## 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!

3:21 am

My spec. textbook is so useless with this topic. Thank you for sharing this video

4:03 am

Thanks! If you find my videos useful then you also might like my new ebook, which is freely downloadable. The link is on my YT Channel (bookboon.com)

6:51 am

THANK YOU SOO MUCH!!!

12:59 pm

thnxx 4 d help… =))

3:52 pm

I am taking a course on Complex Variables in this Fall semester and I really liked your videos!

1:17 pm

Very helpful.

12:36 pm

Thank you!

10:57 pm

Download the free PDF for this lesson http://bookboon.com/en/introduction-to-complex-numbers-ebook

2:45 pm

this is great. thanks!

3:17 am

Cheers!

6:19 pm

You sir saved my life!!

10:44 pm

what kinds of class is this?

12:55 pm

alpha doesnt have to be negative, right? angle of the fan shape is ( beta – alpha ) and not alpha + beta, so i have to say … i am not happy about this

7:32 pm

bad video, you didn't explain why that particular region is the shaded region

1:55 am

2hrs b4 exam starts, first video on sketching regions on the Argand Diagram by UNSW lecturer lol.

6:33 am

Why pi/4 not 3pi/3

4:42 pm

Thankz Chris Brother

4:22 pm

Nice sir

2:24 pm

If we are going to calculate the area of the region, do we assume that the dotted line be the real line and then do the calculations?