WEBVTT mathematics/algebra-2/eaton 00:00:00.000 --> 00:00:01.900 Welcome to Educator.com. 00:00:01.900 --> 00:00:10.500 Today, I will be introducing the concept of imaginary and complex numbers. 00:00:10.500 --> 00:00:21.500 The concept of imaginary numbers is tied in with square roots; so we are going to begin by a review of some of the properties of square roots. 00:00:21.500 --> 00:00:38.800 Recall the product property: the product property and the quotient property--both of these properties can be used to simplify square roots. 00:00:38.800 --> 00:00:54.400 For numbers a and b that are greater than 0, if you have something like this, √ab, that can be broken down into √a, times √b. 00:00:54.400 --> 00:01:02.200 And this allows us to simplify; for example, if you have something like √32x⁵, 00:01:02.200 --> 00:01:08.300 it is not in simplest form, because there are still some perfect squares here under the radical. 00:01:08.300 --> 00:01:16.400 So, the way I can simplify this is to rewrite this as the perfect squares times the other factors. 00:01:16.400 --> 00:01:25.100 Let me rewrite this as the perfect square, which is 16, times 2; so the perfect square of 16 times the other factor, 2. 00:01:25.100 --> 00:01:33.200 And then, for the variable, I have the perfect square, x⁴, times x. 00:01:33.200 --> 00:01:48.100 Now, what this product property allows me to do is then separate out everything like this. 00:01:48.100 --> 00:01:58.300 Once I have that, I can take the square roots of the perfect squares: I have the square root of 16 (is 4); the square root of x⁴ (is x²). 00:01:58.300 --> 00:02:06.000 And then, I am left with 2x, the square root of 2 times the square root of x. 00:02:06.000 --> 00:02:14.300 And by going the other way with the product property, I can put those back together and say what I end up with is 4x² times √2x. 00:02:14.300 --> 00:02:17.500 And this is now in simplest form. 00:02:17.500 --> 00:02:26.700 The quotient property is the same idea, but with division. 00:02:26.700 --> 00:02:34.800 If you have something such as √a/b, you can break that down into √a/√b. 00:02:34.800 --> 00:02:58.100 So again, if I have something such as, let's see, √49x⁴/25, I know that this is equal to √49x⁴/√25. 00:02:58.100 --> 00:03:05.500 This allows me to simplify these into 7x²/5. 00:03:05.500 --> 00:03:12.600 So, we are going to be using these properties of square roots in today's (and in future) lessons. 00:03:12.600 --> 00:03:25.900 OK, a new concept--imaginary numbers: in Algebra I, when we came across something such as √-4, we simply said that it wasn't defined. 00:03:25.900 --> 00:03:29.200 And it is not, if you are looking only at the real number system. 00:03:29.200 --> 00:03:36.400 However, there is another number system called the complex number system; and part of that includes imaginary numbers. 00:03:36.400 --> 00:03:49.800 And what these do is: this allows us to find a solution to something like this. What is this? 00:03:49.800 --> 00:03:58.400 Before we go on, let's look up here: what an imaginary number says is that i equals the square root of -1. 00:03:58.400 --> 00:04:09.000 Looking at that another way: if i equals the square root of negative 1, let's say I square both of these: i² is -1. 00:04:09.000 --> 00:04:14.600 And this is a concept that is going to become important later on, and we are going to use this. 00:04:14.600 --> 00:04:21.900 So, just recall that i² equals -1, and that i equals √-1; both of these are very important. 00:04:21.900 --> 00:04:35.200 So, let's think about an equation such as this: x² = -4 (related to this). 00:04:35.200 --> 00:04:45.700 We can find a solution to this by saying, "OK, x equals the square root of -4"; previously, we couldn't. 00:04:45.700 --> 00:04:57.800 And the reason is: I can break this out into the following: x = √-1(4). 00:04:57.800 --> 00:05:12.700 Using the product property that we just learned, I can say, "OK, x then equals the square root of -1 times the square root of 4." 