1 00:00:00,012 --> 00:00:07,187 So, so let's talk about complex numbers. I cannot emphasize enough how important 2 00:00:07,187 --> 00:00:13,987 complex numbers are in electrical engineering, It's really important, it's 3 00:00:13,987 --> 00:00:20,220 really fundamental, and we're going to exploit the properties of complex numbers 4 00:00:20,220 --> 00:00:22,986 in ways I think you're very surprised to see. 5 00:00:22,986 --> 00:00:27,899 So, as always we're going to define what a complex number is. We'll then learn how 6 00:00:27,899 --> 00:00:32,769 to add, subtract, multiply and divide them, and then we're going to go into the 7 00:00:32,769 --> 00:00:38,014 important topic of the two different ways of writing complex numbers. 8 00:00:38,014 --> 00:00:45,278 It's called Cartesian and Polar forms. Okay, so let's define a complex number, 9 00:00:45,278 --> 00:00:52,176 and this definition, I think, is a little different than you may be used to. 10 00:00:52,176 --> 00:00:59,522 A complex number is an ordered pair of numbers. So, a and b, an ordered pair. 11 00:00:59,522 --> 00:01:06,947 Now, what the ordered term means is that a, b does not equal b, 12 00:01:06,947 --> 00:01:10,221 a. The order matters. 13 00:01:10,221 --> 00:01:18,256 They're not the same thing. And similarly, if you have two complex 14 00:01:18,256 --> 00:01:25,988 numbers that equal each other, what that means is that a equals c and b 15 00:01:25,988 --> 00:01:31,389 equals d. So that's the, what the ordering has to 16 00:01:31,389 --> 00:01:37,035 do with things. And now we need to figure out how to do 17 00:01:37,035 --> 00:01:42,779 arithmetic with these numbers, what the rules of algebra are for, for 18 00:01:42,779 --> 00:01:47,053 them. And lets start with multiplying a complex 19 00:01:47,053 --> 00:01:51,849 number by a scalar. And so, when you do that, what you do is 20 00:01:51,849 --> 00:01:57,674 you multiply each component, the a and b component by the same scalar. 21 00:01:57,674 --> 00:02:00,765 Okay. I think that makes a lot of sense. 22 00:02:00,765 --> 00:02:07,023 If you add them, it's the same as adding the components of the 2 complex numbers. 23 00:02:07,023 --> 00:02:11,057 Again, keeping them separate. I think that follows. 24 00:02:11,057 --> 00:02:17,822 What isn't so obvious is where, where the multiplication and reciprocal formulas 25 00:02:17,822 --> 00:02:21,642 come from. Well it turns that these are absolutely 26 00:02:21,642 --> 00:02:27,407 the right algebraic rules to use when multiplying and dividing and we'll see 27 00:02:27,407 --> 00:02:32,812 what that is in just a second. But first, I want to point out something. 28 00:02:32,812 --> 00:02:39,323 This notation a comma b you should have seen before in analytic geometry. 29 00:02:39,323 --> 00:02:46,187 That's how you put notation you use to represent the location of a point in the 30 00:02:46,187 --> 00:02:50,123 plane. So here we have a point and this is its 31 00:02:50,123 --> 00:02:55,150 location in the plane. Its x component is a, its y component is 32 00:02:55,150 --> 00:03:00,565 b and this is just standard notation and this also helps explain why the 2 33 00:03:00,565 --> 00:03:05,578 components are kept separately in this geometric interpretation. 34 00:03:05,578 --> 00:03:10,741 We're going to call the x component the real part, we're going to call the y 35 00:03:10,741 --> 00:03:15,537 component the imaginary. Okay, and that's just terminology, 36 00:03:15,537 --> 00:03:21,808 but where it comes from, is the following interpretation of the complex number. 37 00:03:21,808 --> 00:03:26,420 We're going to write a complex number in the following way. 38 00:03:26,420 --> 00:03:30,139 a plus jb. J is an indicator of which one is the Y 39 00:03:30,139 --> 00:03:34,987 component, which one is up. So, this a plus jb means I can get to 40 00:03:34,987 --> 00:03:41,409 this point in the plane by going out a along the real part axis, the x axis, if 41 00:03:41,409 --> 00:03:47,568 you will, and then going up b units in the y direction, the imaginary part 42 00:03:47,568 --> 00:03:51,766 direction. And so, this j simply represents which 43 00:03:51,766 --> 00:03:55,782 way is the vertical, the y component direction. 