1 00:00:00,533 --> 00:00:03,769 We're asked to multiple the complex number 1 minus 3 i 2 00:00:03,769 --> 00:00:08,502 times the complex number 2 plus 5 i. And the general idea here 3 00:00:08,502 --> 00:00:10,277 is you can multiply these complex numbers 4 00:00:10,277 --> 00:00:14,662 like you would have multiplied any traditional binomial 5 00:00:14,662 --> 00:00:16,800 you just have to remember this isn't a variable 6 00:00:16,800 --> 00:00:21,800 this is the imaginary unit i, or it's just i 7 00:00:21,800 --> 00:00:23,502 but we can do that in two ways 8 00:00:23,502 --> 00:00:25,390 we can just do distributive properties twice 9 00:00:25,390 --> 00:00:28,800 which I like a little bit more just because it actually, 10 00:00:28,800 --> 00:00:31,400 you're doing it from a fundamental principle, it's nothing new 11 00:00:31,400 --> 00:00:35,852 or you could use FOIL, which you also used when you first multiplied binomials, 12 00:00:35,852 --> 00:00:37,452 and I'll do it both ways. 13 00:00:37,452 --> 00:00:42,400 So you could view this as just a number, 1 minus 3 i 14 00:00:42,400 --> 00:00:46,529 and so we can distribute it over the two numbers inside of this expression. 15 00:00:46,529 --> 00:00:48,302 So when we're multiplying times the entire expression 16 00:00:48,302 --> 00:00:54,379 We can multiply 1 minus 3 i times two, and 1 minus 3 i times 5 i. 17 00:00:54,379 --> 00:00:56,667 So let's do that. So this can be rewritten as 18 00:00:56,667 --> 00:01:04,129 1 minus 3 i times 2, I'll write the 2 out front. 19 00:01:04,129 --> 00:01:13,154 Plus 1 minus 3 i, times 5 i. 20 00:01:13,154 --> 00:01:14,964 All I did is a distributive property here. 21 00:01:14,964 --> 00:01:19,625 All I said is, look if I have a times b plus c, this is the same things as: 22 00:01:19,625 --> 00:01:21,610 ab plus ac. 23 00:01:21,610 --> 00:01:26,098 I just distributed the a on the b or the c. 24 00:01:26,098 --> 00:01:29,687 I just distributed the 1 - minus 3 i on the 2 and the 5 i. 25 00:01:29,687 --> 00:01:31,733 And then I can do it again. I have a 2 now 26 00:01:31,733 --> 00:01:34,200 times 1 minus 3 i. I can distributed it. 27 00:01:34,200 --> 00:01:37,498 2 times 1 is 2. 28 00:01:37,498 --> 00:01:41,718 2 times negative 3 i is negative 6 i 29 00:01:41,718 --> 00:01:43,600 And over here I'll do it again. 30 00:01:43,615 --> 00:01:48,277 5 i times 1, so it's plus, 5 i times one is 5 i 31 00:01:48,277 --> 00:01:52,436 And then 5 i times negative 3 i, so let's be careful here 32 00:01:52,436 --> 00:02:00,667 5 times negative 3 is negative 15, and then I have an i times an i 33 00:02:00,667 --> 00:02:06,200 Right, I'm multiplying. Let me do this over here. Five i times negative 3 i 34 00:02:06,200 --> 00:02:12,887 This is the same thing as 5 times negative 3 times i times i. 35 00:02:12,887 --> 00:02:18,154 So the 5 times the negative 3 is negative 15, and then we have i times i, 36 00:02:18,154 --> 00:02:20,021 which is i squared. 37 00:02:20,021 --> 00:02:26,933 Now we know what i squared is. By definition, i squared is negative 1. 38 00:02:26,933 --> 00:02:29,990 i squared by definition is negative 1. 39 00:02:29,990 --> 00:02:34,692 So negative 15 times negative 1, well that's the same thing as positive 15. 