00:05:12.700 --> 00:05:16.700 This part is easy to handle: I know that the square root of 4 is 2. 00:05:16.700 --> 00:05:26.400 Now, I have a way of handling this: I go back, and I say, "OK, the square root of -1 is i; I have something to define this as." 00:05:26.400 --> 00:05:35.400 So, since the square root of -1 is i, I can simply say that this is i, or x equals 2i. 00:05:35.400 --> 00:05:48.600 Whereas before I couldn't solve this, now I can, because I can pull out this -1 using the product property, and define that as the imaginary number i. 00:05:48.600 --> 00:05:51.500 So, that is what this is saying up here. 00:05:51.500 --> 00:06:04.400 For the positive real number b, the square root of -b² equals bi. 00:06:04.400 --> 00:06:22.800 For the positive real number in this case, it would be 2: b = 2; the square root of -2² is bi; and that gives me this. 00:06:22.800 --> 00:06:33.000 Looking at this using another example: let's say that I am asked to find x = √-9. 00:06:33.000 --> 00:06:43.200 Again, using the product property, I know that this is equal to -1 times 9. 00:06:43.200 --> 00:06:48.000 And the product property allows me to break this up as such. 00:06:48.000 --> 00:06:57.100 This is easy; I know that the square root of 9 is 3; now I have a way to define this as i; √-1 is i, 00:06:57.100 --> 00:07:04.800 so I am going to say this is i3, which is usually written as 3i. 00:07:04.800 --> 00:07:09.400 OK, so when you are in a situation where you need to find a negative square root, 00:07:09.400 --> 00:07:16.500 the thing is to use the product property to pull this square root of -1 out, and then you just need to find the positive square root. 00:07:16.500 --> 00:07:23.700 And the answer is the imaginary number bi. 00:07:23.700 --> 00:07:29.000 Now, I mentioned that imaginary numbers are part of a different number system: we have the real number system, 00:07:29.000 --> 00:07:33.100 and we also have another number system called the complex number system. 00:07:33.100 --> 00:07:43.800 And a complex number is in the form a + bi; and there are two parts to this--a real part and an imaginary part. 00:07:43.800 --> 00:07:55.500 So, this is the real part; bi is the imaginary part; and they are complex--they have two parts. 00:07:55.500 --> 00:08:05.100 For example, I could have 5 + 4i; this is a real number; this is an imaginary number; together it is a complex number. 00:08:05.100 --> 00:08:14.600 Or 8 - 3i: again, the real number is 8; the imaginary is -3i. 00:08:14.600 --> 00:08:29.400 Now, looking a little deeper, let's say I have b = 0; then, what I am going to end up with is a + 0i, which is just a. 00:08:29.400 --> 00:08:34.700 So, this is just a real number; so the real numbers are part of the complex number system. 00:08:34.700 --> 00:08:45.700 For example, if I were to give you 8 + 0i, well, this is going to drop out; and this is just 8, and it is a real number. 00:08:45.700 --> 00:08:49.000 So, when b equals 0, you just end up with a real number. 00:08:49.000 --> 00:08:55.600 Conversely, let a equal 0; then you are going to get 0 + bi. 00:08:55.600 --> 00:09:04.100 This is just a pure imaginary number; this is part of the complex number system, as well--just a pure imaginary number, 00:09:04.100 --> 00:09:18.900 such as (if a is 0) 0 + 4i; 0 drops out, and I just have 4i; and this is just the imaginary part, the imaginary number. 00:09:18.900 --> 00:09:24.500 You can have complex numbers: they have two parts, a real part and an imaginary part. 00:09:24.500 --> 00:09:32.200 If, in the imaginary part, b equals 0, that leaves you with a real number; if a equals 0, it leaves you with a purely imaginary number. 00:09:32.200 --> 00:09:36.700 Or you can have both parts and have a complex number. 00:09:36.700 --> 00:09:43.000 OK, equality: this is pretty straightforward--this just says that, if you have a complex number a + bi, 00:09:43.000 --> 00:09:53.000 it equals c + di if and only if the real parts are equal and the imaginary parts are equal. 