15 - Complex Numbers & the Complex Plane - Free Educational videos for Students in K-12 | Lumos Learning

15 - Complex Numbers & the Complex Plane - Free Educational videos for Students in k-12


15 - Complex Numbers & the Complex Plane - By Math and Science



Transcript
00:00 Hello . Welcome back to algebra . In this lesson
00:03 we're going to talk about something really , really important
00:05 . We're going to talk about the real numbers and
00:08 the imaginary numbers and bring them together and talk about
00:11 the concept of what a complex number is , complex
00:14 number . And so we're gonna talk about complex numbers
00:16 and we're also going to talk about the complex plane
00:19 . Now a lot of students look at complex numbers
00:20 when they first learn it and it looks difficult and
00:23 and hard , but I want to break it down
00:25 for you here and show you how easy it is
00:27 to understand what these things really are and how we
00:29 use them in math . Right ? But more than
00:32 that in the beginning , I want to show you
00:34 an outline of all of the numbers that we use
00:36 in math , starting with complex numbers and how it
00:39 trickles down to all of the real numbers and the
00:41 integers and all the things that we've been using .
00:43 So we're gonna draw like a hierarchy of all of
00:45 the numbers so that you understand where they all fit
00:47 into their place and you'll understand at the end of
00:50 it that the complex number is really the most general
00:53 kind of number that we have in math . It's
00:54 the granddaddy of all of the numbers . And then
00:57 the last half of the class , we're gonna learn
00:59 that complex number can be represented on what we call
01:02 the complex plane . All right . So we're gonna
01:04 do a hierarchy . So up until now , up
01:06 until algebra , all of the numbers that you've ever
01:09 learned in your life are all real numbers , whether
01:12 they're fractions or decimals or negative numbers are positive numbers
01:16 , integers . Uh You know any of those numbers
01:19 ? Those are all real numbers . And then we
01:21 get to this point in algebra where we learn that
01:23 we also have these things called imaginary numbers . So
01:25 the real numbers are just numbers , right ? And
01:27 the imaginary numbers have an eye involved . We talked
01:30 all about that now it turns out that there is
01:33 a more general type of number called a complex number
01:36 and that is a number that has a real part
01:39 right ? A real part of the number but also
01:41 has an imaginary part two . So you see all
01:44 those numbers that you've dealt with like fractions and decimals
01:46 and investors , those are really purely real numbers .
01:49 Just a real part like five potatoes is a purely
01:53 real number . There's no imaginary part to that .
01:55 But then we also talked about these imaginary numbers which
01:58 more generally we should call them purely imaginary numbers .
02:01 In other words five I six I seven half I
02:05 . Those are all purely imaginary because there's no real
02:07 part of those numbers . Those are pure imaginary numbers
02:10 over there . We had the pure real numbers from
02:12 before . If we join them together so that we
02:15 have a real part and an imaginary part , then
02:18 we form the more general number called a complex number
02:22 . And you're gonna learn you're gonna use complex numbers
02:25 to solve the equations from algebra on into more advanced
02:29 math . There never they're never gonna go away .
02:31 Complex numbers are the most general number that we have
02:33 and they pop up in the solutions of tons of
02:36 equations in real life . So let's just draw a
02:38 hierarchy of what these things really are . So we
02:41 have the idea of what we call a complex number
02:48 , a complex number and when we say a complex
02:51 number , it's any number that has a real part
02:54 , but also an imaginary part . So the real
02:56 part would be a for instance , I'm using the
02:58 letter A represent the real part and the imaginary part
03:02 would be some number B times I . So a
03:05 purely real number would just be like five , I'm
03:08 gonna represent that with A in a pure imaginary number
03:11 would be whatever like five I . Or six I
03:13 here I'm just using the letter B . But that's
03:15 purely imaginary . If you stick them together , you
03:18 have what we call a complex number . So let's
03:20 give some examples of complex numbers . Um the number
03:24 four plus five I this is a complex number .
03:28 Now , the first time you look at this it
03:29 looks crazy because you have a plus sign there and
03:32 up until now you're all used to dealing with numbers
03:35 that is just like a single entity like six or
03:38 negative seven or for imaginary numbers like 10 I .
