So, let's start out with what the definition of complex number is. J is equal to the square root of minus 1. A lot of other applications, especially in math, they use i to represent the square root of minus 1. We use J in electrical engineering because we tend to use i for current. So J is equal to the square root of minus 1. That means J square is equal to minus 1. Now, we're going to see 1 over J come up a lot, so we want to find a simple form for that. So 1 over J, we can actually come up with a way of simplifying this. I'm going to multiply by j over j. That gives me j in the numerator and j squared in the denominator. Well that equals -1. So this comes out to -j. So, as something off to the side, I'll write this as 1/j = -j. We use this a lot, so remember that. Now, if I take a complex number, I want to be able to look at it visually. And we do that by plotting it as a vector. The imaginary part along this axis, and the real part along this axis. So if I have, the real part is a and the imaginary part is b, then this vector has a magnitude along here, b, and a magnitude along the real axis of a. When I look at this in rectangular co-ordinates, I have a + jb. In polar coordinates, I would want to find the magnitude of this. So the magnitude of this vector is A squared plus B squared. The square root of that. So I'm going to call that the magnitude of C, where the magnitude is just a squared plus b squared, square root of that. And you can look at this from a trig point of view. So this is essentially a triangle with this value of B, and this side value of A. So, whatever you do with Trig Identities, you can use here. So the magnitude here is A squared plus B squared. The angle Is the arc cosine. There should be an angle right there, theta. In polar coordinates we've got a magnitude in angle theta. This angle theta is right here. That's equal to the arctan of the imaginary part over the real part. For example, if I've got this value here. This complex number three plus j b. The magnitude is going to be three squared plus four squared. The square root of that. that is equal to five. And then the data is equal to the arc tan of the imaginary part, which is four over three, and that equals 53.1. Going from polar to rectangular, again, go back to this and look at trig identities. If I'm given a value A, so this length is A, and I want to convert it back to rectangular, then it's going to be A times C cosine of theta. To give me a, this little a. And Asin(theta) to give me this b. For example, if I've got a complex number in polar form, 10 at an angle of 30, it's going to be 10 times the cosine of 30 plus j times. 10 times the sine of 30. If I add complex numbers, say C1 and C2, then I add their real parts and their imaginary parts separately. So that's the real part gets added and the imaginary part gets added. So the real parts added and the imaginary parts. So, for example I've got ten plus j times two plus ten minus j times two, I add the real parts I get ten, imaginary parts I get minus two. If I want to subtract two complex numbers right here, I do the same thing, I combine the real parts And I combine their imaginary parts, right here, real parts, imaginary parts. So if I want to multiply complex numbers, I actually have a choice. I can do it in rectangular form or I can do it in polar form. If I want to do it in rectangular form, so my complex numbers are written in this sort of form, then I I'd do it as if I'm multiplying polynomials. So for example, I have a times d, which is right there. And then I've got an a times jf, which is right here, jaf. And then I've got a j times bd, so that's j times bd And then I got this final term right here, which is J squared times BF. Well J squared is minus 1 so that's minus bf, which is right there. In polar form it is actually a lot easier to multiply things because I have a Something in this form and this form, I multiply their magnitudes so I just multiply their magnitudes and I add, that should be add, I add their two angles together. So for example if I've got these two expressions in polar form then it's 10 times 4, 40 And then 30 plus a minus 20. So this is 30 plus a minus 20 I add their angles. Now something that's very useful is what we call a complex conjugate which is where every c of j you replace it with a minus j. So this is because and we denote that is a star, so this is a complex conjugate of this number. And if I multiply these two together, I could go back to this form and derive this and show that if I use rectangular multiplication. That if I multiply this value times its complex conjugate, then I get the magnitude squared. This is equal to the magnitude squared. So looking at this example, if C is equal to 10 plus 2 J the magnitude of C is equal to The real part squared. 100 plus the imaginary part squared, which is four square root. So multiplying these together I get c squared, which is just 104 squared. Now dividing I have a choice again. I can divide using rectangular coordinates or polar coordinates. Rectangular coordinates, to divide I have to multiply through the numerator and the denominator by the complex conjugate of denominator. So if I multiply the complex conjugate of the denominator I get the magnitude of the denominator squared. So it's a real part squared plus the imaginary part squared. Right here. And then the numerator, I just have to multiply through this out as we showed above. It's actually easier to divide in polar coordinates because I divide the magnitudes. And I subtract the angles. So, for example, if I've got 10 at an angle of 30 divided by 2 of an angle of 10, I divide the magnitudes, which is 5. I subtract the angles, which is 20. So some final comments here, if I'm adding or subtracting, you must convert to rectangular coordinates. If I'm multiplying or dividing, It's usually easiest to convert to polar, but you could do it with rectangular. For example, if I have something like this, it's kind of messy form. And I want to simplify it? Well, let me look at the numerator first, because I'm adding these two expressions. I have to convert it to rectangular Coordinates first and then I add the phasors. And then I have something that's in rectangular coordinates in the numerator. So I have a rectangular coordinates over rectangular coordinates. At that point I've got a choice. A or B. One is, I can convert everything to polar And then divide or I can multiply the numerator and the denominator but complex conjugate of the denominator and divide that way. Okay so this is quick Tips on complex numbers, these are to give you the basic rules for how to manipulate them, but the best thing for you to do is get a lot of practice with actual numbers. Thank you.