00:09:53.000 --> 00:10:13.000 For example, 4 + 7i equals 4 + 7i, because the real parts are equal (4 = 4, so a = a) and the imaginary parts are equal (b = b; 7 = 7)--straightforward. 00:10:13.000 --> 00:10:25.400 Addition and subtraction: in order to add or subtract complex numbers, combine the real parts and the imaginary parts by addition or subtraction. 00:10:25.400 --> 00:10:36.900 To illustrate this: if you are asked to add 6 + 2i plus 3 + 4i, well, we are told to combine the real parts; 00:10:36.900 --> 00:10:43.200 so I have the real part 6, and I am going to combine that with the real part 3, 00:10:43.200 --> 00:10:53.500 because this is in the form a + bi, where a is real, and then bi is my imaginary part. 00:10:53.500 --> 00:11:02.700 I combine my reals; now the imaginary parts--I have 2i, and I am going to combine that with 4i, 00:11:02.700 --> 00:11:07.300 treating the i like a variable; what you can do is factor it out. 00:11:07.300 --> 00:11:12.300 So, let's pull that i out to give me 2 + 4. 00:11:12.300 --> 00:11:26.300 OK, so this leaves me with 6 + 3 (is 9), plus i times 2 + 4 (is 6), but conventionally, we write it with the number first, and then the i. 00:11:26.300 --> 00:11:32.300 So, this is 9 + 6i, just combining the real parts and the imaginary parts. 00:11:32.300 --> 00:11:49.300 Now, working with subtraction: 5 + 2i minus 4 + 6i; OK, it is often simpler to just get rid of this negative sign 00:11:49.300 --> 00:11:55.500 and rewrite this as adding the opposite, like we have done previously with just the real number system. 00:11:55.500 --> 00:12:08.000 This is plus -4, minus 6i; actually, this should be...no, that is correct. 00:12:08.000 --> 00:12:20.200 OK, 5 + 2i minus 4 + 6i: I am rewriting this as 5 + 2i plus -4 - 6i. 00:12:20.200 --> 00:12:28.300 Now, I need to combine the real parts; so the real part I have here is 5 + -4. 00:12:28.300 --> 00:12:52.500 OK, combining the imaginary parts: I have 2i - 6i; this gives me 5 - 4, plus (I want to factor out the i) i times (2 - 6). 00:12:52.500 --> 00:13:05.000 Now, 5 minus 4 is going to give me 1; plus i--and here I have 2 - 6, so that is -4. 00:13:05.000 --> 00:13:08.400 I am rewriting this as 1 - 4i. 00:13:08.400 --> 00:13:17.000 Again, all I did is changed the signs so that I could just add the opposite of each, plus -4, plus -6i. 00:13:17.000 --> 00:13:30.500 Then, I combined the real parts, which were 5 and -4, to give me 1, and the imaginary parts, which were 2i and -6i, to give me -4i: 1 - 4i. 00:13:30.500 --> 00:13:38.700 OK, the complex plane is like the coordinate plane that we have worked with before, but with some important differences. 00:13:38.700 --> 00:13:44.000 So, before, we worked with the coordinate plane, and of course, we had a horizontal and vertical axis. 00:13:44.000 --> 00:13:46.100 And we had positive and negative numbers on it. 00:13:46.100 --> 00:13:58.500 Well, here the horizontal axis represents the real part of a complex number; so this is the real part, or the real numbers. 00:13:58.500 --> 00:14:09.900 On this vertical axis, we have the imaginary numbers--the imaginary part of the complex number. 00:14:09.900 --> 00:14:21.800 For example, this could be labeled 1, 2, 3, 4, 5, and on--pretty familiar. 00:14:21.800 --> 00:14:39.300 What is different here is the vertical axis: here I am going to have i, 2i, 3i, 4i, and the same on down...-i, -2i, -3i, -4i. 00:14:39.300 --> 00:14:47.600 OK, so thinking about graphing complex numbers on a coordinate plane: 2 + 3i, for example: 00:14:47.600 --> 00:14:56.300 well, the 2 is the real part, so that is going to give me my horizontal coordinate right here, 2. 00:14:56.300 --> 00:15:07.000 Now, the vertical coordinate is 3i; so I have 2 here and 1, 2, 3i up here; so this is 2 + 3i. 00:15:07.000 --> 00:15:15.700 Now, imagine I have something that is just a real number, like 5 + 0i; this is going to drop out, so it is just 5. 