11 - Simplify Expressions with Imaginary Numbers - Part 1 - By Math and Science
Transcript
00:01 | Hello . Welcome back . We're working with imaginary numbers | |
00:03 | . In the last lesson we introduce what an imaginary | |
00:05 | number is . But more importantly I motivated for you | |
00:08 | why we care about imaginary numbers and how useful they | |
00:11 | are in real math , even beyond algebra , in | |
00:14 | real engineering and science and math . So go back | |
00:16 | and watch that last lesson if you haven't already done | |
00:18 | so and this list we're gonna learn how to start | |
00:20 | to simplify expressions that have imaginary numbers . So if | |
00:23 | you remember the imaginary number I we define it to | |
00:26 | be the square root of negative one . And because | |
00:28 | of that , if I square both sides of this | |
00:30 | , when you have the imaginary number I squared then | |
00:33 | it is equal to a real number negative one . | |
00:35 | Both of those facts are equally important for you to | |
00:38 | know you need to know that I is the square | |
00:40 | root of negative one and you also need to know | |
00:42 | that I squared is equal to negative one . So | |
00:45 | let's just train through a bunch of problems and you'll | |
00:46 | see why you need to understand both of those as | |
00:48 | we go along . If you have negative 81 for | |
00:52 | instance , we're gonna take the square root of that | |
00:54 | . What you do is you completely ignore the negative | |
00:56 | sign at first and you take the square root of | |
00:59 | the number . Well , the squared of 81 is | |
01:00 | nine . You could do a factor tree nine times | |
01:02 | nine circle a pair . But you know that it's | |
01:04 | equal to nine . And then because of the negative | |
01:06 | , you're taking the square root of that negative one | |
01:08 | as well , which is I . And it lives | |
01:09 | right behind the number . So the answer is not | |
01:12 | nine , it's nine I . So this is a | |
01:14 | pure imaginary No . nine times bigger than the base | |
01:17 | imaginary number that we have . What if you have | |
01:20 | the problem ? Look negative four times the square root | |
01:23 | of negative 36 . We treat this uh step by | |
01:27 | step as we do with any expression . The negative | |
01:29 | four is going to be multiplied by something and that | |
01:32 | something is the square root of negative 36 . The | |
01:34 | squared of 36 to 6 and the squared of the | |
01:37 | negative one is I . So we actually get six | |
01:39 | I here and so we have negative four times six | |
01:42 | . I It turns out you can multiply imaginary numbers | |
01:45 | just like you multiply any old number . Basically what | |
01:48 | you're doing is you're multiplying coefficient and you almost treat | |
01:51 | this as if it were a variable . So negative | |
01:53 | four times six is negative 24 Times . What time's | |
01:57 | the eye that's there ? You just basically treat it | |
01:59 | like a variable but this is not a real number | |
02:01 | , I is an imaginary number . So this is | |
02:03 | negative 24 times the base imaginary number that we have | |
02:08 | now . What if we have negative 20 ? And | |
02:10 | I would like to take the square root of this | |
02:12 | now for the more complicated ones , that's not a | |
02:15 | perfect square . We have to do a factor tree | |
02:16 | . So go down here and do a factor tree | |
02:18 | . But do not try to write a factor tree | |
02:20 | with negative numbers here . You just ignore the negative | |
02:22 | completely . You say five times four is 20 and | |
02:26 | two times two is four . And you circle the | |
02:28 | twos just like you would always do then you say | |
02:31 | the single too can come out the square root of | |
02:33 | the five will be left over . But because we | |
02:36 | have the square root of negative one , that also | |
02:37 | has to come out as an eye . So we | |
02:40 | write it in front of the radical to I times | |
02:42 | the square root of five . All right . What | |
02:46 | if we had uh three times the square root of | |
02:50 | -8 ? Well , we go and try to do | |
02:52 | a factor tree . You probably know this . We've | |
02:54 | done it enough and you ignore the negative sign . | |
02:56 | We say eight is two times four and four is | |
02:59 | two times two . So we have a pair of | |
03:01 | twos there . And so we can say that we | |
