How to sketch regions in the complex plane


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. 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. 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. 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. 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. 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. 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!

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