1 00:00:00,113 --> 00:00:02,179 We're asked to divide. 2 00:00:02,179 --> 00:00:06,362 And we're dividing six plus three i by seven minus 5i. 3 00:00:06,362 --> 00:00:10,200 And in particular, when I divide this, I want to get another complex number. 4 00:00:10,200 --> 00:00:15,023 So I want to get some real number plus some imaginary number, 5 00:00:15,023 --> 00:00:16,600 so some multiple of i's. 6 00:00:16,600 --> 00:00:18,067 So let's think about how we can do this. 7 00:00:18,067 --> 00:00:20,108 Well, division is the same thing -- 8 00:00:20,108 --> 00:00:26,271 and we rewrite this as six plus three i over seven minus five i. 9 00:00:26,271 --> 00:00:27,971 These are clearly equivalent; 10 00:00:27,987 --> 00:00:33,200 dividing by something is the same thing as a rational expression where that something is in the denominator, 11 00:00:33,200 --> 00:00:34,400 right over here. 12 00:00:34,400 --> 00:00:36,532 And so how do we simplify this? 13 00:00:36,532 --> 00:00:42,469 Well, we have a tool in our toolkit that can make sure that we don't have an imaginary or complex number in the denominator. 14 00:00:42,469 --> 00:00:45,267 And that's the complex conjugate. 15 00:00:45,267 --> 00:00:48,483 If we multiply both the numerator and the denominator of this expression 16 00:00:48,483 --> 00:00:51,067 by the complex conjugate of the denominator, 17 00:00:51,067 --> 00:00:55,933 then we will have a real number in the denominator. 18 00:00:55,933 --> 00:00:57,067 So let's do that. 19 00:00:57,067 --> 00:01:00,800 Let's multiply the numerator and the denominator by the conjugate of this. 20 00:01:00,800 --> 00:01:07,400 So seven PLUS five i. Seven plus five i is the complex conjegate of seven minus five i. 21 00:01:07,400 --> 00:01:12,067 So we're going to multiply it by seven plus five i over seven plus five i. 22 00:01:12,067 --> 00:01:16,400 And anything divided by itself is going to be one 23 00:01:16,400 --> 00:01:19,000 (assuming you're not dealing with zero; zero over zero is undefined). 24 00:01:19,000 --> 00:01:22,152 But seven plus five i over seven plus five i is one. 25 00:01:22,152 --> 00:01:24,800 So we're not changing the value of this. 26 00:01:24,800 --> 00:01:29,067 But what this does is it allows us to get rid of the imaginary part in the denominator. 27 00:01:29,067 --> 00:01:31,333 So let's multiply this out. 28 00:01:31,333 --> 00:01:32,578 Our numerator -- 29 00:01:32,578 --> 00:01:38,467 we just have to multiply every part of this complex number times every part of this complex number. 30 00:01:38,467 --> 00:01:43,000 You can think of it as FOIL if you like; we're really just doing the distributive property twice. 31 00:01:43,000 --> 00:01:49,667 We have six times seven, which is forty two. 32 00:01:49,667 --> 00:01:58,867 And then we have six times five i, which is thirty i. So plus thirty i. 33 00:01:58,867 --> 00:02:09,710 And then we have three i times seven, so that's plus twenty-one i. 34 00:02:09,741 --> 00:02:12,375 And then finally we have three i times five i. 35 00:02:12,375 --> 00:02:14,351 Three times five is fifteen. 36 00:02:14,351 --> 00:02:17,439 But we have i times i, or i squared, which is negative one. 37 00:02:17,439 --> 00:02:22,369 So it would be fifteen times negative one, or minus 15. 38 00:02:22,369 --> 00:02:24,159 So that's our numerator. 39 00:02:24,159 --> 00:02:28,282 And then our denomenator is going to be -- 40 00:02:28,282 --> 00:02:33,067 Well, we have a plus b times a minus b. (You could think of it that way. 41 00:02:33,067 --> 00:02:36,333 Or we could just do what we did up here. Actually, let's just do what we did up here 42 00:02:36,333 --> 00:02:39,242 so you don't have to remember that difference of squares pattern and all that.) 43 00:02:39,242 --> 00:02:42,000 Seven times seven is forty-nine. 44 00:02:42,000 --> 00:02:45,667 Let's think of it in the FOIL way, if that is helpful for you. 45 00:02:45,667 --> 00:02:52,153 So first we did the 7X7. And we can do the outer terms. 7 X 5i is +35i. 46 00:02:52,153 --> 00:02:57,133 Then we can do the inner terms. -5i X 7 is -35i. 47 00:02:57,133 --> 00:02:59,490 These two are going to cancel out. 48 00:02:59,490 --> 00:03:04,250 And then -5i X 5i is -25i^2 ("negative twenty five i squared"). 49 00:03:04,250 --> 00:03:09,893 -25i^2 is the same thing as -25 times -1, so that is +25. 50 00:03:09,893 --> 00:03:11,333 Now let's simplify it. 51 00:03:11,333 --> 00:03:14,235 These guys down here cancel out. 52 00:03:14,235 --> 00:03:19,831 Our denominator simplifies to 49 + 25 is 74. 53 00:03:19,831 --> 00:03:23,000 And our numerator: we can add the real parts, 54 00:03:23,000 --> 00:03:27,238 so we have a 42 and a -15. 55 00:03:27,238 --> 00:03:32,933 Let's see: 42 - 5 would be 37, minus another 10 would be 27. 56 00:03:32,933 --> 00:03:35,829 So that is 27. 57 00:03:35,829 --> 00:03:42,494 And then we're going to add our 30i, plus the 21i -- 58 00:03:42,494 --> 00:03:50,210 so 30 of something plus another 21 of that same something is going to be 51 of that something, 59 00:03:50,210 --> 00:03:55,148 in this case that something is the imaginary unit; it is i. 60 00:03:55,148 --> 00:03:57,200 (We'll do this in magenta; o, I guess that's orange.) 61 00:03:57,200 --> 00:04:00,652 So this is +51i. 62 00:04:00,652 --> 00:04:05,467 And I want to write it in the form of "a+bi," the traditional complex number form. 63 00:04:05,467 --> 00:04:12,006 So this right over here is the same thing as 27/74, 64 00:04:12,006 --> 00:04:32,347 27/74 + 51/74 times i. (I'm going to write that i in that same orange color.) 65 00:04:32,347 --> 00:04:34,359 And we are done. 66 00:04:34,359 --> 00:04:37,084 We have a real part, and we have an imaginary part. 67 00:04:37,084 --> 00:04:41,000 If this last step confuses you a little bit, just remember, if it's helpful for you 68 00:04:41,000 --> 00:04:42,862 that this is the same thing. 69 00:04:42,862 --> 00:04:45,267 We're essentially multiplying both of these terms times 1/74. 70 00:04:45,267 --> 00:04:47,533 We're dividing both of these terms by 74. 71 00:04:47,533 --> 00:04:52,267 And we're distributing the 1/74 times both of these, I guess is one way to think about it. 72 00:04:52,267 --> 00:04:54,600 And that's how we got this thing over here, 73 99:59:59,999 --> 99:59:59,999 where we have a nice real part and a nice imaginary part.