03:04 | ignore that the negative completely . A single to would | |
03:07 | come out . But don't forget the three is out | |
03:08 | here . So we have three times whatever is inside | |
03:11 | of here . The two comes out the two is | |
03:13 | left over . So that stays under the radical . | |
03:16 | But because we're taking the square root of negative , | |
03:17 | that comes out as an eye , which you right | |
03:20 | in front of the radical . Now you have three | |
03:22 | times this quantity . You multiply the numbers giving you | |
03:25 | six I on the outside square root of two . | |
03:28 | This is the final answer . Six I times a | |
03:29 | squared of two . All right . So , you | |
03:32 | see working with imaginary numbers is actually not hard at | |
03:35 | all . Uh As we're trying to show here now | |
03:38 | , we'll switch gears from taking the square root of | |
03:40 | negative numbers . To what happens when we start multiplying | |
03:43 | these negative these imaginary numbers together . What if we | |
03:46 | have to ? I multiply by three . I . | |
03:49 | All right . What you have to know here is | |
03:51 | that when you have uh imaginary imaginary numbers multiplied together | |
03:54 | , you basically pretend that the eye is a variable | |
03:57 | . You all know how to multiply for instance , | |
03:59 | two X times three X . You multiply the numbers | |
04:02 | together and then you multiply the variables together in X | |
04:05 | times X would give you X squared . Right ? | |
04:07 | So we kind of do that initially with this . | |
04:09 | Uh And so you say two times three is six | |
04:12 | . And then I times the eye gives me I | |
04:13 | squared . But we have to take it one more | |
04:15 | step further because we know I squared , we just | |
04:18 | have to remember it in our mind anytime you see | |
04:20 | an eye square , you have to substitute in the | |
04:22 | value of negative one because it's always equal to negative | |
04:25 | one . So what I would say is six times | |
04:27 | negative one . I would write it just like this | |
04:28 | , replacing the I . Square with negative one , | |
04:30 | which gives you negative six . That's the answer negative | |
04:33 | six . So when you have two imaginary numbers multiplying | |
04:36 | together , you can often get a real number back | |
04:39 | . All right . Now what I want to do | |
04:41 | is go over to this board and I want to | |
04:43 | uh do some problems that involve imaginary numbers . But | |
04:46 | we're multiplying radicals . Remember we had entire lessons dealing | |
04:50 | with multiplying radicals together and we want to mix in | |
04:53 | the concept of multiplying radicals when we also have imaginary | |
04:56 | numbers , we're just taking it one step further to | |
04:58 | give you a little more practice . So let's say | |
05:01 | you have the radical square root of seven , multiplied | |
05:04 | by the square root of -7 . Right square to | |
05:07 | seven times square of negative seven . Well , the | |
05:09 | square to seven , I can't really do a factor | |
05:11 | tree for I'm gonna leave it here . But the | |
05:13 | squared of -7 , you now know what is that | |
05:15 | ? Well , the squared of the negative one comes | |
05:18 | out as an eye and the square of the seven | |
05:20 | has to stay behind because they can't simplify that anymore | |
05:23 | . So what you have here is the eye is | |
05:25 | going to flow down in front and you're gonna have | |
05:28 | square root of And when you have two radicals with | |
05:31 | numbers underneath them , you can multiply those together to | |
05:34 | make a square to 49 . And so what you're | |
05:36 | gonna have at the end of the day is I | |
05:38 | let's write it up . Yeah , I times uh | |
05:41 | seven squared of 49 is seven . So maybe I | |
05:43 | time seven . But you always write it with a | |
05:45 | number in front . So you say that seven eyes | |
05:47 | the answer , that's the final answer . Okay , | |
05:51 | What if we have something similar instead of a positive | |
05:55 | ? Hear any negative here ? Let's change it up | |
05:56 | . Where the problem is a little more complicated . | |
05:58 | Let's say you have negative five multiplied by under a | |
06:01 | radical times negative 10 . So again , let's take | |
06:05 | the square root of each of these individually . The | |
06:07 | square root of this is going to be i times | |
06:10 | the square root of five , because I can't really | |
