1 00:00:00,565 --> 00:00:03,573 We're asked to find the conjugate of the complex number 2 00:00:03,589 --> 00:00:05,457 seven minus five i. 3 00:00:05,457 --> 00:00:07,341 And what you're going to find in this video is 4 00:00:07,341 --> 00:00:09,891 finding the conjugate of a complex number is 5 00:00:09,891 --> 00:00:11,119 shockingly easy. 6 00:00:11,119 --> 00:00:13,680 It's really the same as this number, I should say 7 00:00:13,680 --> 00:00:15,161 - be a little bit more particular- 8 00:00:15,161 --> 00:00:18,027 It has the same real parts, so the conjugate of this 9 00:00:18,027 --> 00:00:19,843 is going to have the exact same real part, 10 00:00:19,843 --> 00:00:22,382 but its imaginary part is going to have the opposite sign. 11 00:00:22,382 --> 00:00:25,188 So, instead of having a negative five i, it will have 12 00:00:25,188 --> 00:00:28,279 a positive - a positive- 5 i. 13 00:00:28,279 --> 00:00:31,677 So that right there is the complex conjugate of 14 00:00:31,677 --> 00:00:33,202 seven minus five i. 15 00:00:33,202 --> 00:00:36,128 And some times the notation for doing it that is 16 00:00:36,128 --> 00:00:37,871 -you'll take seven minus five i- 17 00:00:37,871 --> 00:00:43,182 If you have seven minus five i, 18 00:00:43,182 --> 00:00:45,307 and you put a line over it like that, 19 00:00:45,307 --> 00:00:47,523 that means you want the conjugate 20 00:00:47,523 --> 00:00:49,001 of seven minus five i. 21 00:00:49,001 --> 00:00:51,003 And that will equal seven plus five i. 22 00:00:51,003 --> 00:00:53,081 Or sometimes someone will write, 23 00:00:53,081 --> 00:00:57,207 -you'll see z is the variable people often use for complex numbers- 24 00:00:57,207 --> 00:01:02,886 if z is seven minus five i, 25 00:01:02,886 --> 00:01:05,268 then they'll say complex conjugate of z, 26 00:01:05,268 --> 00:01:07,066 -you put that line over the z- 27 00:01:07,066 --> 00:01:09,618 is going to be seven plus five i. 28 00:01:09,618 --> 00:01:13,214 Now you're probably saying ok, fairly straightforward 29 00:01:13,214 --> 00:01:15,476 to find a conjugate of a complex number, 30 00:01:15,476 --> 00:01:17,784 but what is it good for? 31 00:01:17,784 --> 00:01:20,557 And the simplest reason, or the most basic place 32 00:01:20,557 --> 00:01:24,161 where this is useful is when you multiply any imaginary, 33 00:01:24,161 --> 00:01:29,011 any complex number times its conjugate, you're going to get a real number. 34 00:01:29,011 --> 00:01:30,144 And I want to emphasize, 35 00:01:30,144 --> 00:01:33,631 this right here is the conjugate seven plus five, 36 00:01:33,631 --> 00:01:35,882 is the conjugate of seven minus five i. 37 00:01:35,882 --> 00:01:38,763 But seven minus five i is also the conjugate 38 00:01:38,763 --> 00:01:41,005 of seven plus five i, for obvious reasons. 39 00:01:41,005 --> 00:01:42,817 If you started with this, and changed the sign 40 00:01:42,817 --> 00:01:46,119 of the imaginary part, you would get seven minus five i. 41 00:01:46,119 --> 00:01:47,759 They're conjugates of each other. 42 00:01:47,759 --> 00:01:50,973 But let me show you when I multiply complex conjugates, 43 00:01:50,973 --> 00:01:52,430 that I get a real number. 