44 00:03:55,782 --> 00:04:01,076 Okay, so this is an interesting way of writing a vector sum, that's all it is. 45 00:04:01,076 --> 00:04:05,954 The fun comes when you define that indicator j to be the square root of 46 00:04:05,954 --> 00:04:09,522 minus 1. That's where things get interesting, and 47 00:04:09,522 --> 00:04:14,878 this helps us figure out where those multiplication and division rules came 48 00:04:14,878 --> 00:04:19,728 from Before I get into that, I have to point out, and you should be a little 49 00:04:19,728 --> 00:04:23,415 curious, why is j used to represent the square root of mnus 1. 50 00:04:23,415 --> 00:04:27,198 You, you probably learned it as i was the square root of minus 1. 51 00:04:27,198 --> 00:04:32,021 And it's true for world other than electrical engineers use i for the square 52 00:04:32,021 --> 00:04:34,731 root of minus 1. But not electrical engineers. 53 00:04:34,731 --> 00:04:39,460 We're special, we use j and there's a historical reason for that. 54 00:04:39,460 --> 00:04:45,404 It turns out at the beginning of the 19th century, the French physicist Ampere 55 00:04:45,404 --> 00:04:50,453 wrote a fundamental paper on the mathematical physics of electricity and 56 00:04:50,453 --> 00:04:56,039 he referred to current as [FOREIGN], for expressing how much current was flowing. 57 00:04:56,039 --> 00:05:03,791 And so he used the symbol i for current. So, according to him, i is the same as 58 00:05:03,791 --> 00:05:09,251 current. Well, it was toward the end of the 19th 59 00:05:09,251 --> 00:05:17,336 century that the importance of complex numbers came up, and so we can't use i 60 00:05:17,336 --> 00:05:22,084 because electrical engineers throughout the 19th century were using i for 61 00:05:22,084 --> 00:05:25,366 current, so we use the next letter in the alphabet. 62 00:05:25,366 --> 00:05:30,051 So, it's only electrical engineers that use j for the square root of minus 1. 63 00:05:30,051 --> 00:05:35,092 So, since you're in this course, you're going to use j over the square root of 64 00:05:35,092 --> 00:05:37,748 minus 1. And all I can say is, welcome to the 65 00:05:37,748 --> 00:05:42,608 electrical engineering club. You are now, an electrical engineering, 66 00:05:42,608 --> 00:05:45,163 if you use j for the square root of minus 1. 67 00:05:45,163 --> 00:05:49,060 Fun to be in this exclusive group, of really cool people. 68 00:05:49,060 --> 00:05:54,601 Okay, so, lets go back and, look at our, arithmetic rules before, in the light of 69 00:05:54,601 --> 00:05:57,517 thinking about this a plus jb, a vector sum. 70 00:05:57,517 --> 00:06:02,652 So, now multiplication by a scalar. Well, things still make sense. 71 00:06:02,652 --> 00:06:08,687 If you have a scaler multiplying a sum, well, the scalar multiplies each term in 72 00:06:08,687 --> 00:06:12,352 the sum. When you have a sum of sums, you just add 73 00:06:12,352 --> 00:06:17,913 them all up, and then we're going to group them by their x, and y components, 74 00:06:17,913 --> 00:06:22,446 there a real part, and they're measuring for our components. 75 00:06:22,446 --> 00:06:27,120 That makes perfect sense. Now the multiplication rule is going to 76 00:06:27,120 --> 00:06:32,542 come up and make a whole lot more sense. So, you have a product of 2 sums. 77 00:06:32,542 --> 00:06:37,706 Well just use your normal algebraic rules, that you already know. 78 00:06:37,706 --> 00:06:47,768 So, A1 times A2, That's A1A2, of course, and then you get A1 times jB2. Well, 79 00:06:47,768 --> 00:06:56,654 that's this, that's fine. So, now let's say jB1 times A2 There's 80 00:06:56,654 --> 00:07:03,120 that term. That's fine, and the interesting one is 81 00:07:03,120 --> 00:07:09,737 jb1 times jb2. Now, rearranging that multiplication a 82 00:07:09,737 --> 00:07:16,823 little bit, that's got to be j squared times b1, b2 of j squared is the square, 83 00:07:16,823 --> 00:07:22,617 the square root of minus 1, that's minus 1, which means that's where this term 84 00:07:22,617 --> 00:07:28,225 comes from, so in order to use the regular rules of algebra, when you 85 00:07:28,225 --> 00:07:34,180 represent complex numbers in this kind of vector form, this has to be the 86 00:07:34,180 --> 00:07:39,716 multiplication So now where this reciprocal rule comes from, you're 87 00:07:39,716 --> 00:07:44,487 going to have to wait a little bit. I'll get to that in just a second. 