06:12 | simplify this anymore . So the eye comes out when | |
06:15 | you take the square root two times five is 10 | |
06:17 | . I can't simplify that either . So I'm gonna | |
06:19 | multiply that I times a squared of 10 . That's | |
06:22 | what this one is going to be equal to . | |
06:23 | But then when I multiply all this together , what | |
06:25 | I'm going to get is I times I which is | |
06:28 | I squared And I'm gonna have the square to five | |
06:31 | times the square root of 10 . But I can | |
06:32 | multiply under the radical to give me the square to | |
06:34 | 50 , Right Square to 50 . So then I'm | |
06:38 | gonna go over here and do a factor tree for | |
06:40 | 50 . Well , I have two times 25 And | |
06:45 | then 25 is five times five . So I have | |
06:47 | a pair . So there I go , I have | |
06:49 | a pair . And then finally what I'm gonna get | |
06:52 | over here , this I squared is negative one . | |
06:55 | So I'm gonna replace it with a negative one is | |
06:56 | always equal to negative one . And then the square | |
06:59 | to 50 has a single five that comes out and | |
07:02 | they square of two left over because that's what's left | |
07:04 | over . So the negative just multiplies negative five square | |
07:08 | root of two . So the answer is negative five | |
07:10 | times a squared of two . Now I wanted to | |
07:12 | do both of these problems on the board because I | |
07:14 | want to caution you something extremely extremely uh interesting . | |
07:19 | And also it's a gotcha . I really want you | |
07:21 | to be aware of . Here's the punchline anytime you | |
07:23 | have radicals like square roots or whatever with negative numbers | |
07:26 | underneath it . The kind of the order of operations | |
07:29 | or the priority order is I want you to deal | |
07:31 | with each radical separately and then if you're gonna be | |
07:34 | multiplying radicals , then you can do that later . | |
07:36 | So for instance , in this case we had the | |
07:38 | square to seven , we couldn't do anything there . | |
07:40 | What we did is we said , okay , let's | |
07:41 | make this I Time route seven . Then we can | |
07:43 | multiply the radicals and then we continue . Or here | |
07:46 | we turn this one into I . Route five . | |
07:48 | We turn this one into I wrote 10 , then | |
07:50 | we multiplied under the radicals and so on and got | |
07:52 | the answer . Remember many of you will remember what | |
07:55 | we've already learned in the past . That when we | |
07:57 | combine radicals , what we say , I'll put a | |
08:00 | note here . Is that the square root of a | |
08:03 | times the square root of B . Remember any two | |
08:05 | radicals multiplied together was equal to the square root of | |
08:09 | a time speed . Right ? So we did that | |
08:12 | here . Right ? We did seven times seven was | |
08:14 | a square to 49 . We did five times 10 | |
08:16 | was a square to 50 . We've been using this | |
08:18 | rule a lot . What I didn't mention back in | |
08:21 | that lesson because it wasn't relevant until now is that | |
08:24 | this rule is really only supposed to be followed when | |
08:27 | what is under these radicals or positive numbers ? Because | |
08:30 | prior to now having a negative number under the radical | |
08:33 | made no sense . Until now we have imaginary numbers | |
08:35 | of course . So when we combine radicals , the | |
08:38 | A . And the B should be positive in order | |
08:40 | to be able to combine them under a common radical | |
08:42 | like this . So , over here , I'm going | |
08:44 | to mend this rule and I'm gonna say A and | |
08:46 | B positive . Let me show you how you can | |
08:51 | get into trouble if you don't obey this rule . | |
08:53 | Okay , let's go up here to the previous problem | |
08:55 | . The original problem was squared of negative five times | |
08:57 | negative 10 . All right , so , I'm just | |
08:59 | gonna combine these radicals straight away . I'm gonna say | |
09:01 | negative five times negative 10 . It's positive 50 . | |
09:04 | So positive 50 square rid of positive 50 is is | |
09:08 | just going to give me what I have here square | |
09:11 | to 50 , which would just give me five times | |
09:13 | a squared of two . But I would have no | |
09:14 | I squared anywhere . You see if I follow this | |
09:17 | rule that I taught you for radicals With negative numbers | |
09:20 | under the radical when I multiply them , I'm going | |