03:41 Or something . So people look at that plus sign
03:43 and you're like that's crazy , why can't I add
03:45 them together ? Why is there a plus sign there
03:48 ? And that is because the real part of the
03:49 number is four . The imaginary part of the number
03:52 is five or five I right ? But if you
03:55 try to add them together , you really can't do
03:58 the addition because as we know when you add imaginary
04:00 numbers you can't add them unless the eye is present
04:03 in both cases . So we say that this number
04:06 is more general type of number , it has a
04:08 real part of it , but it also has an
04:10 imaginary part of it , right ? And since I
04:13 really can't add them together , I just have to
04:15 leave the plus sign . They're just like X plus
04:17 Y . I can't really add that together either ,
04:19 so I have to leave it X plus y .
04:21 So you need to get used to complex numbers having
04:23 this plus sign here . But really all you need
04:25 to know is there's a real part there's an imaginary
04:27 part . I try to add them but I can't
04:29 really do it so I have to leave it like
04:30 this . Another example of a complex number would be
04:35 for instance um -3 -2 . I will put a
04:41 little cynical in there to tell you that these are
04:43 different things negative three minus two . I so the
04:46 real parts negative three and the imaginary parts negative two
04:49 . I for this number another example would be six
04:53 plus zero times I . So this is kind of
04:55 getting into what I was telling you see this is
04:57 the real , the real part of this is six
05:00 . The imaginary part is actually zero . So every
05:03 number that you've ever known like 567 those are all
05:07 will be called the real numbers but really they're all
05:10 complex numbers but they're complex numbers where the imaginary part
05:14 doesn't exist . So any real number that you have
05:16 is really a complex number with the imaginary part set
05:19 to zero . So I'm just showing you that that
05:21 these are complex numbers two . And then another crazy
05:24 example you can have , for instance , negative two
05:26 times I times the square root of three . This
05:31 is an imaginary number . We call that a purely
05:33 imaginary number , but it's also a complex number because
05:37 there's an implied zero here , there is no real
05:39 part of this is purely imaginary . So the idea
05:42 of a complex number is it's any number where you
05:44 have a real part and an imaginary part , the
05:47 real part sits by itself . The imaginary part is
05:50 linked by a plus sign , as we have in
05:51 these first two examples , but of course I can
05:53 set the imaginary part equal to zero , or I
05:56 can set the real part equal to zero , but
05:58 there's still complex numbers . So really any number possible
06:01 is always a complex number because any number will always
06:04 be able to be ridden as a real part in
06:06 an imaginary part . And you might say why do
06:08 I care about this real part imaginary part ? Remember
06:11 the last lesson I gave you examples of how imaginary
06:14 numbers pop up in applications , you know in science
06:18 and math you just can't get away from them .
06:20 And I kind of fibbed a little bit , it's
06:23 not just the imaginary numbers that pop up , it's
06:25 really the concept of a complex number is what really
06:27 pops up all the time . So in three or
06:29 four years when you solve a really complicated equation and
06:32 calculus you might get a complex answer , write a
06:35 real part and an imaginary part . But as we
06:38 combine these complex numbers together in different functions , we
06:41 might get a real answer out of the whole thing
06:43 . As we discussed in the last lesson , we
06:44 talked about uh with when I talked about real uh
06:48 when I talked about imaginary number , sorry about that
06:51 . So the complex number is the most general thing
06:53 . All right . And that's why it sits at
06:56 the top of the food chain . So I'm gonna
06:58 draw a little box around this . So underneath ,
07:02 the idea of a complex number is what we talked
07:05 about . We have the real numbers and we have
07:07 the imaginary numbers . They are what live as Children
07:10 under the granddaddy , which is the complex number that
07:12 encompasses everything . So under here let me go and
07:16 draw another little arrow down because it lives kind of
07:19 underneath here underneath complex number . And on the other
07:22 board we'll do the other side . We have what
07:24 we call the pure imaginary numbers . What numbers pure
07:35 imaginary those are numbers ? Would just basically would just
07:37 I Right . And so what kind of examples would
07:41 that be ? Well , it's it's gonna be ,
07:42 for instance , A plus B I . Which is
07:45 a general form of a complex number . However ,
07:47 with a set to zero and be not equal to
07:52 zero . In other words , it's a complex number
07:54 , but will reset the real part to zero and
07:56 then this part is non zero . So to give
07:58 you some examples of that , some concrete examples ,