88 00:07:44,487 --> 00:07:50,299 Well, now we come to one of the most important results mathematical results 89 00:07:50,299 --> 00:07:56,994 for this course called Euler's Formula. Euler was an 18th century mathematician, 90 00:07:56,994 --> 00:08:01,404 the best and most famous mathematician after Newton. 91 00:08:01,404 --> 00:08:07,502 And he noticed the following formula. So take an exponential, e to the cx, 92 00:08:07,502 --> 00:08:13,110 and write it in Taylor series. And what Euler wondered was, what is, 93 00:08:13,110 --> 00:08:17,407 what happens when c is the square root of minus 1, it is, j. 94 00:08:17,407 --> 00:08:23,061 Well, he didn't know what e to the jx was, but one way of looking at it is to 95 00:08:23,061 --> 00:08:27,032 use the Taylor series formula and see what you get. 96 00:08:27,032 --> 00:08:32,674 So here are the calculations that he made. So he just plugged in j for ce, 97 00:08:32,674 --> 00:08:39,552 everywhere in the formula, and just did the usual rules that we just talked 98 00:08:39,552 --> 00:08:43,922 about. So let's look at the even powered terms. 99 00:08:43,922 --> 00:08:48,367 1 is 1, okay. This term, that's j squared times x 100 00:08:48,367 --> 00:08:51,952 squared. Well j squared is minus 1, so that's 101 00:08:51,952 --> 00:08:57,117 where this term comes from. j to the 4, that's j squared times j 102 00:08:57,117 --> 00:09:04,197 squared, so that's plus 1. So that's where that term comes from. 103 00:09:04,197 --> 00:09:09,662 Okay, let's look at the odd powered terms. 104 00:09:09,662 --> 00:09:14,542 j x, okay, there's j x. j cubed is j times j squared. 105 00:09:14,542 --> 00:09:19,092 So that's j times minus 1. So we get a minus x cubed term here. 106 00:09:19,092 --> 00:09:27,222 So, I think you can continue this on and what the Taylor series says is that the 107 00:09:27,222 --> 00:09:35,859 real part, the part with the x component consist of all of the Even powered terms, 108 00:09:35,859 --> 00:09:41,739 with an alternating sign. The imaginary part, consists of the odd 109 00:09:41,739 --> 00:09:49,587 powered terms, with an alternating sign. Well, low and behold Euler noticed, right 110 00:09:49,587 --> 00:09:56,442 away, because he's a smart mathemetician That it has to be true, that e to the jx 111 00:09:56,442 --> 00:10:01,281 is cosine x plus j sine x. Because that is the Taylor series for 112 00:10:01,281 --> 00:10:05,183 cosine. And that is the Taylor series for sine. 113 00:10:05,183 --> 00:10:11,771 It has to be true of all, everything is going to work together and hang together, 114 00:10:11,771 --> 00:10:15,672 and be a coherent mathematical formulation. 115 00:10:15,672 --> 00:10:23,962 So, and say another way, he showed that re to the j theta, has to be r cosine 116 00:10:23,962 --> 00:10:30,892 theta times jr sine theta. Well that should be very strike a 117 00:10:30,892 --> 00:10:36,922 familiar note, cause this looks like polar form. 118 00:10:36,922 --> 00:10:42,193 Remember this is a plus jb. Hmm, that sure looks like a polar form, 119 00:10:42,193 --> 00:10:47,969 and sure enough, you go back to geometry, that's exactly what it is. 120 00:10:47,969 --> 00:10:54,826 So, here's our representation of a point in a plane, using the regular ordered 121 00:10:54,826 --> 00:10:59,916 pair notation, which you can also think about a point in 122 00:10:59,916 --> 00:11:06,752 the plane in polar form. so there is a distance from the origin, the point is 123 00:11:06,752 --> 00:11:11,456 some distance from the origin, and at some angle. 