00:15:15.700 --> 00:15:24.400 Therefore, it is going to be right on the x-axis; I am going to have 5, and then for the vertical plane it is just 0; so, this is 5 + 0i. 00:15:24.400 --> 00:15:31.700 I may also have a pure imaginary number: maybe I have something like 0 + 2i, so there is no real part to it. 00:15:31.700 --> 00:15:38.000 The real part is just going to be 0; the imaginary part is going to be right here at 2i. 00:15:38.000 --> 00:15:47.700 OK, so to graph a complex number, you find the real part on this horizontal axis, and the imaginary part on the vertical axis. 00:15:47.700 --> 00:15:57.300 Pure real numbers go on the horizontal; pure imaginary numbers go on the vertical axis. 00:15:57.300 --> 00:16:02.400 To multiply complex numbers, you treat them just like any two binomials. 00:16:02.400 --> 00:16:10.500 For example, if you are asked to multiply 3 + 4i and 2 - 5i, I am going to use FOIL; 00:16:10.500 --> 00:16:17.400 and just treat the i's like variables, just as you have in the past, as far as multiplication goes. 00:16:17.400 --> 00:16:46.800 Now, I multiply out 3 times 2 (First); then my Outers (and that is 3 times -5i); and then the Inners (+ 4i times 2), and then Last (that is 4i times -5i). 00:16:46.800 --> 00:16:49.400 Now, in a minute, we are going to get into some differences. 00:16:49.400 --> 00:16:52.200 But in these first steps, really, you are just treating it like binomials. 00:16:52.200 --> 00:16:54.700 When you go to simplify, there are differences, though. 00:16:54.700 --> 00:16:57.900 But for the multiplication part, it is familiar territory. 00:16:57.900 --> 00:17:09.600 OK, so working this out: 3 times 2 is 6; 3 times -5i is -15i; plus 4i times 2--that is plus 8i; 00:17:09.600 --> 00:17:20.500 and then, I have 4i times -5i; so this is going to give me -20, and i times i is going to be i². 00:17:20.500 --> 00:17:24.400 Simplify just as you always have: combine like terms. 00:17:24.400 --> 00:17:31.400 I have a 6, and then I have a -15i; and I can combine that with 8i to get -7i. 00:17:31.400 --> 00:17:36.600 OK, -20i²: now, you might think you are done, but you are actually not-- 00:17:36.600 --> 00:17:42.600 recall from before that i equals the square root of -1. 00:17:42.600 --> 00:17:48.100 Well, if I square both sides, I mentioned that you would get i² = -1. 00:17:48.100 --> 00:17:54.000 This helps us to simplify with multiplication, because, since i² equals -1, 00:17:54.000 --> 00:18:02.100 I can say 6 - 7i - 20, and I am going to substitute in -1. 00:18:02.100 --> 00:18:14.700 So, I get 6 - 7i; and a negative and a negative is a positive; now, I can simplify...-7i + 26. 00:18:14.700 --> 00:18:19.500 So again, I proceeded with my multiplication, just as I would any two binomials. 00:18:19.500 --> 00:18:24.400 The difference came when I went to simplify, because i² is -1; 00:18:24.400 --> 00:18:29.600 so that actually allowed me to further simplify, because it got rid of that imaginary number. 00:18:29.600 --> 00:18:37.100 I still have an imaginary number here, though, so my result is going to be a complex number, -7i + 26. 00:18:37.100 --> 00:18:45.600 OK, division is a little bit more complicated, but it calls upon some familiar concepts from before. 00:18:45.600 --> 00:18:49.900 With imaginary numbers, we can have what are called complex conjugates. 00:18:49.900 --> 00:18:55.600 Before I go into this, recall the idea of conjugates when we talked about radicals. 00:18:55.600 --> 00:19:07.400 Remember that we said something like √3 + √x has the conjugate √3 - √x. 00:19:07.400 --> 00:19:17.400 And you might recall that we used these conjugate pairs when we were dealing with situations where we had radicals in the denominators. 00:19:17.400 --> 00:19:26.900 We used conjugate pairs, and we would multiply the numerator and the denominator by its conjugate to get rid of radicals in the denominator. 00:19:26.900 --> 00:19:40.500 Here, what I want to do is: now, I want to get rid of complex numbers in the denominator, so that I can divide. 