00:00Hi guys, okay, and this
00:01lesson I want to introduce
00:02you to complex numbers. Now
00:05the first thing I want
00:06to talk to you about,
00:07or I want to bring
00:08you back to the number
00:11sets. So let's go back
00:13to the natural numbers. So
00:17the natural numbers are the
00:18counting numbers. That is back
00:20in the day. People when
00:21they first created the invented
00:23numbers, they said, right, well,
00:25let's count how many, how
00:27many animals
00:28holes are over there in
00:28that field, or how many
00:30days is it gone since
00:32this thing happened, or how
00:34many children do you have?
00:35We need it to count.
00:37So this was the natural
00:38numbers. There's a little bit
00:40of debate whether zero is
00:41a natural number, but these
00:42are the whole numbers. Zero,
00:44one, two, three, four, five,
00:45six, seven, eight, nine, ten.
00:47True, whatever. But then, at
00:50some point, someone decides what
00:51hang on. If I have
00:53a situation like this, two
00:54minus five equals
00:56I don't have an answer
01:00that fits into my natural
01:03number set. So I need
01:05to expand the number system
01:07and they came up with
01:09integers. integers we use z,
01:13the next little line there.
01:16So these are the integers.
01:18And the integers are the
01:21whole numbers, but they include
01:23negative numbers.
01:24Then again, someone had another
01:26problem saying, hang on, not
01:28everything we deal with is
01:30a whole number, what if
01:30I have a half or
01:3423 .7 or whatever. These
01:38are not whole numbers, they're
01:40not integers, so I need
01:41to again expand my number
01:42system into, we'll create rational
01:44numbers, really. Use a queue,
01:46these are, this is a
01:49funny -looking queue.
01:52these are the rational numbers.
01:54So, uh, rational numbers are
01:56any number, any number, a
01:58number that can be written
01:58as a over b, where
02:00a and b are integers.
02:02So these are rational numbers.
02:04And then, so I say,
02:05well, hang on, not all
02:07numbers can be written like
02:08this. What about root two?
02:10What about pi? We can't
02:12write them like this. So
02:14again, I need something else.
02:16Then we came up with,
02:17I say, we, but I
02:19had nothing to do with
02:19it. We came up
02:20with real numbers, fantastic. And
02:24then what happened? Well, some
02:28mathematicians back in the 1600s
02:32were going around solving equations
02:34or whatever. They came across,
02:35well, let me give you
02:37a simplified version. They had
02:39an equation like this, x
02:41squared, x squared plus nine,
02:47x squared plus nine,
02:48equals zero. What's the solution?
02:52Well, we can say x
02:54squared is obviously negative nine,
02:56and then x has to
02:57be plus or minus, we
02:58like the square root of
03:00negative nine. But that's the
03:03problem. We can't do the
03:04square root of negative nine.
03:06So what do we do?
03:08Well, what they did was
03:11they came up with a
03:13way to solve this. They
03:16created what I'm sure many
03:19of you know, the square
03:20root of negative, the square
03:23root of negative one is
03:25equal to i. So they
03:27said let the square root
03:28of negative one equal i.
03:30And therefore the square root
03:31of negative nine would actually
03:33be three i because it's
03:35the square root of nine
03:36times the square root of
03:38negative one, which is three
03:40times i, that's three i.
03:42And the square root of
03:44negative five would be root
03:47five i so these were
03:50called imaginary numbers imaginary well
03:57they're not real so i
03:58guess they're imaginary someone i
04:00think it was actually day
04:01cart who all the famous
04:03when i was researching this
04:05all the famous mathematicians come
04:07up uh... or either gals
04:09i think it was the
04:10character who actually came up
04:12with
04:12imaginary numbers. Okay, so these
04:15are imaginary numbers. I, 3,
04:18I, root 5, I, 10,
04:19I, a million, I, whatever.
04:21If there's an I in
04:21it, it's an imaginary number.
04:25Now, what are complex numbers?
04:26Well, let me give you
04:27another quadratic to think about.
04:29Let's say I have x
04:31minus, let's go with x
04:35minus 3 squared plus 4
04:39equals 0. That's solve this.
04:40So I have x minus
04:423 squared equals negative 4.
04:46x minus 3 is equal
04:47to the square root or
04:49plus or minus the square
04:50root of negative 4. x
04:52is equal to 3 plus
04:55or minus from this 2i.
05:00So these 3 plus 2i
05:04or 3 minus 2i
05:08these are called complex numbers.