08:01 we know what these guys are . These are just
08:02 the imaginary numbers three . I that's a purely imaginary
08:06 number , negative two . I that's a purely imaginary
08:08 number . One half I that's a purely imaginary number
08:12 . So you can have fractions of course involved ,
08:16 I times the square root of three . That's a
08:18 pure imaginary number . Don't let the square root get
08:20 in your way . I mean , it's still just
08:22 an imaginary number . Right ? Of course we can
08:25 let's go down below here , we could have negative
08:26 something crazy too , I times the square root of
08:28 seven . So any number of radicals , negative positive
08:32 fractions doesn't matter if you have an eye and it's
08:35 just purely imaginary . We call that obviously the purely
08:39 imaginary numbers . So let me draw a box around
08:41 all of this stuff . All right . So ,
08:44 if this describes what lives under the complex number ,
08:48 one subset of complex numbers , are there purely imaginary
08:51 numbers and the other subset are gonna be the purely
08:53 real numbers ? So , we're gonna go over and
08:56 draw a little era this way and I'm going to
08:59 kind of continue it here . Let me see how
09:01 I'm gonna do this . Yeah , let me go
09:04 right over here , make a way over to the
09:06 center of the board . I'll come down right here
09:08 and I'm gonna show the other half of what lives
09:10 under a complex number . We call these pure real
09:16 numbers . Actually you don't even need the word pure
09:21 . They're just real numbers . Right ? So what
09:23 would be , what would that look like ? Well
09:26 , it's gonna look like the general form of the
09:28 complex number A plus B . I , but B
09:33 is gonna equal to zero and A is going to
09:36 equal not equal to zero . So in other words
09:38 , it's a complex number where there's no imaginary part
09:41 . Only a real part . That's why we call
09:42 it a purely real number or just a real number
09:45 . All right . So what are examples of real
09:47 numbers Now ? When I say real numbers , it's
09:49 literally anything on the number line . Any number ,
09:52 integers , non integers , fractions , decimals , whatever
09:55 . As long as it lives on that real line
09:58 , it's called a real number . Right ? So
10:00 it's literally anything you can think of as far as
10:03 like basic number . So for instance the number six
10:05 , That's a real number . The number square root
10:07 of two . That's a crazy radical . That's that's
10:10 a real number . The number two pi pi goes
10:12 on forever as far as the decimal places . But
10:14 it's still a real number lives on the number line
10:17 . Negative numbers are totally fine that live on the
10:19 number line . So like negative one half , fractions
10:21 are totally fine too . -4 . That's a negative
10:25 whole numbers or negative energy on the number line .
10:28 And let's go down here and say um Yeah ,
10:32 let's say two points six to when they bar over
10:36 the two , you see , this is 2.6 to
10:40 to to to to to to to to to to
10:41 that repeats forever . Right ? So if it repeats
10:44 forever , if it doesn't repeat for if if this
10:47 guy has a is a repeating decimal or if it's
10:51 like square root of two or pi where the decimals
10:53 never repeat . So this is basically rational or irrational
10:56 . It doesn't matter . Everything is contained under the
10:59 umbrella of a real number . If it lives on
11:02 the number line whether or not it's a repeating decimal
11:06 or not whatever negative positive basically it's a real number
11:11 . So you see what we have is the complex
11:13 numbers of the granddaddy because they contain real and imaginary
11:17 parts . But as a subset of the complex numbers
11:19 we have just the real part which is pretty much
11:21 any number you can think of on the number line
11:23 . And then we have the pure imaginary parts which
11:26 are any number that would live on , kind of
11:27 like the imaginary number line , which we'll talk about
11:30 in a minute , but pretty much fractions , decimals
11:32 , negative , positive , whatever . As long as
11:34 there's an I in there , it's a purely imaginary
11:36 number . Now let's go a step further . Let's
11:38 take a look at what lives under the real numbers
11:41 . Real numbers are in general broken up into two
11:43 parts , right ? We have the rational numbers which
11:47 can be written as fractions and we have the irrational
11:49 numbers . Those are the general to general kinds of
11:52 real numbers we can have because all numbers that live
11:54 on the number line can either be written as a
11:56 fraction or not written as a fraction . So those
11:59 are the rational and the irrational . So we have
12:01 two categories . So the first one it's called rational
12:07 numbers and I know you've all heard about rational numbers
12:12 , we've discussed it many times ourselves . Uh in
12:15 this class , the rational numbers can be written as
12:17 fractions . The number six can be written as a
12:19 rational number because it can be written as 6/1 .