00:19:40.500 --> 00:19:44.300 So, I need to think about conjugates for complex numbers. 00:19:44.300 --> 00:19:50.000 And with complex numbers, it is the same idea: you just reverse the sign before the second term. 00:19:50.000 --> 00:20:00.200 So, if I had a + bi, its conjugate is going to be a - bi. 00:20:00.200 --> 00:20:07.500 For example, if I have 4 + 5i, its conjugate is 4 - 5i. 00:20:07.500 --> 00:20:15.600 To divide complex numbers, multiply both the divisor and the dividend by the conjugate of the divisor. 00:20:15.600 --> 00:20:27.300 In other words, if I have 1 + 2i, 3 - i, here is my dividend; now, what is my conjugate? 00:20:27.300 --> 00:20:41.900 I have 3 - i; the conjugate is going to be 3 + i; so I need to multiply both the divisor and the dividend by 3 + i. 00:20:41.900 --> 00:20:52.200 OK, so I am going to multiply this by 3 + i and this by 3 + i. 00:20:52.200 --> 00:21:06.700 Using my techniques for multiplying two binomials, I am going to get 3, and then Outer terms--that is i; 00:21:06.700 --> 00:21:16.800 the Inner terms--that is going to give me 6i; and then my Last terms: 2i²--using FOIL, just like we always have. 00:21:16.800 --> 00:21:39.400 OK, now in the denominator: 3 times 3 is 9; Outer terms--that is positive 3i; Inner terms: -3i; Last: -i². 00:21:39.400 --> 00:21:56.500 So, this gives me (simplifying) 3; i + 6i is 7i; plus 2i²; over 9; 3i - 3i...that drops out; minus i². 00:21:56.500 --> 00:22:06.300 Now, just recall for a second the important concept that i² equals -1. 00:22:06.300 --> 00:22:19.000 Coming down here, this is going to give me 3 + 7i + 2; and I am going to substitute in -1 here and -1 there; that is going to give me -1. 00:22:19.000 --> 00:22:44.700 In the denominator: 9 minus -1 squared; this is going to give me 3 + 7i - 2, over 9 minus -1², which is 1. 00:22:44.700 --> 00:22:58.400 Correction: this is not squared--this is simply -1: 9 minus -1, so a negative and a negative is going to give me a positive. 00:22:58.400 --> 00:23:05.000 Again, i squared is equal to -1, so this entire term would just be -1. 00:23:05.000 --> 00:23:13.700 Simplifying 3 - 2 gives me 1 + 7i, over 9 + 1 (is 10). 00:23:13.700 --> 00:23:22.300 So, you see what happened: by multiplying both the numerator and the denominator by the complex conjugate, 00:23:22.300 --> 00:23:29.600 I was able to eliminate the complex number in the denominator. 00:23:29.600 --> 00:23:40.500 Now, just to show you a little bit of a shortcut: when you multiply a complex number by its conjugate, you get a² + b². 00:23:40.500 --> 00:23:50.300 So, if I multiply a + bi times a - bi, I am actually going to get a² + b². 00:23:50.300 --> 00:23:56.500 And that would have allowed me to save a lot of work and a possible mistake down here, because the more work, the more chance of a mistake. 00:23:56.500 --> 00:24:06.700 If you look at it this way, if I have 3 - i here, a equals 3, and b equals -1. 00:24:06.700 --> 00:24:13.200 So, I could just say that what I am going to end up with, if I multiply 3 - i times 3 + i (the complex conjugates) 00:24:13.200 --> 00:24:21.300 is a², which is 3², plus -1², or 9 + 1, which equals 10. 00:24:21.300 --> 00:24:27.100 And that is a good shortcut to use: I multiplied it out just to show you how this term drops out, and this term-- 00:24:27.100 --> 00:24:31.700 you get rid of the imaginary number, and you just end up with the real number down here. 00:24:31.700 --> 00:24:36.900 But it is really a good idea to use shortcuts when you can; it will save you time and mistakes. 00:24:36.900 --> 00:24:48.900 So again, to divide, you multiply both the divisor and the dividend by the conjugate of the divisor. 00:24:48.900 --> 00:24:58.000 OK, in our first example, we are going to use some of the concepts of properties of square roots, and also of complex numbers. 00:24:58.000 --> 00:25:08.600 Using the product property, I can rewrite this so that I can factor out the perfect squares and deal with the imaginary number. 