40 00:02:34,692 --> 00:02:40,154 So this can be rewritten as 2 minus 6 i plus 5 i 41 00:02:40,154 --> 00:02:44,164 Negative 15 times negative 1 is positive 15 42 00:02:44,164 --> 00:02:46,985 Now we can add the real parts 43 00:02:46,985 --> 00:02:50,492 We have a 2 and we have a positive 15 44 00:02:50,492 --> 00:02:55,282 So 2 plus 15, and we can add the imaginary parts 45 00:02:55,282 --> 00:03:00,867 We have a negative 6, or negative 6 i I should say 46 00:03:00,867 --> 00:03:05,533 and then we have plus 5 i. So plus 5 i 47 00:03:05,533 --> 00:03:09,348 And 2 plus 15 is 17 48 00:03:09,348 --> 00:03:12,087 And if I have negative 6 of something plus 5 of that something 49 00:03:12,087 --> 00:03:13,110 What do I have? 50 00:03:13,110 --> 00:03:17,338 Or if I have 5 of that something, and I take 6 of that something away, 51 00:03:17,338 --> 00:03:19,308 then I have negative 1 of that something. 52 00:03:19,308 --> 00:03:22,533 Negative 6 i plus 5 i is negative 1 i 53 00:03:22,533 --> 00:03:25,333 Or I could just say minus i. So in this way 54 00:03:25,333 --> 00:03:27,462 I just multiplied these two expressions 55 00:03:27,462 --> 00:03:29,462 or these complex numbers, really 56 00:03:29,462 --> 00:03:32,600 I multiplied them just using the distributive property twice. 57 00:03:32,600 --> 00:03:34,969 You could also do it using FOIL. 58 00:03:34,969 --> 00:03:36,933 And I'll do that right now really fast. 59 00:03:36,933 --> 00:03:40,467 It is a little bit faster, but it's a little bit mechanical 60 00:03:40,467 --> 00:03:43,067 So you might forget why you're doing it in the first place 61 00:03:43,067 --> 00:03:46,215 But at the end of the day, you are doing the same thing here 62 00:03:46,215 --> 00:03:49,600 You're essentially multiplying every term of this first number 63 00:03:49,600 --> 00:03:50,656 or every part of this first number, 64 00:03:50,656 --> 00:03:52,871 times every part of the second number. 65 00:03:52,871 --> 00:03:55,467 And FOIL just makes sure that we're doing it. 66 00:03:55,467 --> 00:03:56,477 Let me just write FOIL out here, 67 00:03:56,477 --> 00:03:59,467 which I'm not a huge fan of, but I'll do it 68 00:03:59,467 --> 00:04:00,667 just in case that's the way you're learning it. 69 00:04:00,667 --> 00:04:03,871 So FOIL says let's do the first numbers, let's multiply the first numbers 70 00:04:03,871 --> 00:04:08,415 so that's going to be the 1 times the 2. So 1 times the 2. 71 00:04:08,415 --> 00:04:11,200 That is the "F" in FOIL. 72 00:04:11,200 --> 00:04:13,471 Then it says, let's multiply the outer numbers 73 00:04:13,471 --> 00:04:16,395 times each other, so that's 1 times 5 i. 74 00:04:16,425 --> 00:04:21,867 So plus 1 times 5 i. This is the "O" in FOIL. 75 00:04:21,867 --> 00:04:25,333 The outer numbers. Then we do the inners numbers. 76 00:04:25,333 --> 00:04:30,467 Negative 3 i times 2. So this is negative 3 i times 2. 77 00:04:30,467 --> 00:04:32,308 This is, those are the inner numbers. 78 00:04:32,308 --> 00:04:34,431 And then we do the last numbers. 79 00:04:34,431 --> 00:04:39,431 Negative 3 i times 5 i. So negative 3 i times 5 i. 80 00:04:39,431 --> 00:04:43,400 These are the last numbers. That's all that FOIL is telling us. 81 00:04:43,400 --> 00:04:48,136 It's just making sure we're multiplying every part of this number, times every part of that number. 82 00:04:48,136 --> 00:04:52,031 And then when we simplify it: 1 times 2 is 2 83 00:04:52,031 --> 00:04:58,200 1 times 5 i is 5 i. Negative 3 i times 2 is negative 6 i 84 00:04:58,200 --> 00:05:03,795 And negative 3 i times 5 i, well we already figured out what that was, 85 00:05:03,795 --> 00:05:07,533 negative 3 i times 5 i turns out to be 15. 86 00:05:07,533 --> 00:05:11,667 Negative 3 times 5 is negative 15. But i times i is negative 1 87 00:05:11,667 --> 00:05:16,933 negative 15 times negative 1 is positive 15. 88 00:05:16,933 --> 00:05:21,800 Add to the real parts: 2 plus 15, you get 17 89 00:05:21,800 --> 00:05:28,071 Add to the imaginary parts: you have 5 i minus 6 i. You get negative i 90 00:05:28,071 --> 99:59:59,999 And once again, you get the exact same answer.