12:22 Which is a fraction right ? Of course the number
12:25 negative . 3/4 is a fraction itself negative . Doesn't
12:28 matter if it can be written as a fraction ,
12:29 It's still a rational number . What about certain decimals
12:32 ? Let's look at 1.25 . Well you might say
12:35 well that's not a fraction . But I can write
12:36 this as a fraction . I can go in a
12:38 calculator and figure out something divided by something to give
12:41 me 1.25 . And then as a final example ,
12:45 Let's say negative 6-7 with a repeating decimal bar over
12:51 the 27 So this is negative 6.27 to 7 to
12:54 7 to 7 to 7 to 7 to seven .
12:57 Right ? So the basic idea here is let me
13:00 circle this and then we'll get back to it .
13:01 So what live under the real numbers are the rational
13:06 numbers and then we're gonna talk in just a second
13:08 about the irrational numbers . So it's obvious that this
13:11 is a rational number because it can be written as
13:13 a fraction . It's obvious that this is because it's
13:15 a fraction these two trips students left basically if a
13:19 if a decimal can be written if a decimal is
13:21 truncated like 1.25 or 3.75 , then you can always
13:26 write it as a fraction always . That's something you
13:28 just , you learn as you kind of work through
13:30 the problems anytime it's a truncated decimal that stops ,
13:34 it can always be written as a fraction . So
13:35 they're all rational . Also if a decimal goes on
13:39 and on forever , even if it goes on and
13:40 on and on forever . But in a repeating way
13:43 like to to to to to that's a repeating pattern
13:46 or 27 to 7 to seven forever Or 399399399399 .
13:52 If it repeats in any pattern forever and ever then
13:55 it can always be written as a fraction . So
13:57 it's rational . But the flip side of this is
14:00 we have other numbers that are not rational and we
14:02 call those irrational numbers and I know that you know
14:11 what these are , we've talked about them before .
14:13 They're the very special numbers that cannot be written as
14:15 a fraction . Right ? So those are numbers like
14:18 pi no matter what anyone has ever told you ,
14:20 you cannot write Pie is a fraction of what I
14:23 say . When I say written as a fraction ,
14:25 I mean written as a ratio of whole numbers like
14:28 three and four . Those kinds of numbers that when
14:31 I say written as a fraction that's what I mean
14:33 . Alright . The number negative pi over four ,
14:37 Pie itself is irrational . So of course prior before
14:39 is also irrational . The number negative square root of
14:43 two . If you actually take square root of two
14:45 and calculate what it is , there's no repeating patterns
14:48 of the decimals . The decimal just goes on and
14:50 on and on forever . Yeah Some more examples ,
14:54 the cubed root of five , that's irrational And the
14:59 number E which we're gonna talk about later . When
15:01 we talk about logarithms and exponential equations E is a
15:04 special number , it's about 2.71 but it has a
15:07 decimal that goes on and on forever , it never
15:10 ends . And there's also no pattern to the decimal
15:13 . So the basic idea is irrational numbers cannot be
15:16 written as fractions because they have decimals that go for
15:19 on forever with no pattern . So if there is
15:22 no pattern to the decimals that go on and on
15:24 forever , then there's no way to write it as
15:27 a fraction . Whereas these , if they have truncated
15:29 decimals or decimals that go on forever with a pattern
15:32 , you can write them like this in terms of
15:35 a fraction . So we call them rational . So
15:38 these are the two main kinds of numbers that live
15:40 underneath the concept of the real numbers . That's pretty
15:43 much all the numbers that can live on the number
15:45 line . Either they can be written as a fraction
15:47 or they cannot be written as a fraction because the
15:49 decimals go on and on forever with no pattern notice
15:52 , negative positive fraction . All that stuff is uh
15:54 negative positive is okay . It's just the decimal repeating
15:58 nature of it determines if it can be written as
16:00 a fraction or not . So that's the end of
16:02 the road for the irrational numbers . But we still
16:04 can break down the rational numbers a little bit more
16:07 if we want to . Right ? So I'm gonna
16:10 go here and I'm gonna draw another branch right here
16:14 . So underneath the rational numbers you can have the
16:16 fractions and you can also have the integers . That's
16:19 how we can break that up some more . So
16:21 we have fractions . Yeah right ? What are some
16:25 examples of fractions ? I know that you all know
16:27 what fractions are but you know , we have things
16:29 like one half that's fraction . We have 3/4 negative
16:32 , 3/4 negative positive . Doesn't matter . It's a
16:34 fraction . 6/13 that's a fraction . So of course
16:38 that's one half of the of the types of things
16:41 that can live under the rational number umbrella . The
16:44 fractions But we also have the integers , right ?
