (-1 + 2i).

The way to do that would be simply to add the 3 and

the -1, 3 + -1 = 2.

And then add the 5 and the 2, and that's a 7i.

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So from here to here, that distance is exactly the same as from here to here,

which corresponds to the length u.

And on the other hand, the same is true on the y-axis.

So addition of complex numbers corresponds to a vector addition.

The modulus of a complex number is the distance of that number from the origin,

or the length of the vector from the origin to the point z.

So again, if I draw my complex plane,

and suppose I have a point z = x + iy.

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And this is the x coordinate and here's the y coordinate.

And suppose I'm asking, how far is z from the origin?

So I'm asking for this length right there.

How long is that?

And I'm going to call that the modulus of z, that length.

Well, I can see there's a right triangle right here.

And in order to find the length of the hypotenuse in that right triangle,

all I need to do is I square the lengths of the two legs.

But how long are those legs?

This leg right here has length x squared, well, that's the square of this leg.

This leg has length y, and so y squared.

And together that gives me the length of the hypotenuse squared.

So z squared is x squared plus y squared, or

in other words, z itself is the square root of x squared plus y squared.

Now how do we multiply complex numbers?

Here's the motivation.

Let's pretend for a second that they behave just like normal

numbers that we know from elementary school.

So suppose I have number x + iy and I wanted to multiply that with u + iv.

And suppose all the regular rules held,

so I can just distribute this out the way I'm used to.

Then I would get x times u by multiplying this

number with this number, plus ixv, from here.

And then I get iyu And an i squared yv.

So, we get exactly these four terms.

Now, if i indeed has the property that i squared is -1,

then this should really just be a -1 here.

Now, this is a motivation, so this is the thought that goes

into this following definition.

If this is supposed to be true, then the part of this number

that is revealed will consist of this xu because there's no i in there, and

this number right here, because i² is -1, so -yv is also a real number.

So, the real part of the product of these complex numbers should be xu, -yv and

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So, let's look at an example.

Suppose we wanted to multiply 3 + 4i by -1 + 7i,

so what I have to do according to this formula,

I have to take the x and multiply it with the u that gives me a -3.

Then, I have to take the y, just the 4,

multiply it by v, and put a negative sign in front of the products.

So, 4 times 7 is 28, -28, then I'm going to get the complex part by taking

x and multiplying it with v.

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Then, I simplify ( -3- 28 ) is -31 and (21- 4) is 17.

So, the result of this product is again a complex number, and it is -31 + 17i.

Now, one can check that with this definition

the usual properties of multiplication still hold,

even though we're now multiplying complex numbers and not just real numbers.

And those are, for example that z1 times z2 times z3, if you want to multiply

three numbers, it doesn't matter if you multiply the first and

the second first, then, take the result and multiply it by the third or,

you multiply the second and the third and then, the first one times that result.

This is called Associativity, you can move these blank piece around and

decide which multiplication get's executed first.

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Multiplication is also communitive which means it doesn't matter if

I write this factor first and this factor, or if I write this the other way around.

So in fact, this would be the same thing as (-1 + 7i) (3 + 4i).

You should check this out.

Get out a piece of paper and

apply this formula to this product and verify that,

indeed, you do get the same result, -31 + 17i.

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Finally, our multiplication is distributed.

So, if I have three complex numbers and I have to multiply Z1 by the sum of Z2 and

Z3, I could also multiply this out and multiply Z1 and

Z2 and multiply Z1 and Z3, and add them up.

That's called the Distributive Law, and all of these laws still hold.

Now, what exactly is i, that number i?

So far, it's kind of just a symbol that's been sitting there.

Well, i is certainly the same as 0 + 1i.

And so, I can now calculate i squared because i

is 0 plus 1 times i and the other i is also 0 plus 1 times i and

I can use the rule of multiplication that answers to find to find out what this is.

So, according to my rule of multiplication, I have to take this 0 and

that 0 and multiply them and that gives me this number right here,

then I have to take this one.

And that one, multiply them and put a negative sign in front of it.

That's, what I get right there.

Then, I have to take this 0, and that 1, and multiply them.

And finally, I have to take this 1, and

that 0, and multiply them, and these two form the imaginary part together.

So, 0 minus 1 is -1, and all the rest is 0.

So, i squared is indeed -1.

So the multiplication we define has the property that we wanted it to have,

i squared=-1.