00:25:08.600 --> 00:25:23.600 I have a -1, times 36, times 2; so this is factoring out this -72, so that I have factored out the negative part, and I have factored out the perfect square. 00:25:23.600 --> 00:25:27.500 x² and y⁴ are also perfect squares. 00:25:27.500 --> 00:25:36.900 The product property tells me that this is equal to this. 00:25:36.900 --> 00:25:46.300 This allows me to simplify: recall that i equals the square root of -1, so instead of writing this, I am going to write it as i. 00:25:46.300 --> 00:25:52.200 The square root of 36 is 6; I can't simplify √2 any further. 00:25:52.200 --> 00:26:02.400 Now, let's look at x²; be careful with this, because what they are asking for is the principal, or positive, square root. 00:26:02.400 --> 00:26:09.600 To make this more concrete--how you have to handle this--let's think about if I was told that x² equals 4. 00:26:09.600 --> 00:26:17.100 If I were to take the square root, well, the square roots of that are +2 and -2. 00:26:17.100 --> 00:26:24.400 And the reason is because -2 squared equals 4, and 2 squared equals 4. 00:26:24.400 --> 00:26:29.900 But when I use the radical sign here, what I really want--I am saying I want the principal, or positive, square root. 00:26:29.900 --> 00:26:36.300 So, in order to ensure that I am expressing that, I need to use absolute value bars. 00:26:36.300 --> 00:26:41.100 If I wanted the square root that is a principal square root, I could say it is the absolute value of x. 00:26:41.100 --> 00:26:46.600 And since x equals +2 or -2, the absolute value of x here would just be 2. 00:26:46.600 --> 00:26:53.700 OK, now y⁴, actually...the square root of that is y², and I don't need absolute value bars. 00:26:53.700 --> 00:27:01.200 And let's think about why: I don't know what y is; let's say y stands for -3. 00:27:01.200 --> 00:27:08.600 Well, when I take y², I would get a positive number. 00:27:08.600 --> 00:27:19.500 So, it doesn't matter if y is negative; it doesn't matter if y is positive, because y² will always be positive. 00:27:19.500 --> 00:27:24.600 Since y² is always positive, I don't need to specify absolute value, 00:27:24.600 --> 00:27:30.200 whereas x could be negative, so I do need to specify an absolute value. 00:27:30.200 --> 00:27:38.700 OK, I am just rewriting this as 6i√2|x|y². 00:27:38.700 --> 00:27:54.300 Simplifying this using properties of square roots and the properties of imaginary numbers...knowing that i is √-1 allowed me to simplify this. 00:27:54.300 --> 00:28:02.300 OK, Example 2 involves addition and subtraction of imaginary numbers. 00:28:02.300 --> 00:28:14.200 Simplifying: the first step, to keep my signs straight, is going to be to change this to addition, thus pushing this negative sign inside the parentheses. 00:28:14.200 --> 00:28:21.600 I am going to take the opposite of -3, which is 3, and the opposite of -4i, which is positive 4i. 00:28:21.600 --> 00:28:30.300 Now, remember that, to add complex numbers, you add the real parts to each other, and the imaginary parts. 00:28:30.300 --> 00:28:37.100 So, let's look at what I have for real parts: I have 4; I have 7; and I have 3. 00:28:37.100 --> 00:28:41.800 OK, I am adding the real parts; I am combining those; and I am combining the imaginary parts. 00:28:41.800 --> 00:28:53.500 Here, I have -3i; I have 2i; and I have 4i. 00:28:53.500 --> 00:29:01.400 Let's factor out the i, so all I have to do is add these real numbers in here. 00:29:01.400 --> 00:29:11.200 7, 4, and 3 is simply 14; plus i, times -3, 2, and 4; so that is 6 minus 3, which is 3. 00:29:11.200 --> 00:29:22.100 I am rewriting this as 14 + 3i; again, working with complex numbers, adding and subtracting, 00:29:22.100 --> 00:29:28.800 you add the real parts to each other and the imaginary parts to each other. 