16:48 Because we say six is a rational number . Also
16:50 it can be written as 6/1 , but obviously it's
16:53 different than the fraction . So we can have the
16:55 fractions . We can also have the integers . So
16:58 what we can do is let me come down here
17:01 and say , well let's go down here and let's
17:03 break this into integers . What are integers ? Well
17:10 , integers are basically whole numbers negative and positive .
17:15 Whole numbers that live on the number line and including
17:18 the number zero . So for instance , negative six
17:20 is an integer . You know that negative two's and
17:22 insecure zero actually is an integer and the number four
17:25 is an integer . The number 17 is an integer
17:27 . So , you see there basically whole numbers including
17:30 zero , but they can be negative and positive .
17:33 We call those interviews . All right . And then
17:36 we're going to box that up . We're almost done
17:39 with this . And then we'll move along along but
17:42 underneath the integers what can live under this . Well
17:45 under this , we have the negative integers . We
17:48 have the positive integers . And then we have the
17:49 very special number called zero , which is kind of
17:52 in the middle . So you can break that up
17:54 into those three cases . So let's go over here
17:57 and we have the negative integers . Right ? And
18:06 we have the very special # zero . And we
18:10 have the positive ones other than zero like 123456789 Those
18:15 things we have a special name for those those are
18:17 called and natural numbers . So let's just write a
18:24 couple of quick examples to show what these are .
18:26 The negative images as you might guess would be things
18:29 like negative five negative two negative one , negative 30
18:33 for whatever . Zero is a very special number .
18:35 It's the only one that lives right there . And
18:37 the natural number is everything left over on the positive
18:40 side . 123 for dot dot die . Basically the
18:44 positive kind of whole numbers over there . So we're
18:47 gonna circle this right and we're gonna circle this guy
18:55 and we're gonna circle this guy and we have kind
19:02 of a special name for zero . And the natural
19:05 numbers that kind of can be you probably heard of
19:09 the term hole uh , numbers because these are the
19:16 numbers that are kind of like the counting numbers ,
19:17 including you also have zero there . So this is
19:20 the general idea . Why did I spend all the
19:22 time to put this on the board ? Because I
19:23 want you to understand the hierarchy of how important a
19:26 complex number is . You learn when you're a kid
19:29 how to count on your fingers . Those are the
19:30 natural numbers . And then you introduced the concept of
19:33 zero , which is nothing at all . Right .
19:35 It's not really something you can count . But it's
19:37 a concept that we learn when we're really young .
19:39 We call these the whole numbers but they're really two
19:41 different things . These are the accounting numbers that we
19:44 can count . Then we have of course zero .
19:45 Then you learn the concept of the negative number ,
19:47 which is really weird when you first learn it .
19:49 But we learn about that as being kind of like
19:52 when I owe you something , you know that's what
19:54 a negative number is . Those all fall under the
19:56 umbrella of what we call integers , which is zero
19:59 . The positive whole numbers and the negative whole numbers
20:02 , right ? But integers are only half the story
20:06 of the rational numbers . We have integers and we
20:08 also have fractions . You learn about fractions which are
20:10 basically parts of the integers , essentially fractional parts of
20:14 the integers . Together they make up the concept of
20:17 irrational number which are basically any number that can be
20:20 written as a fraction . And then of course you
20:22 have the irrational numbers which can be written as fractions
20:25 . Those are involved the special numbers like pi square
20:27 roots of numbers , Q groups of certain numbers E
20:30 and things like that , decimals that never ever repeating
20:33 a pattern . And then those live under the impression
20:36 umbrella of the real number . So every all of
20:38 these numbers from here down live on a number line
20:40 somewhere with a purely real part you see here is
20:44 an irrational number . Here's an irrational number and all
20:47 the other ones are rational , but they all live
20:49 under the number line . And then we have ,
20:51 we've learned now in algebra the imaginary numbers , which
20:54 are totally totally separate ball of wax numbers that exist
20:58 on a totally different number line . And they have
21:00 their own kind of life over there , but they
21:03 fall under the general umbrella of what we call a
21:05 complex number . So the complex number lives at the
21:08 very tippy top of the pyramid . It literally is
21:11 all of the possible numbers that we know to exist
21:14 . And so we used them to solve equations .
