13 - Add and Multiply Imaginary Numbers - Part 1 - By Math and Science
Transcript
00:01 | Hello . Welcome back . We're now going to move | |
00:03 | from the general idea of simplifying expressions that have imaginary | |
00:06 | numbers to having expressions where we have to add or | |
00:10 | subtract or even multiply and divide imaginary numbers that we're | |
00:13 | gonna work on here . So it's easiest to do | |
00:15 | these kinds of things just by showing you examples rather | |
00:17 | than just trying to explain it . So let's do | |
00:19 | one . Let's say we have negative 25 we'll take | |
00:21 | the square of this . We're going to add to | |
00:24 | that the square root of negative 36 . So the | |
00:26 | first thing we do is we always tackle the radicals | |
00:29 | first . Now we know what the square to 25 | |
00:31 | is and we know what the squared of 36 is | |
00:33 | . So this is going to evaluate to five times | |
00:36 | I because we have that negative in there . So | |
00:38 | that gets evaluated as a square there and give you | |
00:40 | five I . And this one is going to give | |
00:42 | you a six I for the same reason squared of | |
00:45 | this is six squared of the negative one . As | |
00:47 | I now when you're to this step you're adding to | |
00:49 | imaginary numbers together . And my rule of thumb is | |
00:52 | exactly the same as what we've done for any kind | |
00:54 | of expression . You can only add imaginary numbers together | |
00:57 | if they both have an eye , if there if | |
00:59 | one of them has an eye and one of them | |
01:01 | doesn't then you can add them . Like you can't | |
01:03 | add two plus five . I I mean you can't | |
01:06 | combine them into some simpler things , you just can't | |
01:09 | add them . So just like variables , Everything has | |
01:11 | to match I and I match . Of course we | |
01:13 | can add that . We'll get 11 times by and | |
01:17 | that's the final answer . All right . What if | |
01:20 | we have uh negative 25 , take the square of | |
01:23 | this and we're gonna multiply it now by uh -36 | |
01:28 | . So we have a similar deal except instead of | |
01:30 | the addition , we're gonna multiply . So this we | |
01:32 | already know evaluates to five times I and this we | |
01:35 | already know evolved evaluates to six times I same sort | |
01:39 | of deal with multiplication . You treated exactly as you've | |
01:41 | done any multiplication with variables . So if this were | |
01:45 | five x times six X , you would say is | |
01:47 | 30 X squared . Well , in this case it's | |
01:50 | not an ex you say it's 30 I square because | |
01:53 | I times I But now , you know that I | |
01:55 | squared is always equal to negative one . So you | |
01:58 | just say that this is equal to negative one . | |
02:00 | And so the answer is negative 30 . The answer | |
02:03 | is negative 30 . All right . These are kind | |
02:07 | of fun problems . We're just going to cruise through | |
02:08 | . And what if we have three times the square | |
02:11 | root of negative two minus the square root of negative | |
02:15 | 50 . Now here we have to work a little | |
02:17 | harder because this square root has a larger number inside | |
02:20 | . So let's go off to the side and figure | |
02:22 | out what the factor tree for 50 looks like . | |
02:25 | And it's going to be , let's say five times | |
02:26 | 10 and 10 is five times two . And of | |
02:29 | course , you see the , the pair that we | |
02:31 | have right here . So for the first one here | |
02:34 | , yes , uh , the negative too , we | |
02:37 | still have a three times whatever is inside here . | |
02:39 | The square of negative two is going to be i | |
02:42 | times the square of two . Why ? Because the | |
02:45 | square of the negative one comes out as an eye | |
02:47 | and the squared of two is left over . Then | |
02:49 | this we have a subtraction here and the square root | |
02:51 | of negative 50 means we have an eye that comes | |
02:54 | out because of the squared of negative one . But | |
02:56 | the actual square root is five times the square root | |
02:59 | of two . Now , typically the way we write | |
03:01 | it as follows three times this quantity , we multiply | |
03:04 | what's on the outside of the radical three I times | |
03:07 | the square root of two . And then here we | |
03:09 | have we don't write it as I five . Typically | |
03:12 | write it as five times I times the square root | |
03:13 | of two . Now remember back to radicals you can | |
03:16 | only add or subtract radicals when you have exactly the | |
03:19 | same radical . In this case we do we have | |
03:21 | a matching thing so we can add or subtract what's | |
03:23 | in front and again we can only add or subtract | |
03:25 | these if we have both have an eye and they | |
03:28 | do both have an eye . So here it reduces | |
03:30 | to saying what is 3 -5 , you know that's | |
03:33 | equal to negative two . And then you have the | |
03:35 | eye coming along and you have the square root of | |
03:37 | two coming along negative two times I times the square | |
03:39 | of two . So let's take this general problem template | |