So, what is i cubed?

Well, i cubed is i squared time i by the associativity we just realized,

it doesn't matter which one i multiply first, i squared is -1.

So, i cubed is -1 times i, so it's -i.

I to the 4th?

Well, I could write that as i squared times i squared.

Each of these i squared, as we just showed, is -1.

So that's -1 times -1, and -1.

What's i to the 5th?

Well, that's i to the 4th times i.

I to the 4th we just showed as 1, so it's 1 times i, which is just i.

I to the 6, well that is i to the 5th times i.

I to the 5th was i, i squared is -1.

And it keeps going, so what have we noticed?

I is i, i squared to the- 1, i cubed is- 1 before this 1.

And then, these four numbers keep repeating.

So i, i squared is -1, i cubed is- i,

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So, how do you divide complex numbers?

Suppose we're given a z = x + i y and a w = u + i v, so two complex numbers.

How do I divide z by w?

And of course I'm going to need w to be non 0, I can't divide by 0.

So, here's a great trick and I'll show that to you with actual numbers.

Z over w would be x plus i y over u plus i v.

Let's do this whole calculation with actual numbers.

Suppose z is 3 plus 4 1 and

so, suppose w is -1 + 7i and I wanted to calculate this quotient.

So, I wanted to calculate 3 + 4i and

divide that by -1 + 7i.

How do I do that?

What the idea is to multiply top and

bottom by this number u-iv, so almost the same as the bottom but

I'm multiplying it by the bottom number with the plus replaced by a minus.

So, in our example here I would

have still my top (3 + 4i) / -1 + 7i,

but now, I would multiply top and

bottom by- 1- 7 i.

So, it's always the same at the bottom.

But I'm replacing this plus here with a negative sign.

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Now, when I multiply this out,

on top I have a regular multiplication of two complex numbers.

So I'm going to get here, 3 times -1.

It's -3 Then 4 times negative 7 is negative 28, but I'm

going to have to make it a plus, because that's how the multiplication goes.

And then plus i times, and I need to multiply the 3 by the negative 7,

so that's negative 21, and the 4 by the negative 1, so minus 4.

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plus i times and here I have a minus one half.

It could have pulled the minus sign out and put it right here, but

this looks exactly like our complex numbers look.

In other words,

when I divide complex numbers, the result is again a complex number.

And the trick we used was to make the denominator real.

And how did we do that?

We make the denominator real by multiplying the denominator (u

+ iv) by (u- iv).

We also multiply the numerator by that so that we would have equality.

So we just multiply top and bottom by (u- iv) but

the trick was that that made the imaginary part cancel out.

And so the denominator became purely real, so that we could then separate real and

imaginary part of the result.

So here's the result.

It's a formula for the quotient of two complex numbers.

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In particular, if we just wanted to calculate 1 over z, so not z over w, but

just 1 over a complex number is 1 / x + iy and again we would do the same thing.

We would multiple top and bottom by x- iy and if we do that and

the bottom the imaginary parts would cancel out and

we would just end up with x squared + y squared.

That's unless, of course, z is not 0.

So, 0 plus 0i would be i.

Couldn't be dividing by that.

That this is 1/z.

So this quantity x- iy were we just replaced the plus with the minus,

that was really important in the previous calculation.

It is so important that we give it a special name.

So if z equals x + iy is a complex number, then we call that

complex number would be replaced the plus with a minus the complex conjugate of z.

And it's denoted with a z with a bar on top of it, do you see that here?

So zbar is the complex conjugate of z.

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So the complex conjugate is the number z, but

it's reflected with respect to the x-axis.

Now here's a property of complex numbers.

If I take the complex conjugate of a sum of two complex numbers,

I might just as well take the conjugate of z first and then the conjugate of w,

and then add those two numbers up.

It's a property one easily verifies.

The absolute value of the conjugate is the same as the absolute value of Z.

We can kind of see that in my picture, right here.

But this length right there is the length of Z and down here

is the length of the conjugate of Z and because this is just a reflection of that.

These, of course, have the same length.

Large as you see, in my picture.

But because this is a reflection they have the same length.

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Now here's a really neat property.

Z times its conjugate.

That's what we were making off in the calculation of the quotient

This x plus iy times x minus iy.

When I multiply through, I get x times x, which is x squared.