21:16 They have very practical applications , but that's generally the
21:19 idea of what a complex number is . And a
21:21 complex number will always have a real part in a
21:24 complex number will always have an imaginary part and denoted
21:26 by the eye . So now we want to talk
21:28 about this thing called the complex plane . A lot
21:30 of books will just throw you into the complex plane
21:33 and say , here it is , enjoy it .
21:34 And it kind of gives students a little bit of
21:36 heartburn . So I want to break it up just
21:38 a tiny bit slower . So you really understand what
21:40 the complex plane is . So here we have this
21:43 thing called complex numbers . I want you to kind
21:45 of keep that in the back of your mind .
21:47 Kind of like just think we're gonna get back to
21:49 it , but just keep in the back of your
21:50 mind . First , I want to talk about something
21:52 , you know more about the purely real numbers .
21:55 You know about purely real numbers . We've been using
21:57 them forever and ever . All right , so let's
22:00 talk about that for a second . We have real
22:03 numbers . Numbers can be graphed on a number line
22:15 , write a regular old number line . Nothing fancy
22:17 nothing . Crazy . Right ? So let's do that
22:20 real quick . Let's just draw a quick little number
22:21 line here . Go down memory lane . We did
22:24 these a long time ago . We put it ,
22:26 we say zero exists here . And let me put
22:29 a tick mark , your tick mark here , tick
22:32 mark , your couple tick marks here . Now because
22:35 we're talking about real numbers , you all know that
22:37 this number line is real numbers , but I'm just
22:39 gonna put the word or the little letters R .
22:41 E . Over here just to remind you that this
22:43 is really we're just talking about the real numbers .
22:45 This is , you know , numbers that we've been
22:46 dealing with all of our life . But the point
22:49 is is you can graph numbers on the number line
22:51 . Any real number can be graphed on the number
22:53 line . So for instance , the number one ,
22:54 we can just put it right there and say it
22:56 lives right there in the number line , Right ?
22:59 The # three is a real number . We can
23:00 just say it lives right there in the number line
23:03 . The number negative too exist right there . We
23:06 say it lives on the number line . Of course
23:07 the number of zeros on the number line . And
23:09 then of course all the fractions and everything else can
23:11 live there too . We can say this is one
23:12 half it exists on the number line . If I
23:15 want to put pie on the number line , it's
23:16 going to be 3.14 to live a little bit to
23:19 the right of three Square root of two is on
23:21 here . 141 is squared of two of the live
23:24 around here . You get the idea , no matter
23:26 what number you want to plot on the number line
23:28 . You just draw the horizontal number line , tick
23:30 marks . And then you put the point on that
23:34 line anywhere you want . It is going to find
23:36 a home somewhere on this line . Every real number
23:39 that you know about . All right . But then
23:41 we have the idea of the imaginary numbers . So
23:45 we have this idea of imaginary numbers . We know
23:47 that that's going to come into play some sort of
23:49 way . But here's the deal . If we start
23:51 plotting all of the imaginary numbers on this same graph
23:54 , it's gonna get really confusing because if I start
23:56 plotting You know the number one and then also the
23:59 number two I on the same exact number line .
24:02 I mean you could do it but then you're gonna
24:04 get really confused as to which numbers on that line
24:06 are real and which numbers on that line are imaginary
24:09 . But we would like to plot the imaginary number
24:11 . So what do we do ? So what we
24:13 do is we graph the imaginary numbers on a vertical
24:27 . Why in other words we just turn the line
24:29 sideways . So instead of doing a horizontal we draw
24:31 a totally separate line like this . Why do we
24:34 draw it up and down ? Well we wanted to
24:36 look different than this one because we're gonna end up
24:38 writing the imaginary numbers . So we're gonna put the
24:41 word I am there to remind me that what I
24:43 am actually plotting here is imaginary numbers . So here
24:46 I have my zero point on this . Okay and
24:49 I can put tick marks on this line . Just
24:51 like I have tick marks here . But you see
24:54 if I want to plot numbers on this line they
24:56 have to be of course imaginary . Right ? So
24:58 let's plot a couple of numbers . So here's one
25:00 too . But this is not to this point is
25:03 not to this point is to I because this is
25:06 not a number line with real numbers , it's a
25:08 number line only of imaginary numbers . And this is
25:11 not the number two . This is number two I
25:13 Right . And then of course you could say negative
25:16 one , negative two , negative three , negative four
25:18 . Here's a point . Let's just plot it right
25:19 here , what's this ? It's not negative four ,
25:21 it's negative four I Right . And then of course
25:24 I can plot something in between zero and one here
25:27 and this would be negative one half , but it's
25:29 not negative one half . Its negative one half .