09:22 | to get positive square to 50 and I'm gonna get | |
09:24 | I'm gonna get a positive answer here instead of a | |
09:26 | negative answer . So what we want to do is | |
09:29 | follow the rule that's fine . But just turn each | |
09:32 | radical into its imaginary number first before you combine any | |
09:35 | radicals together . Now it turns out in this 17 | |
09:38 | times negative seven is negative 49 . So if you | |
09:41 | take the square of -49 , you're actually still gonna | |
09:44 | get seven I back . So technically it does work | |
09:47 | if one of them is positive and one of them | |
09:49 | is negative and all that . But really the rule | |
09:51 | of thumb I want you to remember because it's the | |
09:52 | easiest thing to remember is that you can combine radicals | |
09:55 | like this as long as you have positive numbers underneath | |
09:58 | . Okay , no problem . And the second rule | |
10:00 | is if you have negative numbers under a radical , | |
10:02 | always always always deal with those radicals and make them | |
10:06 | into imaginary numbers first . Like we did in both | |
10:08 | of these problems before combining anything in the final answer | |
10:11 | . Otherwise you might run into problems getting getting the | |
10:14 | wrong sign of your answer . I wanted to caution | |
10:17 | that to you . Okay , so let's move on | |
10:20 | now that we've got all the kind of the basics | |
10:23 | out of the way and we're gonna crank through a | |
10:25 | bunch of additional problems just to give you practice what | |
10:28 | if we have seven times I as a quantity and | |
10:31 | we're gonna square that . Well we're going to the | |
10:33 | square is gonna apply to the seven and then we'll | |
10:35 | also apply to the eye . So to be seven | |
10:37 | squared I squared but seven times seven is 49 I | |
10:42 | squared is always negative one so that I can multiply | |
10:45 | those and say the answer is negative 49 . This | |
10:48 | is the answer . All right . Next problem . | |
10:53 | We're just gonna crank through a bunch of the none | |
10:54 | are really any harder than the other . What if | |
10:56 | I have negative I quantity squared ? A lot of | |
10:58 | students get tripped up by this . But you can | |
11:00 | think of it , you can bust it on out | |
11:02 | if you like . And think of it as well | |
11:03 | . This is negative I times negative I because it's | |
11:06 | the quantity that's squared . So it's this times itself | |
11:09 | . But then you know that negative times negative is | |
11:11 | positive and you know that items I . Z squared | |
11:14 | . So you really get a positive I . Squared | |
11:16 | . But you know that I squared is negative one | |
11:18 | . So that's gonna be the final answer . That's | |
11:20 | negative one . All right . Another way you can | |
11:22 | do of course you can do what I've done here | |
11:24 | . But you can all always think of things different | |
11:26 | ways . Or you can write or think uh as | |
11:30 | follows this negative I hear can be written as negative | |
11:32 | one times I write negative negative one times I is | |
11:36 | negative negative I and that whole thing can be squared | |
11:39 | . This is what this is equal to . Then | |
11:41 | . You can say , well the square would then | |
11:43 | apply to the -1 . And then it would also | |
11:46 | apply to the i this is going to give you | |
11:48 | a positive one , but this is going to give | |
11:50 | you negative one . And so that's the final answer | |
11:52 | . So if you want to think of it like | |
11:53 | that or if you want to multiply them together , | |
11:55 | you're gonna get the same answer . Of course . | |
11:57 | Both ways moving right along what if we have I | |
12:02 | times the square root of two , quantity squared . | |
12:06 | Again , the square applies to the eye and the | |
12:08 | square applies to the square root of two separately . | |
12:11 | So square root of two gets its own square here | |
12:14 | , it just goes in and applies to everything . | |
12:16 | But this gives me a negative one and this the | |
12:19 | square cancels with the square root just giving me a | |
12:21 | too , so I get negative too . It's very | |
12:23 | important when you're doing this stuff to write it all | |
12:25 | down . So you notice I didn't go in here | |
12:27 | and say , oh this is negative one and just | |
12:28 | do too many things . I wrote it all down | |
12:30 | so that I wouldn't make any sign errors . All | |
