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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