25:30 I you see you see the idea this point right
25:33 ? Here is not the number three it's three I
25:35 . So we have the idea that if you have
25:37 purely real numbers , you just plot them on a
25:39 horizontal number line like this . And if you have
25:42 purely imaginary numbers you just plot them on a vertical
25:46 bar like this . We use the word Ari to
25:48 tell me I'm only plotting real numbers here . We
25:51 don't need to do that when you're in basic math
25:52 . But now that we're here we need to to
25:54 show what we're plotting . And then we use the
25:56 letter . I am to tell me that I'm only
25:58 plotting imaginary numbers . And when I plot the number
26:00 , I need to make sure and put the eyes
26:02 here to remind me that I'm not just plotting real
26:04 numbers , I'm plotting imaginary numbers . Right ? So
26:07 then if we know how to plot the real numbers
26:09 here and we know how to plot the imaginary numbers
26:12 here , how do we plot complex numbers ? Because
26:15 complex numbers have a real part , but they also
26:18 have an imaginary part . So here's the punchline of
26:21 the whole thing . We use a vertical line and
26:25 a horizontal line to form a plane we call it
26:27 the complex plane . And all of the complex numbers
26:31 live in the complex plane and can be plotted on
26:34 the complex plane . So here's your handy dandy complex
26:37 plane right here , notice that we have a real
26:40 axis , all of the purely real numbers that you
26:43 want to plot in your entire life you've ever done
26:45 , you just plot them on this real access down
26:47 here . Right ? So in previous math classes ,
26:50 you didn't even know about the imaginary axis . So
26:52 you just poof disappear it and you only look at
26:54 this right here . But now we know about imaginary
26:57 numbers . So the purely imaginary numbers are plotted purely
27:00 on this axis right here , just as we have
27:01 done because they only have imaginary parts . But as
27:05 you now know , the complex numbers have real parts
27:07 and imaginary parts . So what we're gonna do is
27:10 we're going to plot a few , a few imaginary
27:14 numbers . So let's go and do that . What
27:17 would uh let's just pick something , let's take something
27:20 easy right here . So we have 12 Let's put
27:21 a plot , man , let's do it in um
27:23 let's do it in a different color . What would
27:26 this point ? B right here , here's a point
27:28 right here . Well , the real part is this
27:31 so there is no real part at all , but
27:33 the imaginary part is three . I so the way
27:35 you would write this down , as you would say
27:37 , this complex number is zero plus three , five
27:41 because it has a real part and an imaginary part
27:44 that makes it a complex number . The real part
27:46 is just zero . So it's a purely imaginary thing
27:48 that lives right there . All right , um What
27:52 would this point ? B right here , let's go
27:54 up here . What would this point ? B Right
27:56 here ? Well , the real part is negative one
27:59 and the imaginary part is for So really what I
28:02 would put is negative one plus four , I negative
28:07 one plus four . I this is the complex number
28:10 that's associated with this point right here . So crank
28:12 in right along . What would a number over here
28:14 be ? Let's go and do it not right here
28:16 . The real part is negative three . So I
28:18 have negative three but the imaginary part is actually zero
28:22 because there is no imaginary part because it's living right
28:24 there in the axis . So this is negative three
28:26 plus zero . I what would this point ? Right
28:30 here , B well , it's got a real part
28:33 of three and it's got an imaginary part of just
28:35 a positive one . I so this is gonna be
28:37 three plus I . Or three plus one . I
28:40 however you want to look at it . Um What
28:43 about this one on the access down here ? Well
28:45 it's got a real part of two and it doesn't
28:47 have any imaginary part at all . So this is
28:49 gonna be two plus zero . I like this .