12:33 | right . What if I have negative one times the | |
12:35 | square root of three , quantity squared . This can | |
12:39 | go and apply to the negative . I'm sorry , | |
12:41 | negative I this is supposed to be negative . I | |
12:42 | tend to square the thing . It can go and | |
12:45 | apply to the negative I And then it can apply | |
12:48 | to the square root of three , quantity squared . | |
12:52 | Now what is this ? This is going to be | |
12:54 | negative items negative I write which is going to give | |
12:57 | me positive I squared . And then this cancels the | |
13:01 | square in the square root here . But this is | |
13:03 | a negative one , times three . So really I | |
13:05 | get a negative three , that's the final answer there | |
13:08 | . So up until now we've had no real fractions | |
13:10 | involved . We have just had either things being squared | |
13:13 | or imaginary numbers multiplied by another imaginary number . But | |
13:17 | what if we change the game a little bit and | |
13:19 | say what if I have negative to over ? I | |
13:23 | uh and I want to simplify that . Well you | |
13:25 | might look at that and say well it's already simplified | |
13:27 | , right ? Because it's you know , it's just | |
13:29 | a negative two on the top and I on the | |
13:31 | bottom , there's not much else I can do . | |
13:32 | Here's another rule of thumb I need to throw at | |
13:34 | you remember when we were simplifying radical expressions we said | |
13:38 | we never ever want a radical in the denominator of | |
13:41 | a fraction . We always want to get rid of | |
13:42 | it by by doing multiplying by the conjugate or whatever | |
13:46 | to get rid of the radicals . So the same | |
13:48 | thing is true of imaginary numbers because when you think | |
13:50 | about it , this is a radical . I mean | |
13:52 | this is a square root of negative one . So | |
13:53 | we don't want that in the bottom . Just like | |
13:54 | we don't want any radical in the bottom . So | |
13:57 | what we have to do is multiply this by something | |
14:00 | -2 over I . And it's very very simple . | |
14:03 | All you do is you multiply by the imaginary part | |
14:05 | that you have in the top and the bottom . | |
14:08 | Because all you're doing is multiplying by one . But | |
14:11 | you see what's going to happen when you multiply the | |
14:13 | bottoms , you're gonna get I squared . When you | |
14:15 | multiply the top , you're gonna get negative two . | |
14:17 | I but then on the bottom you know that I | |
14:20 | squared is just negative one . And these divide away | |
14:22 | and give you a positive to this is the real | |
14:25 | answer . Two times I this two times I is | |
14:28 | exactly the same thing . Is this negative two divided | |
14:30 | by I They are the same thing . But we | |
14:33 | consider this to be more simplified because there's no imaginary | |
14:36 | number in the bottom , which means there's no radical | |
14:38 | in the bottom . So we want to get we | |
14:39 | want to do that and we always clear it the | |
14:41 | same way . We just multiply by whatever the imaginary | |
14:44 | part is over itself . Final problem . Uh What | |
14:50 | if we have 8/3 times I and I say simplify | |
14:54 | that same sort of thing . I want to get | |
14:56 | rid of the eye that's in the bottom there . | |
14:57 | So I'm gonna rewrite my problem and of course I | |
15:01 | could multiply by three . I over three I I | |
15:04 | could do that . I mean it's going to be | |
15:05 | giving the same answer , but really I'm only required | |
15:07 | to multiply by the imaginary the imaginary part , whatever | |
15:10 | it is , divided by itself . Right ? I'll | |
15:12 | probably do it both ways just to show you here | |
15:14 | . But let's just go and do this when I'm | |
15:16 | gonna get here is eight times I on the top | |
15:18 | and on the bottom I'll get three I squared right | |
15:22 | ? But I know what three I squared is equal | |
15:24 | to have a I on the appeared this is negative | |
15:26 | one , I squared is negative one . So really | |
15:28 | have a negative three on the bottom here . So | |
15:30 | the answer that you would really circle on your test | |
15:32 | , you can float this negative sign in front eight | |
15:34 | I over three . This is what I would write | |
15:36 | and this is considered to be more simplified than what | |
15:39 | I had here . But I want to caution you | |