03:43 | that we have and go over here and change this | |
03:45 | to multiplication . So what we get , So that | |
03:47 | would be three times the square root of negative two | |
03:51 | , multiplied by negative square root of negative 50 . | |
03:57 | So we want to multiply that and see what we | |
03:59 | get . So here the same exact thing . This | |
04:01 | is gonna be three times I square root of two | |
04:05 | . Right ? Then we have inside of here we | |
04:08 | have a negative one . Let's do it like this | |
04:10 | , let's put this negative one in parentheses . Because | |
04:12 | here we're going to have the eye is going to | |
04:14 | come out And the square to 50 . We already | |
04:16 | figured out the square to 50 is five times a | |
04:19 | squared of two . So it's gonna be five square | |
04:22 | root of two like this . Make sure you understand | |
04:24 | the negative here just comes from what was above here | |
04:27 | . This comes out squared of negative one is I | |
04:29 | . And then this comes from the square to 50 | |
04:32 | being five times squared of two . And all this | |
04:34 | stuff is multiplied together . So to clean it up | |
04:36 | a little bit , I'm gonna have three times I'm | |
04:38 | squared of two . That's gonna be in the first | |
04:41 | term . Second term is negative . Then I have | |
04:43 | them to flip this around to be negative five times | |
04:45 | I times the square root of two . And I | |
04:48 | have to multiply these . So I multiply what's outside | |
04:51 | of the radical , negative times positive negative three times | |
04:54 | five is 15 items eyes I squared and then the | |
04:58 | square root of the of these guys is the square | |
05:01 | root of two times two is four . And multiply | |
05:03 | what's under the radical . All right . And so | |
05:06 | what I'm going to have here is -15 . But | |
05:08 | don't forget I square is always equal to negative wine | |
05:10 | . And the square root of four is too . | |
05:12 | So I have negative times negative positive and then 15 | |
05:16 | times two is 30 . So the answer is positive | |
05:18 | 30 . You know , it looks difficult . But | |
05:21 | the secret to doing all of these problems is to | |
05:24 | write down every single step , notice that I didn't | |
05:26 | skip any steps . I didn't try to to make | |
05:30 | this I times five squared or two and then multiply | |
05:33 | by the negative one all in the same step . | |
05:35 | I wrote it all down because when you solve enough | |
05:37 | of these problems like I have , you will find | |
05:39 | that you will make mistakes with imaginary numbers and mistakes | |
05:42 | always come from trying to do too many things in | |
05:44 | one step . So write down your steps . It's | |
05:46 | not that lunch guys write it down so you don't | |
05:48 | get the wrong answer each . Alright , let's let's | |
05:52 | crank through Um a couple more . I'm gonna work | |
05:55 | one . I think over to the side here because | |
05:58 | I want to work its companion down below . What | |
06:01 | if we have over here ? What if I have | |
06:04 | to times the square root of negative 24 minus the | |
06:09 | square root of negative 54 . All right . So | |
06:12 | I have to do some factor trees . I have | |
06:15 | to do some factor trees . Well , I have | |
06:16 | 24 . Right , So let's go do that one | |
06:18 | first 24 . What is that ? That's six times | |
06:21 | four . And you all know that four is two | |
06:22 | times two . And you know that six is three | |
06:24 | times too . So I have a nice pair and | |
06:26 | that's orphaned . Left over 54 can be written as | |
06:30 | nine times six . This is three times two and | |
06:34 | nine is three times three . So here's my pair | |
06:36 | here . So I have both of the square roots | |
06:37 | figured out . And so I've kind of clouded up | |
06:42 | my space here , apologize for that . So let's | |
06:43 | continue down here . This is going to be two | |
06:45 | times whatever this is , The square root of the | |
06:49 | negative part is going to give you the eye and | |
06:51 | the square root of the 24 is going to be | |
06:53 | a two coming out and then what's left over is | |
06:55 | three times two is six . That's what's left over | |
06:57 | under the radical . The minus sign comes from between | |
07:00 | there and then this is going to evaluate from here | |
07:03 | . But the square root of the negative means I | |
07:05 | have like I-54 , a single three is going to | |
07:08 | come out with a square of six leftover underneath it | |
07:11 | . So I need to clean up these terms a | |
07:13 | little bit two times two is four . So it's | |
07:15 | gonna be four I square root of six minus . | |
07:18 | I'm gonna flip this around 23 I square root of | |
07:21 | six and I have a square root of six matching | |
07:23 | . So I can subtract them . I have eyes | |
07:25 | matching so I can subtract them . What is four | |
07:27 | minus three ? It's just one . So it's gonna | |
07:29 | be one I times the square root of six that | |
07:32 | I times square to six . That's the final answer | |
07:35 | . So now what I want to do is take | |