28:52 That's what that's equal to . Let's go and plot
28:55 some down here and see what they look like .
28:56 What if we go way down here and put a
28:58 point down here what is that point signified down there
29:02 ? Well the real part is nothing and the imaginary
29:05 parts negative four so it's zero plus what you say
29:08 zero plus a negative for but really you want to
29:10 write it as zero minus four . I you were
29:14 right at zero minus four . I and then let's
29:18 see if we do this . Let's point right here
29:20 positive real part of one and negative two . I
29:23 in the imaginary part means one minus two . I
29:27 the real part being one . The imaginary part being
29:30 the negative to I . And then the last one
29:31 I'm gonna do is going to be this one right
29:34 here . The real parts negative one . But the
29:36 imaginary parts negative too . I so it'll be negative
29:39 1 -2 I negative one minus two . I so
29:44 let me just double check myself . So this point
29:47 is negative one plus four . I that's good .
29:50 This one is zero plus three . I that's good
29:53 . This one is three plus positive one . I
29:55 that's good . This one's two plus zero . I
29:58 that's good . This one's negative three plus zero .
30:01 I that's good . This is -1 -2 . I
30:04 that's good . This is positive 1 -2 . I
30:07 that's good . And this is zero . And the
30:09 real part negative for I for the imaginary part .
30:12 So there you go . That's the idea of a
30:14 complex plane . Now when students first learn the complex
30:16 plane it looks crazy . How can you have a
30:18 number that lives on the plane ? I mean a
30:21 lot of you already know that we use xy planes
30:23 to represent points on a plane all the time .
30:26 But I want you to be really careful though because
30:28 in the past when I told you to plot the
30:30 point , X comma y what you're doing is you're
30:33 basically looking at the inputs of a function X .
30:37 And the outputs coming out of the function we call
30:39 it why ? Or you can think of it as
30:40 F . Of X . So X and Y .
30:43 But the X . Is the input value of a
30:45 number of a function and the output is why ?
30:48 And we do plot them as coordinate pairs on a
30:50 graph just like this . But that is different than
30:53 what this is because that is plotting input versus output
30:57 . And you're you're basically plotting what the function looks
30:59 like as a function of input and output input to
31:03 the to the mathematical machine of the function and the
31:05 output that comes out the other side , that's what
31:07 we usually applauding on the on the xy graph .
31:10 But here we're not plotting any inputs or outputs were
31:13 saying that these numbers actually have two pieces to them
31:16 . They have a real piece of an imaginary peace
31:18 . And there's no way that you can draw those
31:20 on a single number line because they have two parts
31:22 . So we have to draw two totally separate axes
31:25 . So before you learn about imaginary numbers , the
31:27 only number line you know about is this one the
31:30 real one ? Right ? But then when you do
31:31 learn about imaginary numbers , you learn that there's another
31:34 number line called the imaginary number line , all the
31:36 imaginary numbers live on here . But any number that's
31:39 not purely real or purely imaginary will live somewhere else
31:43 on this plane . So this thing is not an
31:45 input output of a function going on . This is
31:48 literally saying these complex numbers have two parts that we
31:51 have to represent . So we write them in what
31:54 we call the complex plane . Now graphing them behaves
31:56 exactly the same as graphing any X . Y pair
31:59 . I've tried to show you here so it's not
32:01 magical but that's what it is . It's really showing
32:04 you that these these numbers have kind of two dimensions
32:07 to them . You can think of it that way
32:08 . They have two dimensions . A real and imaginary
32:10 part means they have two dimensions . All right ,
32:13 So that's all I really want to talk about now
32:15 . We introduce the concept of a complex number ,
32:17 how all of the numbers are related to one another
32:20 . So that's really it . There's no more numbers
32:22 that we know about and that's what we have the
32:24 hierarchy of the complex numbers . And then we talked
32:26 about the complex plane and you're gonna be dealing with
32:29 this later in algebra , but you'll be dealing with
32:31 it even more in advanced pre calculus calculus . Advanced
32:35 calculus and things like that because the complex numbers really
32:38 are something we use to solve real equations for real
32:42 applications in science and engineering . So make sure you
32:44 understand this . Follow me on to the next lesson
32:46 . We're going to learn how to simplify expressions that
32:49 involve complex numbers . To wrap up this concept in
32:52 algebra .
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