00:29:28.800 --> 00:29:38.300 In this example, we are multiplying some complex numbers; and you handle these just as you handle any other binomials, using FOIL. 00:29:38.300 --> 00:29:47.500 Multiply out the First terms: that is 4 times 3; the Outer terms: 4 times 6i; the Inner terms-- 00:29:47.500 --> 00:30:01.900 this is going to give me + -5i, times 3; and then the Last terms: -5i times 6i. 00:30:01.900 --> 00:30:24.600 Finish our multiplication, and then simplify: 4 times 3--that is 12, plus 24i; -5i times 3 is -15i; -5i times 6i is going to give me -30i². 00:30:24.600 --> 00:30:34.000 Simplify a bit more to get 12; I can combine these two imaginary numbers; 24i - 15i is positive 9i. 00:30:34.000 --> 00:30:37.300 I can take it one step further with the simplification. 00:30:37.300 --> 00:30:58.900 Recall that i² equals -1; so, since i² = -1, and I have i² right here, I can substitute -1 right here. 00:30:58.900 --> 00:31:17.200 12 + 9i - 30(-1) gives me 12 + 9i...a negative and a negative is a positive, so that gives me + 30. 00:31:17.200 --> 00:31:24.800 Now, I can combine these two: 30 + 12 gives me 42 + 9i. 00:31:24.800 --> 00:31:28.300 Multiply these out just like any two binomials. 00:31:28.300 --> 00:31:38.400 Then, I got to this point; I combined like terms; and right here, I stopped and realized that i² is equal to -1. 00:31:38.400 --> 00:31:48.800 So, I substituted that here, which turned this into a real number that I added to 12; and this is my answer. 00:31:48.800 --> 00:31:52.000 OK, simplify: now we are working with division. 00:31:52.000 --> 00:32:12.200 Remember that, in order to divide, what I need is to multiply the divisor and the dividend by the conjugate of the divisor. 00:32:12.200 --> 00:32:21.300 I am looking here at 4 + 2i; its complex conjugate is 4 - 2i. 00:32:21.300 --> 00:32:33.200 I need to multiply this numerator and denominator by 4 - 2i. 00:32:33.200 --> 00:32:37.800 OK, in the numerator, I am going to go ahead and do my FOIL. 00:32:37.800 --> 00:32:51.200 First gives me 2(4), which is 8; Outer terms--this is 2(-2i)--that is -4i. 00:32:51.200 --> 00:33:03.700 Inner terms: -3i(4) is -12i; Last terms: -3i(-2i) is + 6i². 00:33:03.700 --> 00:33:13.800 Now, in the denominator, I could use FOIL and multiply it out; or I could remember that, if I multiply complex conjugates, 00:33:13.800 --> 00:33:22.800 a + bi times a - bi, what I am going to end up with is a² + b². 00:33:22.800 --> 00:33:32.500 Now, looking at 4 + 2i here, a equals 4 and b equals 2; so let me just take that shortcut 00:33:32.500 --> 00:33:43.500 and say that I then have a² (which is 4, so 4²), plus b² (which is 2²). 00:33:43.500 --> 00:33:56.200 OK, now, simplifying the numerator a bit further gives me 8; -4i - 12i is -16i; plus 6i². 00:33:56.200 --> 00:34:02.700 In the denominator, 4² is 16, and 2² is 4. 00:34:02.700 --> 00:34:19.600 OK, recall that i² is -1; so I am going to substitute -1 here. 00:34:19.600 --> 00:34:34.200 In the denominator, I just have 16 + 4 is 20; this gives me 8 - 16i; this is -6 over 20; 00:34:34.200 --> 00:34:43.100 I can simplify a bit more, because 8 - 6 is 2; this is 2 - 16i, over 20--that is my solution. 00:34:43.100 --> 00:34:49.000 OK, so in order to simplify this, I took the conjugate of the denominator (which is 4 - 2i), 00:34:49.000 --> 00:34:58.000 and I multiplied both the divisor and the dividend by this complex conjugate. 00:34:58.000 --> 00:35:07.300 In the denominator, it was easy, because I just said, "OK, multiplying these conjugates gives me a² + b²." 00:35:07.300 --> 00:35:11.800 So, that is 4² is 16, and 2² is 4, to get 20. 00:35:11.800 --> 00:35:21.200 In the denominator, I used FOIL; I multiplied these out, just as I normally would, to get this. 00:35:21.200 --> 00:35:28.500 I combined like terms to get 8 - 16i + 6i²; and then, I said, "OK, i² is -1," 00:35:28.500 --> 00:35:38.500 allowing me to simplify this into -6 and combining 8 - 6 to get 2 - 16i over 20. 00:35:38.500 --> 00:35:44.900 That concludes this session of Educator.com introducing complex numbers and imaginary numbers. 00:35:44.900 --> 00:35:45.000 I will see you next time!