Hello everyone. Welcome to Week 9 part 2 of Carmen's Core Concepts,
my name is Carmen Bruni,
and in this video series, we talk about the weekly concepts in Math 135.
So part 1 of Week 9, that was RSA.
In part 2, we're going to talk about complex numbers.
Okay so what is a complex number? So a complex number
is an expression of the form x plus y i, where x and y are real and
i is the imaginary unit.
So i on its own has no meaning. i is just
an object, okay?
Denote the set of complex numbers by the set
as follows: x plus y i where x and y are real numbers.
Some examples are 1 plus 2i, 3i, square root of 13 plus pi i,
2, which is the same as 2 plus 0i.
So something to think about is that the real numbers are
contained inside the complex numbers,
so are the rational numbers, so are the integers and so on and so forth.
So a lot of people
when they first see complex numbers, they define i as the square root of minus 1.
I'm not going to do this and there's a lot of good reasons why we shouldn't be doing this,
and one of the reasons is that,
that doesn't really make sense, right? I mean when we
say like the square root of 4 what we mean is the
positive number such that
when I multiply it by itself I get 2…
or I get 4, sorry, which is 2.
But I don't want to define i like that
because it's not a positive number, it’s…
it's an object, okay?
So if all I'm doing is saying that i is some object,
then what am I going to do with this? Well we'll see.
At the moment, right now, C is just a set of things, it's formed…
this plus sign is actually - has no meaning as of now, okay,
it's just really a symbol.
It's denoted by x with the plus symbol
and then y times i, okay?
Now what we're going to do is we're going to give the plus sign some meaning,
and we're going to give multiplication some meaning.
So we're going to turn the complex numbers into a ring.
The last note: 2 complex numbers, z equals x plus y i and
w equals u plus v i are equal if and only if x equals u and y equals v.
Let's turn C into a ring and a field, okay?
So we turn C into a commutative ring by defining
an addition and actually a subtraction operation,
and how do we do this? So we're going to define x plus y i
plus or minus u plus v i to be
x plus or minus u, as a real number,
plus y plus or minus v, as a real number, times i.
And that way we get addition and we get subtraction.
Notice that this also defines the additive inverse, right?
So we have the addition structure, and the multiplication structure:
x plus y i times u plus v i, that's going to be defined to be
x u minus y v
plus x v plus
u y all multiplied by i.
That is the structure that we are giving our set C,
and this will make it a commutative ring. Okay?
Something to note now, this multiplication operation above,
well if we had 0 plus i times 0 plus i, we can see that
the left hand side would be denoted by i squared,
and the right hand side would tell us that
the value is minus 1.
So this object that we defined with these two addition and multiplication operations,
tell us that this new object, i, has the property that i squared is minus 1.
So something to note is now that we have…
we now have solutions to the polynomial, let's say,
x squared plus 1 equals 0 over the complex numbers. The solution is i,
actually it’s plus or minus i, both of those are valid.
So something to think about, okay?
So this is one of the reasons - oh I didn't mention the motivation
for defining complex numbers,
but one of the motivations is to try to give algebraic expressions…
roots.
So to try to give all algebraic expression solutions,
and we'll see this in the upcoming weeks why this actually does this.
Now that we know that i squared is minus 1, we can actually remember this
operation here, the multiplication operation, just
by multiplying it out how we would before.
We would multiply x times u, and then we would take x times v i and then
y i times u and then y i times v i.
And if we do that, we actually see that the…
under the…
under the new fact that i squared is minus 1, we now see that
this y v i squared goes back to the
x u minus y v,
and then x v plus u y i is the second term there.
So here we see that multiplication is actually something that we know here,
and addition is not too hard either. It’s basically common factoring the i.
So addition and multiplication are actually very natural operations here.
We note that C is a field, so how do we note that it's a field? So
we know that the inverse of any non-zero complex number,
x plus y i, is actually defined as x over
x squared plus y squared, minus y over x squared plus y squared, times i.
If you multiply this by x plus y i,
you will see that the answer is 1.
I will leave that as an exercise though for you to check out.
This is the hard way to think about the inverse. We'll learn in a bit
that we can look at inverses using
an operation called the modulus of a complex number,
and that will help us to remember the inverse formula here.
But for now, let's just - I'm just going to state it
and I'll leave it as an exercise for you to check,
that if I multiply by x plus y i I get 1.
Complex conjugation, this is a very useful operation but…
before - I mean if you just look at it, it seems very simple.
A complex conjugation of a complex number z equals x plus y i
is z bar, which is defined to be x minus y i. So we just
negate the imaginary component.
So if you think about this graphically, if you have a vector…
or a complex number in the
real-imaginary plane,
the complex conjugate is just its reflection about
the line y equals x, so something like that is what it would look like.
Properties of conjugates, these are very easy to prove
so I'm only going to prove a couple here. They're not hard,
one way to do this is to just introduce coordinates, so write z as
x plus y i, write w as u plus v i
and then just go through the operations. So add them,
then take the conjugate, is the same as taking the conjugate and then adding them,
and so on and so forth.
What's this one supposed to be? Is it z z bar? Or z bar…
ah there should be a multiplication sign here, but we can't see it.
The conjugate of z times w is supposed to be the
the conjugate of z times the conjugate of w.
I'll be sure to correct this on the thing. I think there's a…no, I don't know.
I mean there should be a multiplication sign, that's really what we want.
If I do the process of complex conjugation twice, I get z.
z plus z bar, so z plus its conjugate, is 2 times the real part,
and z minus its conjugate is 2 times the imagin- 2i times the imaginary part.
So again, all these can be done with just introducing coordinates.
So here's an example of 3, for example: z bar bar is
x plus y i bar bar, which is the bar of x minus y i,
which again - so if we're taking the minus sign and negating it,
then we just get plus, and that's the same as z.
Similarly with addition and subtraction.
So if you introduce coordinates, it's not too hard.
By coordinates here, I mean the x plus y i.
And this is what we call the standard form of a complex number.
We'll see at the end the polar form is coming up.
So that’s properties of conjugates here.