07:37 | this problem right here with these numbers and sort of | |
07:39 | change it where this turns into a multiplication and just | |
07:41 | get some practice working . It's kind of companion there | |
07:44 | and we've done a lot of the work already . | |
07:46 | So it'll be pretty easy . So the problem is | |
07:48 | like this two times the square root of negative 24 | |
07:54 | . Multiply by negative square root negative 54 . Okay | |
08:01 | . Like that . What do we have left ? | |
08:04 | Uh what do we have left here ? So here | |
08:06 | we have this . So we already figured this out | |
08:08 | from last time . But let's write it down again | |
08:10 | . The 24 is going to come out to be | |
08:12 | two times the square root of six . So it's | |
08:14 | gonna be two times I from this times the square | |
08:17 | root of six exactly as it was before . But | |
08:19 | we're gonna multiply that by what we have here but | |
08:21 | we have a negative . So I'm gonna kind of | |
08:23 | like put a negative one I guess here I'll put | |
08:26 | in parentheses because I have to evaluate this which was | |
08:29 | three times the square root of six . But I | |
08:30 | had this negative so it's three times I times the | |
08:33 | square of six . Exactly as it was before . | |
08:36 | Right ? But then I have that negative here . | |
08:38 | So I want to make sure I'm not screw that | |
08:39 | up . So this will be four times I times | |
08:41 | the square root of six Inside here . It'll be | |
08:44 | negative three I square root six . Okay now and | |
08:49 | multiply the coefficients in front three times four is 12 | |
08:52 | but it's gonna be negative . So negative 12 items | |
08:56 | . I as I squared . Don't forget that . | |
08:58 | And then I'm gonna have the square root of six | |
09:00 | times six , which is 36 . All right . | |
09:03 | So , I'm gonna have the -12 . I square | |
09:06 | is always equal to negative one squared of six or | |
09:09 | 36 is six . And I can just now multiply | |
09:11 | between get positive and what I'm going to get is | |
09:14 | 12 times six is 72 . So positive 72 is | |
09:17 | the final answer . So obviously , I mean these | |
09:19 | problems are mostly there to give you practice more than | |
09:22 | anything . But obviously if you change something from a | |
09:24 | subtraction multiplication , you're gonna get a radically different answer | |
09:28 | . Here is a real number is an answer . | |
09:29 | Here is an imaginary number is an answer . When | |
09:31 | I did the subtraction right ? And a similar thing | |
09:34 | when I did the subtraction , I got kind of | |
09:36 | an imaginary result here . But when I changed it | |
09:38 | to multiplication , I got a real result . Basically | |
09:40 | when you multiply imaginary numbers together , you get real | |
09:43 | answers and that's kind of something you're just gonna have | |
09:45 | to get used to . Not something you have to | |
09:48 | memorize , but it's something that you'll get used to | |
09:50 | sing . All right . We have not too much | |
09:53 | more actually , just a few more . Um but | |
09:56 | they're kind of fun little short problems . What about | |
09:58 | I times the square root of 18 plus the square | |
10:03 | root of negative eight . So the first thing we | |
10:05 | have to do is go off to the side and | |
10:06 | figure out what the square roots really are . We've | |
10:08 | done 18 so many times , but it's nine times | |
10:11 | two and three times three . And then the eight | |
10:15 | , you all know by now is two times four | |
10:18 | , two times 2 . Here's your pair . So | |
10:20 | I just want to write them down . So we're | |
10:21 | all on the same page here . It's I times | |
10:24 | this , so it's gonna be I times with on | |
10:27 | the inside which is noticed . There's no negative number | |
10:29 | here . Right ? So it's not gonna be an | |
10:31 | imaginary answer here , but it's gonna be three times | |
10:33 | a squared of two . Three route to added to | |
10:37 | that . This is going to be two times the | |
10:39 | square root of two . But I have an eye | |
10:41 | there . So it's gonna be two times I times | |
10:43 | the square root of two because the square of the | |
10:45 | negative makes an I come out like that . So | |
10:49 | let me clean this up a little bit would be | |
10:50 | three I root two plus two . I route to | |
10:55 | . Now I can add them because I have a | |
10:57 | matching square root and I do have a matching imaginary | |
11:00 | number . So what I'm gonna have is five ISA | |
11:02 | coefficient to the square root of 25 I times square | |
11:05 | of two . That's the final answer . All right | |
11:10 | now , let's change this addition to multiplication just to | |
11:12 | kind of get practice . What if I have i | |
11:14 | times the square root of 18 ? Uh Let's do | |
11:17 | it like this , multiplied by the square of negative | |
11:19 | eight . Now we've already done the radical part . | |
11:21 | So what we have is i times what was a | |
11:23 | squared of 18 ? It was three times a squared | |
11:25 | of 23 route to . Then over here I have | |
11:30 | the squirt of negative eight which we already figured out | |