15:41 | that here . I multiply by I over I but | |
15:43 | you can of course do it , you can do | |
15:45 | it . Uh As follows , you can say 8/3 | |
15:47 | . I I can multiply by three . I over | |
15:50 | three . I a lot of students will do this | |
15:51 | . Just multiplying by the whole denominator . It works | |
15:53 | fine , too . Okay . Eight times three is | |
15:55 | 24 times I this is three times three is nine | |
15:59 | I squared . All right . So , what I'm | |
16:03 | gonna have is 24 . Uh I over And this | |
16:07 | is negative one times nine . So negative nine negative | |
16:11 | nine . And then you have to do some simplification | |
16:13 | of fractions , right ? Because 24 if you divided | |
16:15 | by three is gonna give you eight . So , | |
16:17 | you have the negative sign floating up in front . | |
16:19 | So divide this by three . You're gonna get eight | |
16:21 | . I divide this by three . You're gonna get | |
16:23 | three . It's gonna match exactly what we had before | |
16:26 | . So , you can multiply by whatever you want | |
16:28 | . As long as you have the imaginary part and | |
16:30 | the bottom , it's going to clear the I . | |
16:32 | Which is what you care about . So what you | |
16:33 | really want to do for any radical expression is get | |
16:36 | rid of the radical on the bottom . And because | |
16:38 | imaginary numbers are radicals basically squared of negative one . | |
16:42 | You want to get rid of any imaginary numbers that | |
16:43 | are in the bottom , anytime you multiply by whatever | |
16:46 | the imaginary part is in the bottom , you will | |
16:48 | always get rid of it as we have done here | |
16:51 | . All right , One last thing I want to | |
16:52 | talk to you about , some students will some teachers | |
16:55 | will teach this . I don't really like to teach | |
16:56 | this , but you're going to notice it over time | |
16:58 | . Notice that I divided I had an eye in | |
17:00 | the bottom and I don't like eyes in the bottom | |
17:02 | so I multiplied . And then I brought it upstairs | |
17:05 | . Notice what really happened when I brought the eye | |
17:07 | upstairs . It became too I or let me put | |
17:10 | it a different way compare this to the answer . | |
17:12 | The eye moves in the process of multiplying it moves | |
17:15 | upstairs . But in the process of it everything gets | |
17:18 | multiplied by a negative one . So this turned positive | |
17:21 | . Same thing happened here . This i when you | |
17:23 | multiply it , what ended up happening is that sort | |
17:25 | of moved upstairs . But in the process that multiplied | |
17:28 | by a negative . So the way I want you | |
17:31 | to solve your problems . I always want you to | |
17:32 | multiply by the imaginary number over itself or whatever to | |
17:36 | clear the denominator as as we're doing here . But | |
17:39 | in the back of your mind I want you to | |
17:41 | because you might be talked this , you can kind | |
17:43 | of think of just grabbing that I and moving it | |
17:46 | upstairs . But in the process you have to put | |
17:48 | a negative sign in front of the whole fraction . | |
17:50 | So if you didn't want to do all this multiplication | |
17:52 | , if you as you get more practice you can | |
17:54 | think of saying okay I'm gonna grab this , I'm | |
17:55 | gonna move it upstairs but then I'm gonna negate the | |
17:57 | whole thing here . I'm gonna grab this , I | |
17:59 | I'm gonna move it upstairs but I'm gonna negate the | |
18:01 | whole thing . So that's always true . Anytime you | |
18:04 | multiply to clear the imaginary number and pure imaginary number | |
18:08 | in the bottom , grab that negative one , move | |
18:10 | it upstairs and you negate the whole thing . But | |
18:12 | in the beginning I want you to show your work | |
18:14 | so you know where it's coming from . So this | |
18:16 | was just part one of several parts . We're simplifying | |
18:19 | expressions that involve imaginary numbers . Make sure you consult | |
18:22 | every one of these and follow me on to the | |
18:23 | next lesson . We're going to increase the complexity of | |
18:25 | these expressions so you get good practice , so follow | |
18:28 | me on and we'll do that right now . |
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11 - Simplify Expressions with Imaginary Numbers - Part 1 is a free educational video by Math and Science.
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