I get y times minus y, which is minus y squared,

but together with a negative sign that I get from the i squared, so plus y squared.

And I get x times minus iy, and x times plus iy.

That cancels out.

So, I get x squared plus y squared.

Which is their length of z squared?

So z times its conjugate is actually the length of z squared.

And that means we can lead by the formula for 1 over z.

1 over z now becomes, multiply the top and bottom by z conjugate.

So z times z conjugate, and in the numerator of z conjugate and

the denominator simplifies to length of z squared, so

1 over z is the conjugate of z divided by the length of z squared.

Now when is it the case that the conjugate of the number is

equal to the number itself?

We learned that the conjugate is simply a reflection with respect to the x axis.

So when is the reflection with respect to the x axis the same as the number itself.

So when is this number equal to its own reflection?

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Now here's another interesting thing to note.

If you add a number and

its complex conjugate, what you end up doing is (x + iy) + (x- iy).

So you add the x and the x, you get 2x, and

then you add the complex parts, the iy and the -iy, they cancel each other out.

So you get something real.

In other words if I divide this equation by two I get the x which is the real

part of z.

Is equal to Z plus its conjugate over two.

So we find a nice formula for finding the real part of a complex number.

We add the number to its conjugate.

And similarly if you take a complex number and

subtract the conjugate then you get two I times the imaginary part.

So if you want the imaginary part by itself,

without the I, you're going to have to divide by 2i.

So this is also a nice formula.

Now you can verify that the length of z times w is the length of z times

length of w.

So length in multiplication behave well with respect to each other.

You can also verify that the conjugate of

Z over W is the conjugate of Z over the conjugate of W.

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Then this is the real part of Z, this length right there.

It's the real part of Z.

Whereas this length right here is the normal, the length of z.

And this length right there is the imaginary part of z.

So this is the real part of z and that's the imaginary part of z, and

since they form a triangle I see that this length here is certainly greater than or

equal to either one of those two lengths.

And that's what's expressed here.

The real part of z is less than or

equal to the absolute value of z and the same is true for the imaginary part.

And because that's true in absolute value we get the other

inequality easily as well.

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And if I look over here,

this length here that was the length of z, and here I see the length of w.

Because it's the same vector as this one down there.

And again these form a triangle,

my triangle didn't really meet very nicely over here.

These three lengths form a triangle and

the shortest path from the origin to z+w is certainly this diagonal and

the slightly longer path is to go along z first and then along w.

So z+w is greater than or equal to the direct path z+w.

That's what the triangle inequality says.

There's a reverse triangle inequality that follows from the triangle inequality.

It's |z- w| is greater than or equal to |z|- |w|, won't go into much detail there.

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Finally, a little outlook into the future.

This is the fundamental theorem of algebra, you may have heard of it.

Suppose we have complex numbers a0, a1 through an, and

this an, this last one, is a nonzero number.

If you look at this polynomial, so for example,

if you'd like to have an example in mind,

you could look up the polynomial p of z equals 5 z

cubed plus 4 z squared minus 2 z plus 14.

So in this case, 3 would be the largest n that we have here,

so a3 would be 5, a2 is 4, a1 is negative 2, and z0 is 14.

So this is called a degree 3 polynomial, and

this theorem says that such a polynomial has n roots.

So in this case it has three roots in Cs of 3 ps for

which p of z can be equal to 0.

And it can be factored as an times Z minus the first group,

Z minus the second group and so forth.

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We'll be able to prove this theorem later on in this course.

Now, in R this was totally not true.

This is a special fact about complex numbers.

For example, look at polynomial p (x) = x squared + 1.

So we have a degree two polynomial, but we're only considering it for

real numbers and not for complex numbers.

This never, even crosses the x axis.

So, if I graph this polynomial x squared plus one is always above the x axis.

It never crosses the x axis, so it's never equal to zero and so

we can't factor it either, it has no real roots.

But, if you looked at that as a polynomial in C.

So z squared + 1 instead of x squared + 1, and we're allowed to

not only plug in real numbers for the x, but also complex numbers.

Then I can find two complex numbers, namely i and negative i, for

which this term right here is actually equal to 0.

i squared plus 1 is 0 and minus i squared plus 1 is also zero.

[INAUDIBLE] And so it actually factors.

So this is a theorem about complex numbers and

we will be able to prove this later on using complex analysis in this course.