11:32 | was two times I Times the square root of two | |
11:36 | . So it's to route to but then there's also | |
11:38 | an eye in there and now I have to multiply | |
11:40 | things together . So what ends up happening is you | |
11:42 | multiply the coefficients here . So three times two is | |
11:45 | six items , eyes I squared And then the radicals | |
11:50 | multiply two times 2 is four goes under the radical | |
11:55 | . And so what I'm going to end up happening | |
11:57 | having is six . This becomes a negative one . | |
12:00 | This becomes a two and then I have 12 . | |
12:02 | And so you have a negative 12 And that's the | |
12:05 | final answer . Negative 12 . All right . Obviously | |
12:09 | , same sort of thing . You add these radicals | |
12:11 | together . You've got an imaginary answer whenever I multiply | |
12:13 | them . I got a real answer . Mhm . | |
12:15 | All right . Let's um that's two more . Two | |
12:20 | more which are basically cousins of one another . So | |
12:22 | they're almost the same problem . Um what if I | |
12:25 | have i times the square root of -98 . Subtract | |
12:31 | Square Root of Pososos 98 . So obviously , I'm | |
12:34 | gonna need to know how to take the square root | |
12:36 | of 98 . So let's go over here and say | |
12:38 | 98 . We know that two times 50 is 100 | |
12:41 | . So we know that two times 49 has to | |
12:43 | be 98 . And this can be seven times seven | |
12:45 | . And there's my pair . That's all I really | |
12:47 | need because it's the only radical I have . So | |
12:50 | , what I'm going to have here is it's gonna | |
12:51 | be I and on the inside is gonna be squared | |
12:54 | of 98 . Was seven times the square root of | |
12:56 | two . But because of this , I have an | |
12:57 | eye involved . So seven times I square up to | |
13:02 | right then I have a subtraction . And then this | |
13:04 | is going to be the exact same radical . Seven | |
13:06 | times a squared of two . Of course , there's | |
13:08 | no I because this was not square root of a | |
13:10 | negative number . So let's multiply with I in . | |
13:14 | So you're gonna have seven and the nytimes eyes , | |
13:16 | I squared square root of two minus seven from the | |
13:19 | square root of two . But this is negative one | |
13:22 | . So seven times negative one route to minus seven | |
13:26 | route to . So you're gonna get negative seven route | |
13:29 | to minus seven route to . Now you have a | |
13:32 | matching radical and you can just add the coefficient . | |
13:35 | So negative seven minus seven . Negative 14 square root | |
13:39 | two . That's the final answer . Negative 14 squared | |
13:42 | of two . Yeah . All right . Final problem | |
13:46 | that we're gonna have is gonna be a basically an | |
13:48 | exact copy of this one with multiplication . So I | |
13:52 | square root negative 98 multiplied . Bye On the Inside | |
13:58 | Negative Square root 98 . We have all the radicals | |
14:02 | . We know what the squared of 98 equals . | |
14:04 | Uh So here we have I times what do we | |
14:08 | have squared of negative 98 ? It's seven times the | |
14:11 | square root of two . But that's going to have | |
14:12 | an eye involved . So it's seven I square root | |
14:15 | of two , multiplied by . This is gonna be | |
14:18 | negative seven square root to write . Because the negative | |
14:21 | from here , seven squared of two is what the | |
14:23 | radicals equal to . Okay . And so what I'm | |
14:26 | gonna have is items I squared . So seven I | |
14:30 | squared route to And let's just wrap it up like | |
14:35 | this . And the negative seven route to hear this | |
14:38 | I squared is negative one . So it's going to | |
14:41 | be negative seven square of two times negative seven square | |
14:46 | or two . So you see the negative times negative | |
14:48 | gives you positive seven times seven is 49 square root | |
14:52 | of two times two , which is four . So | |
14:54 | you have 49 times two . So what do you | |
14:57 | get ? 49 times two ? Is 98 . Just | |
14:59 | double checking my answer . You get a positive nine | |
15:01 | game for the answer . None of these are hard | |
15:04 | . I want to do enough problems with you though | |
15:06 | so that you get comfortable with it . What you're | |
15:08 | doing is you're basically having to remember that the square | |
15:10 | root of a negative number involves I write and you | |
15:13 | also have to remember that I squared is negative one | |
15:15 | . Other than that you treat I as a variable | |
15:18 | as far as adding , like terms as far as | |
15:21 | exponents , things like that is basically treated like a | |
15:23 | variable , although you know it's not a variable . | |
15:25 | You have to substitute I squared being negative one . | |
15:28 | And so that can sometimes make the answer real even | |
15:31 | when your problem was imaginary . So follow me on | |
15:33 | to the next lesson , we're gonna continue multiplying and | |
15:36 | adding and dividing these imaginary numbers to give you more | |
15:38 | practice right now . |
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