124 00:11:11,456 --> 00:11:15,414 Well we know how to convert between the two. 125 00:11:15,414 --> 00:11:21,051 We know that the a component, the x component, is r cosine theta, 126 00:11:21,051 --> 00:11:27,454 and that the vertical component, the y component, the imaginary component here 127 00:11:27,454 --> 00:11:32,005 is r sine theta. So that's where the rule I derived on the 128 00:11:32,005 --> 00:11:35,617 previous slide, is all consistent with this. 129 00:11:35,617 --> 00:11:40,342 All consistent with this geometric picture of representing something 130 00:11:40,342 --> 00:11:45,292 Cartesian coordinates or polar form. It turns out to hop between the two, 131 00:11:45,292 --> 00:11:50,142 Euler's formula had better be true or else something is very very wrong. 132 00:11:50,142 --> 00:11:54,578 So, you know, from your geometry that you 133 00:11:54,578 --> 00:12:01,907 find the radius, just by using the Pythagorean theorem, and the angle is 134 00:12:01,907 --> 00:12:07,729 just the arc tangent of the ratio of the y and x components. 135 00:12:07,729 --> 00:12:14,792 Okay, so, we now have 2 ways of representing the same complex number. 136 00:12:14,792 --> 00:12:22,124 One is Cartesian form, a plus jb. One is Polar form, re to the j-theta and these 137 00:12:22,124 --> 00:12:26,792 rules are how you would go from one to another. 138 00:12:26,792 --> 00:12:33,890 Go from Polar form back to Cartesian using these rules, go from Cartesian form 139 00:12:33,890 --> 00:12:37,854 to Polar form using these rules. And we're going to do that a lot. 140 00:12:37,854 --> 00:12:42,651 It turns out that certain calculations are better done in one form than another, 141 00:12:42,651 --> 00:12:47,127 and you need to be able to go back and forth, and these are the rules that they 142 00:12:47,127 --> 00:12:50,149 come from. But these aren't just arbitrary rules, 143 00:12:50,149 --> 00:12:53,639 they have to do with the geometry of the 2 d point, Okay, 144 00:12:53,639 --> 00:12:58,662 well now this might help explain how some of these weird and [INAUDIBLE] 145 00:12:58,662 --> 00:13:02,158 multiplication and reciprocal rules come from. 146 00:13:02,158 --> 00:13:07,318 Well let's check multiplication rule when things are written in polar form. 147 00:13:07,318 --> 00:13:11,718 So we have two polar form complex numbers, and a little bit of 148 00:13:11,718 --> 00:13:17,019 rearrangement gives us this. And note that you're multiplying 2 149 00:13:17,019 --> 00:13:21,570 exponentials together, their exponents add. 150 00:13:21,570 --> 00:13:25,656 So that's where this comes from. Okay, well. 151 00:13:25,656 --> 00:13:32,158 This is just a special, form. I'm going to convert it to Cartesian form 152 00:13:32,158 --> 00:13:36,512 using our rules. Okay, that's where that comes from. 153 00:13:36,512 --> 00:13:42,027 And now I'm going to use the cosine of a sum of angles, and the sine of a sum of 154 00:13:42,027 --> 00:13:45,907 our angles, formulas that you'd learn in 155 00:13:45,907 --> 00:13:49,022 trigonometry. So, that's where this is. 156 00:13:49,022 --> 00:13:53,580 This term corresponds to the cosine of the sum of angles, 157 00:13:53,580 --> 00:13:57,735 and this turn, comes from the sine of of a sum of angles, 158 00:13:57,735 --> 00:14:02,923 and we're using the J to keep us, straight about which one goes where. 159 00:14:02,923 --> 00:14:08,883 And I've rearranged things a little bit to put the r1 with theta 1, and the r2 160 00:14:08,883 --> 00:14:13,181 with the theta 2. Well, identifying terms, that is just a 161 00:14:13,181 --> 00:14:20,981 1, that's just a 2, etc., and sure enough we come up with the same application that 162 00:14:20,981 --> 00:14:25,473 we derived for Cartesian form. So that's, that works fine. 163 00:14:25,473 --> 00:14:30,454 Now the reciprocal rule can be derived rather easily. 