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Video Source: www.youtube.com/watch?v=8gtnZ5xSLuE
Free ebook http://bookboon.com/en/introduction-t... • 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 • 74 • 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.
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