05:12Complex numbers, they have a
05:14real part and an imaginary
05:16part. Now the imaginary numbers
05:21are complex numbers and remember,
05:24well, maybe you don't remember,
05:26but all natural numbers are
05:28integers, all integers are rational
05:31numbers, all rational numbers are
05:33real numbers and all real
05:35numbers are complex
05:36So that's something that not
05:39everybody is aware of. So
05:41complex, let me draw that
05:42probably, is a C with
05:44a funny looking thing here
05:46like this. There. So the
05:51all real numbers are complex
05:53numbers. So for example, if
05:53you have the number six,
05:58it's just six plus zero
06:02I, it just has no
06:03imaginary part.
06:04when you have a real
06:05part and an imaginary part
06:06which can be zero, you
06:08have a complex number and
06:11similarly here that can be
06:12written as zero plus i.
06:15That is a complex number.
06:17Okay, that's the first kind
06:19of little bit of the
06:20introduction. The next thing I
06:21want to show you is
06:22how we display these complex
06:27numbers, how we can visualize
06:29them. So let me
06:33introduce to you something called
06:36argand diagram. So this is,
06:40I'm going to draw it
06:41here, an argand, argand diagram.
06:49So what an argand diagram
06:50is, is it's a set
06:54of axes like this. And
06:58I won't say the x
07:00-axis and the
07:01x's, but the real axis,
07:05which is the horizontal axis,
07:07and we just write real
07:08like that, or e, and
07:11the horizontal axis is the
07:13real axis, and the vertical
07:16axis is the imaginary axis.
07:19So the way we actually
07:22visualize or draw complex numbers
07:25is a bit like
07:29Well, once you study vectors,
07:30this might make a little
07:31more sense, but certainly you
07:33know how to draw or
07:34how to write a point
07:37and a Cartesian plane. So
07:39here, imagine I have three
07:40plus two i. The complex
07:42number, z is often, let's
07:44say I have z1 equals
07:46three plus two i. Let's
07:51say I have z2 is
07:52equal to three minus two
07:54i, these ones. Let's just
07:56make up a few other
07:56ones.
07:57to have minus five minus
08:01four i. And let's say
08:06I have minus, I don't
08:13know, three i. So where
08:17do we put these? So
08:17three plus two i. Let's
08:19just imagine that we have
08:21our numbers on the
08:25axis. So let's say I
08:27have one, two, three. So
08:29three plus two, either real
08:31part is three. So I
08:32go along three and the
08:33imaginary part is two. So
08:35I go up two. So
08:36this is Z one is
08:39equal to three plus two.
08:42I now you may see
08:43these written as points. Let
08:46me be a point in
08:47an argon diagram or you
08:50sometimes see these as vectors.
08:51Now again, you may or
08:52may not have
08:53studied vectors yet, but a
08:56vector is just like this.
08:59So it's a line that
09:00goes from zero. That's not
09:02what a vector is, but
09:03in this case, that's how
09:04I'm going to display. It
09:04just goes from zero to
09:06three plus two. I know
09:07this becomes useful when we're
09:10looking at the sizes of
09:12these complex numbers and different
09:15geometrical features that we can
09:18apply to them. So this
09:19is Z1, Z1,
09:21That two would be three
09:22minus two eyes, so three
09:24minus two eyes would be
09:25down here. This is z,
09:28two. Z three would be
09:31minus five minus four eyes,
09:33so minus five. Let's say
09:36minus three minus four minus
09:37five minus four eye, we're
09:40down here. This would be
09:42z, three. And then z
09:46four would be minus three
09:49eye.
09:49So let's say here because
09:52it's zero along the real
09:54part and down minus three
09:56i. So this would be
09:58z four. Now again, they
10:00can be points are vectors.
10:04So usually in the iB
10:05they're written as vectors. I'm
10:07going to draw these lines
10:08in here like this. And
10:12like so. Okay.
10:17So look, that is a
10:18very, very basic introduction to
10:21complex numbers. Of course, I'm
10:23going to do many more
10:23lessons where it gets more
10:25tricky. The next lesson I'm
10:27going to do is just
10:27looking at operations, how we
10:29add, subtract, multiply, divide complex
10:31numbers. And then it gets
10:33very interesting. It does, really
10:37does. So I'll see you
10:39in the next lesson. Make
10:40sure that at the very,
10:42very least, you understand that
10:44the square root of negative
10:45one
10:45is i and the square
10:47root of negative nine is
10:48three i and that we
10:51display complex numbers on an
10:53argon diagram like so.