164 00:14:30,454 --> 00:14:34,905 So we're writing Z in, in polar form, that's going to be that whole reciprocal 165 00:14:36,122 --> 00:14:41,162 of an, this is 1 over r times e to the minus j theta, 166 00:14:41,162 --> 00:14:48,187 because, e to the x, and then take the reciprocal of that, why is e to the minus 167 00:14:48,187 --> 00:14:52,373 x. So, that, this has to be true. 168 00:14:52,373 --> 00:15:00,291 Well, now convert that to Cartesian form. So, e to the j theta is just cosine of 169 00:15:00,291 --> 00:15:07,435 theta, I'm sorry, cosine of minus theta is cosine of theta That's where that 170 00:15:07,435 --> 00:15:10,601 comes from. The sine of minus theta is minus the sine 171 00:15:10,601 --> 00:15:15,502 of theta, and that's where this comes from. 172 00:15:15,502 --> 00:15:26,362 So, that's really quite nice, and now you see where this comes from, because if I 173 00:15:26,362 --> 00:15:33,147 multiply top and bottom of this expression by r, 174 00:15:33,147 --> 00:15:40,269 we see that R cosine theta, well that's A, and R sine theta is E, and R squared 175 00:15:40,269 --> 00:15:45,778 is just A squared plus B squared. So that's where that reciprocal rule 176 00:15:45,778 --> 00:15:50,181 comes from. It's most easily derived in polar form, 177 00:15:50,181 --> 00:15:56,872 and now everything hangs together. We now understand all the algebra that 178 00:15:56,872 --> 00:16:01,070 there is. Now there's one special thing that comes 179 00:16:01,070 --> 00:16:07,486 up in complex numbers, which is a new thing, and that's the complex conjugate. 180 00:16:07,486 --> 00:16:11,803 It's a very easy operation, but it's not algebraic. 181 00:16:11,803 --> 00:16:17,778 So, the complex conjugate of z, which we denote by z star, 182 00:16:17,778 --> 00:16:24,898 what it does is it flips the sign of the imaginary part. 183 00:16:24,898 --> 00:16:30,306 So, z is equal to a plus jb and z conjugate is a minus jb. 184 00:16:30,306 --> 00:16:37,422 In Cartesian form, and in polar form, what that does is it changes the sign of 185 00:16:37,422 --> 00:16:41,592 the angel, flips it, and that's what the conjugate is. 186 00:16:41,592 --> 00:16:47,432 And the conjugate, it turns out, is mathematically important because of 187 00:16:47,432 --> 00:16:53,876 following reasons. You can now use, say that z times e conjugate is equal to 188 00:16:53,876 --> 00:17:01,066 the radius squared. So, that was e times e conjugate, well 189 00:17:01,066 --> 00:17:08,231 that's just squared. So you can derive, if you will, from R, 190 00:17:08,231 --> 00:17:15,527 just from looking at Z. And also, we can define what's called the 191 00:17:15,527 --> 00:17:19,613 magnitude of a complex number, is equal to r. 192 00:17:19,613 --> 00:17:24,754 So, another word for the radius is called the magnitude, 193 00:17:24,754 --> 00:17:29,712 and the magnitude squared is just r squared is z times. 194 00:17:29,712 --> 00:17:33,365 Z conjugate. It's this formula that's going to be 195 00:17:33,365 --> 00:17:39,384 very, very important for us, in talking about, complex numbers mathematically. 196 00:17:39,384 --> 00:17:45,121 You can also express the real part and imaginary parts, of a complex number 197 00:17:45,121 --> 00:17:51,397 using the com, the complex conjugate. So, the real part of z, which we'll 198 00:17:51,397 --> 00:17:58,702 denote by this, the x component, if you will, is equal to z plus z, conjugate. 199 00:17:58,702 --> 00:18:04,212 So, that's a pretty easy thing to see. So you have a plus jb. 200 00:18:04,212 --> 00:18:10,238 I'm going to add to that a minus jb. Well, when you do that, these, these 201 00:18:10,238 --> 00:18:14,509 parts cancel, and you're left with 2 times a, 202 00:18:14,509 --> 00:18:19,694 so you divide by 2. And a similar thing happens when you 203 00:18:19,694 --> 00:18:24,505 subtract. Now what happens, is that the real parts 204 00:18:24,505 --> 00:18:31,555 cancel, and you're left with the two imaginary parts adding, and you have to 205 00:18:31,555 --> 00:18:36,493 divide by 2J to get back to B. So this equals a and this equals b. 206 00:18:36,493 --> 00:18:41,922 So now we can derive the Cartesian components just by referring to the 207 00:18:41,922 --> 00:18:49,759 original complex number, by simply saying real part, and the imaginary part, by 208 00:18:49,759 --> 00:18:54,598 saying the imaginary part and get them by using this formula. 