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Hi again everyone. In this video we are going to continue our introduction to complex numbers
and in particular we are going to sketch a region in the complex plane. Now let us motivate our
study, why are complex numbers important? Well, complex numbers is basically an extension of
the real numbers you saw at school,and as I have mentioned in previous videos complex
numbers, although quite abstract, they find useful applications in many many places in science,
engineering and technology. So here I have listed a few, and basically a good understanding of
how to work with complex numbers, in this case how to sketch regions, well this knowledge then
gives us the power to solve more interesting, challenging and important problems. 00:00-00:59.
So todays video is very basic and we can look at this particular example:sketch the region in the
complex plane defined by all those complex numbers 'z' that satisfy the following inequalities.
Okay well, I am going to discuss the general case first and then we are going to solve this
particular problem. 01:00-01:18.
So suppose 'z nought' is a complex number, 'a' is a positive real number and α and β, (I guess
I really should have a less than here) and α and β are real numbers that are between -π and
π. Well, this will represent a wedge in the complex plane now I am using the term wedge very,
very loosely there. Let me give you a picture and you will see what I mean. So here is our axes.
Right so, first we move to 'z nought', so it is here, and we draw a circle around 'z nought' with
radius a. Okay alright, so what we would like to do is include all those complex numbers that
surround the point 'z nought' within this radius and let us bring this condition into play well, draw,
consider a horizontal line that is parallel to the real axis. We want to rotate, I guess, between
the angle α and β so we are just going to assume that α is negative in this case and β is
positive. So our region is somewhere in here. 01:19-03:25.
Now we have to be a bit careful because here I have got a strictly less than, so actually I am
going to put some dotted lines in there and the other places we have less than or equals to, so
we can actually include those edges. So here is our general region of interest, so we do not
include this edge here and you can see I have put in some dotted line there. Okay, well our
problem is similar but you see I have got a strictly less than here, so I would change this edge
to a dashed edge because I do not include that edge. So let us have a look at c and see what
we can do here now, just before we do that you can see, well if α is -π and β is positive π, well,
this will extend all the way around there and β will β will extend all around there so we
actually get a disc. So you can see how I am using the word wedge quite liberally there. so for
our problem 'z nought' equals 2i, a equals 1, α equals 0 and β equals 3 π/4. So let us construct
our wedge and put it all together. 03:25-05:06.
Alright, so let us go up to 2i and draw a circle around 2i with radius 1. So notice I am not going
to include the edge here so I am going to draw a dotted line. Okay, alright so let us look at our
angles now so in this case α is 0 so I can just go straight here because remember I draw what I
consider a horizontal line and I want to rotate. Okay so the first condition says that I do not do
any rotation. The second conditions I rotate around 3 π/4 radians, and because I have less
than or equals to I do not need to draw dotted line here. So basically I want to show you that
there is an angle π/4 radians down here. Okay, so where is our region? Well, it will be in here.
So we have not included this edge, we have included that edge but we do include this edge
here. Okay so let us look at the bigger picture. 05:06-06:53.
The first piece of information is that do not be daunted when trying to graph this complicated
regions. it is slow at first but but you need to work through the problem systematically and with
some practice you will recognize the mathematical expressions for regions in the complex
plane very quickly. Now a good idea, if you have time is after you come up with your region, test
one or two points in the region to see if they possess the desired property. So for example, I
could choose the points say, 2.5i which just lies there and then test these inequalities to see if
they hold just as a backup. 06:54-07:31.
Now I am going to leave you with a couple of examples that I want you to try. it is important
when you watch this video just do not watch the video and be passive and expect that you can
understand everything. The important way to understand mathematics is to do mathematics so,
have a go at these problems sketch the regions in the complex plane associated with these
conditions and enjoy! 07:32-08:00.