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Complex numbers MA4001 Engineering Mathematics 1 Lecture 3 Polar form Dr. Sarah Mitchell Autumn 2014 We can see that cos x/r and sin y/r so that we can represent x and y as x r cos y r sin and thus z x iy r cos i sin. This the polar form of the complex number z. Modulus of z From Pythagoras theorem we know that r which is clearly real and x2 y2 zz The parameter r is called the modulus of z and is denoted by z. Note that if z is real z is the absolute value of z. Arguments of z The directed...

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This video is on complex numbers and this is the first of a series of video son this topic and IN#39’m going to explain this from the ground upstarting with the very basic ideas of what complex numbers are and why we need them and then go on to explain how we can use them to solve problems that we otherwisewouldn'’t be able to solve this aimed stat algebra 2 level of difficulty so you×39’re in algebra 2 class, and you need some help with complex numbers this video should be helpful or if you're Ina more advanced class and need to review this concept or clarify some ideas in your own mind about this concept this video should be helpful to let me start by saying that complex numbers despite the name really aren't all that complicated you just need to understand the basic idea, and I'm going to explain this from the beginning and IN×39’ll start with an example that shows where complex numbers come in handy or not just handy but are in fact unnecessary 2x squared plus 9 equals 5 take a look at this problem this is an ordinary algebra problem so let's try to tackle it using the basic techniques of algebra which basically says we can do pretty much anything we want to do to the equations long as we do exactly the same thing to each side, so it stays balanced now my goal here is to solve for X I want to isolate X I want to manipulate this equation, so it eventually says x equals something, so clearly I need to subtract9 from each side and the plus 9 and negative 9 cancel out that leaves me 2 squared on the left and on the right Have 5 minus 4 which is negative 4, and then I divide by 2 on each side and the twos cancel, and I'm left with x squared equals negative 2 and then to solve for to get X completely isolated I need to take the square root of both sides so on the left the square root of x squared is just X and on the right when I take the square root of negative 2 I end up with plus or minus done#39’t forget the plus or minus plus or minus the square root of negative 2 now at this point you might think ah that#39’s the problem we can'have negative numbers the radical and in algebra one classes we typically get to a situation like this, and we say that there's no real solution and the terminology here is meaningful when we say there's no real solution we mean their#39’s no real number that is a solution to this equationthere'’s no real number that we can plug inn like then think of this no real number that we can plug in right therefor X and end up with negative 2 as result it turns out there's no real number solution, but there is a complex number solution and when you think Oreal numbers just think of the number line so if you imagine 0 here and we putout little tick marks, but the numbers don't have to be right on the tick marks remember you could have the number 2 or the number negative 1 and 1/2 or the number pi any number on the number lines a real number and that#39’s why we Callie the real number line and that×39;basically the...

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