209 00:18:54,598 --> 00:19:00,085 And that's going to be very useful for us in doing some operations on complex 210 00:19:00,085 --> 00:19:03,436 numbers. So, how do you use all this, to do 211 00:19:03,436 --> 00:19:07,660 calculations? What I tend to do is add and subtract in 212 00:19:07,660 --> 00:19:12,184 Cartesian form, and I multiply and divide in either form. 213 00:19:12,184 --> 00:19:18,291 It depends on the details of what I do. I tend to use Polar form especially for 214 00:19:18,291 --> 00:19:21,935 division, and you'll see that in a second. 215 00:19:21,935 --> 00:19:28,092 And then I convert between the, between Polar and Cartesian form, depending on 216 00:19:28,092 --> 00:19:31,764 what's going to be needed. So here's a little example. 217 00:19:31,764 --> 00:19:35,490 Suppose we want to divide these two complex numbers. 218 00:19:35,490 --> 00:19:40,971 the easiest way to see this, I could use the reciprocal formal, but the easier 219 00:19:40,971 --> 00:19:44,742 thing to do, is to notice you can play a little, 220 00:19:44,742 --> 00:19:53,217 play a little mathematical trick. So a plus jb divided by c plus jd, I can 221 00:19:53,217 --> 00:20:03,692 multiply top and bottom by the conjugate of the denominator. 222 00:20:03,692 --> 00:20:07,927 I keep writing v, I don't know why, that's a d. 223 00:20:07,927 --> 00:20:14,047 Okay, I'm multiplying by 1. I'm writing 1 in a fancy way. Well, the 224 00:20:14,047 --> 00:20:20,017 denominator, is the magnitude squared of that complex number, 225 00:20:20,017 --> 00:20:27,368 and then I just have to multiply the two top things together, where you can see 226 00:20:27,368 --> 00:20:31,495 where it comes from. Because 5, is 1 squared plus 2 squared, 227 00:20:31,495 --> 00:20:38,529 and there's the, conjugate part. And I just multiply them together, and 228 00:20:38,529 --> 00:20:45,216 when the smoke clears, you get 3-J. So, that's a, easy way to do a division. 229 00:20:45,216 --> 00:20:49,569 In this case in Cartesian form. If it's in Polar form, it's, if these 230 00:20:49,569 --> 00:20:55,039 original numbers were in Polar form, I would have stayed in Polar form and just 231 00:20:55,039 --> 00:20:58,484 done it kind of like, because it's very, very easy. 232 00:20:58,484 --> 00:21:03,143 Now another thing that comes up is the magnitude of a ratio. 233 00:21:03,143 --> 00:21:09,230 And what I want to point out is that the magnitude of a ratio is equal to the 234 00:21:09,230 --> 00:21:14,715 ratio of the magnitudes. And, I think this is pretty easy to see 235 00:21:14,715 --> 00:21:20,252 if you think about this in polar form, you can derive this yourself. 236 00:21:20,252 --> 00:21:27,516 So the magnitude of the, numerator is square root of 1 squared plus 1 squared. 237 00:21:27,516 --> 00:21:35,829 So you get, 2, square root of 2. And as we've already seen, the square 238 00:21:35,829 --> 00:21:40,353 root of 1 squared plus 2 squared is the square root of 5. 239 00:21:40,353 --> 00:21:44,589 And so, a magnitude of this come, the number is 240 00:21:44,589 --> 00:21:50,920 the square root of 2 over 5, and if you take, you apply, you find the magnitude 241 00:21:50,920 --> 00:21:57,806 of this number, which I asked you to do, I think you will get the square root of 2 242 00:21:57,806 --> 00:21:58,240 over 5. And just do better. 243 00:21:59,250 --> 00:22:04,100 Alright, so complex numbers, it turns out, are going to simplify our 244 00:22:04,100 --> 00:22:08,051 calculations immensely. I cannot stress that enough. 245 00:22:08,051 --> 00:22:14,046 It is extremely important for signal and system theory in electrical engineering. 246 00:22:14,046 --> 00:22:19,088 Why that is is not obvious yet. Just stay tuned, and we'll get to it very 247 00:22:19,088 --> 00:22:20,017 very quickly.