44 00:01:52,430 --> 00:01:56,025 So lets multiply seven minus five i times seven plus five i. 45 00:01:56,025 --> 00:01:58,392 And I will do that in blue. 46 00:01:58,392 --> 00:02:03,609 Seven minus five i times seven plus five i. 47 00:02:03,609 --> 00:02:06,484 And remember, whenever you multiply these expressions, 48 00:02:06,484 --> 00:02:08,893 you really just have to multiply every term times each other. 49 00:02:08,893 --> 00:02:10,267 You could do the distributive property twice. 50 00:02:10,267 --> 00:02:13,137 You could do something like foil, to remind yourself 51 00:02:13,137 --> 00:02:16,715 every part of this complex number times 52 00:02:16,715 --> 00:02:18,524 every part of this complex number. 53 00:02:18,524 --> 00:02:20,248 So let's just do it any which way. 54 00:02:20,248 --> 00:02:23,751 So you have seven times seven, which is forty-nine. 55 00:02:23,751 --> 00:02:31,044 Seven times five i, which is thirty-five i. 56 00:02:31,044 --> 00:02:33,628 Then you have negative five i times seven, 57 00:02:33,628 --> 00:02:35,420 which is negative thirty-five i. 58 00:02:35,420 --> 00:02:37,518 You can see the imaginary part is canceling out. 59 00:02:37,518 --> 00:02:40,950 Then you have negative five i time positive five i. 60 00:02:40,950 --> 00:02:45,076 Well that's negative twenty-five i-squared. 61 00:02:45,076 --> 00:02:46,622 And negative twenty-five i-squared, 62 00:02:46,622 --> 00:02:48,172 -remember i-squared is negative one- 63 00:02:48,172 --> 00:02:50,768 So negative twenty five i-squared, 64 00:02:50,768 --> 00:02:51,570 -let me write this down- 65 00:02:51,570 --> 00:02:58,464 negative five i, times five i, is negative twenty-five times i-squared. 66 00:02:58,464 --> 00:03:01,707 i-squared is negative one, so negative twenty-five times negative one 67 00:03:01,707 --> 00:03:03,710 is positive twenty-five. 68 00:03:03,710 --> 00:03:06,375 And these two guys over here cancel out, 69 00:03:06,375 --> 00:03:09,789 and we're just left with forty-nine plus twenty-five. 70 00:03:09,789 --> 00:03:11,433 Let's see, fifty plus twenty-five is seventy-five. 71 00:03:11,433 --> 00:03:12,916 So, this is just seventy-four. 72 00:03:12,916 --> 00:03:15,999 So, we are just left with the real number seventy-four. 73 00:03:15,999 --> 00:03:17,327 Another way to do it, 74 00:03:17,327 --> 00:03:19,263 -you don't even have to do all this distributive property- 75 00:03:19,263 --> 00:03:20,983 You might just recognize this looks like. 76 00:03:20,983 --> 00:03:24,349 This is something plus something times 77 00:03:24,349 --> 00:03:25,856 -or something minus something- 78 00:03:25,856 --> 00:03:27,594 times that same thing plus something. 79 00:03:27,594 --> 00:03:30,060 And we know this pattern from our early algebra. 80 00:03:30,060 --> 00:03:34,148 That a plus b -a plus b- times a minus b 81 00:03:34,148 --> 00:03:38,047 is equal to a-squared minus b-squared. 82 00:03:38,047 --> 00:03:39,761 Is equal to a diference of squares. 83 00:03:39,761 --> 00:03:43,931 And so in this case a is seven, a-squared is forty-nine. 84 00:03:43,931 --> 00:03:47,407 And b, b in this case is five i. 85 00:03:47,407 --> 00:03:51,770 b-squared is five i-squared, which twenty-five i-squared, 86 00:03:51,770 --> 00:03:54,988 which is negative twenty-five, and we're subtracting that. 87 00:03:54,988 --> 00:03:57,215 So it's going to be positive plus-twenty-five. 88 00:03:57,215 --> 00:03:59,553 You add 'em together, you get seventy-four.