16 - Add and Multiply Complex Numbers - Part 1 - Free Educational videos for Students in K-12

16 - Add and Multiply Complex Numbers - Part 1 - By Math and Science

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00:00	Hello . Welcome back . The title of this lesson
00:02	is called adding and multiplying complex numbers . So in
00:05	the last lesson we introduced the concept of a complex
00:08	number . We talked about the hierarchy of numbers and
00:10	how the complex numbers are the most general kind of
00:12	number that we have that live kind of at the
00:14	top of the hierarchy . And then we talked about
00:16	the complex plane where we can plot how the complex
00:20	numbers actually look . So you should now know that
00:22	complex number always has a real part and an imaginary
00:26	part . And so the two of those together for
00:28	what we call a complex number . Now , if
00:30	you have two complex numbers , one over here and
00:32	one over here , the only way those complex numbers
00:34	are actually equal to one another is if the real
00:37	parts are equal and also if the imaginary parts are
00:40	equal . So let's go through a few examples of
00:42	what I mean by that . And then we're gonna
00:44	get into simplifying some expressions involving multiplying and adding complex
00:48	numbers . So let's say for instance um I tell
00:52	you I mean this is kind of gonna be a
00:53	little bit trivial but I just wanna make sure you
00:54	understand if you have a complex number two plus three
00:57	I you have that over on the left hand and
01:00	you have another person or somebody has some another complex
01:04	number two plus three . I and I ask you
01:06	are these numbers equivalent ? I think you would all
01:08	tell me yes . How do you know they're equivalent
01:11	? Well the only way these are actually true ,
01:13	truly equivalent is if the real parts match and if
01:17	the imaginary parts match , if there's any mismatch they're
01:19	not equal . So those are actually equal . Um
01:23	if somebody has negative 2 -4 I and on the
01:27	other hand they have negative two minus four . I
01:29	you see what I mean ? It's pretty simple here
01:31	. The negative two's match completely . The negative four
01:33	as match of these are equal . Those are complex
01:35	numbers that are equal . If somebody have six minus
01:39	ISA solution to an equation and somebody else has six
01:41	minus ISA solution to an equation , the real parts
01:44	and the imaginary parts . Exactly . Or a match
01:46	. So that's a solution or excuse me not a
01:49	solution there exactly equivalent to one another . And then
01:52	the last one here is even if you involve fractions
01:55	, it's the same thing . Uh one half minus
01:58	three I over two . And then somebody else has
02:00	a complex number , one half minus three . I
02:02	over to the one half match and the negative three
02:06	I over to completely matched . So these guys are
02:07	completely equivalent to one another , but let's change it
02:11	up a little bit and say , well what if
02:12	I give you ? Um if one person has two
02:15	plus three ISA solution and then I have two minus
02:19	three . I A lot of students will just say
02:21	they're equal because the twos matching the three I matches
02:24	, but notice the sign is different , these are
02:26	different . So these are not equivalent to one another
02:28	. So you need to know when to complex numbers
02:31	are equivalent . And the only reason that they are
02:32	is if they completely and totally match , including the
02:36	sign in the middle , what if you have negative
02:39	two plus I on the one hand and someone has
02:42	to plus I over here ? Well , the negative
02:45	two does not match the to the eyes match ,
02:48	but the two and the negative two don't . So
02:49	these are not equivalent to one another . Moving right
02:52	along , what if we have 1/2 -2 I .
02:56	And on the other hand , somebody has 1/2 -2
03:01	i . One half . Well these are actually equivalent
03:04	, let's say we have let's make it like this
03:06	one half plus two . I Well these are not
03:09	equivalent so that we have the one half matches .
03:11	But this is a negative two I this is a
03:13	positive two I . There . And then if we
03:15	have so they don't they're not equivalent . What if
03:17	we have 3/4 plus one half I . And so
03:21	many year has uh one half plus 3/4 high .
03:26	Sometimes students get confused and they'll say well this real
03:29	part matches this imaginary part in this imaginary part matches
03:32	this real part . So they're equal . Well of
03:33	course they're not equal . The only way they're equal
03:35	is if the real parts are equal . And then
03:37	separately from that the imaginary parts have to be equal
03:39	. So that's pretty trivial stuff . But I want
03:41	to give a couple of examples . Still let you
03:43	know because none of us are born understanding what a
03:45	complex number is . The only way they can be
03:47	equal is if the real parts exactly match and the
03:50	imaginary parts exactly match , including the sign that has
03:53	to match too . So now we're gonna do some
03:56	examples of very simple addition and multiplication of complex numbers
04:00	. So if I have a complex number nine plus
04:03	two , I And I have another complex number ,
04:07	1 -7 , 9 . And I'd like to add
04:10	them together . What do I do ? Well ,
04:11	basically what you do is you kind of pretend that
04:14	this i is basically like a variable because you know
04:17	how to handle this if I is just a variable
04:19	, like if I was X and this was ex
04:21	you already know how to do it because you know
04:24	about like terms , you can only add things together
04:26	if they're like terms . So you can add the
04:29	coefficients of I together and you can add just the
04:32	numbers together . But of course i is not really
04:35	a variable , but you treat it the same way
04:36	. The only way you can add those imaginary parts
04:39	is if they both have an I the only way
04:40	you can add the numbers is if they're just pure
04:42	numbers , you can't really add real parts directly to
04:45	imaginary parts because they're not like terms when you think
04:48	about it . So what we have here is we're
04:52	gonna add these real parts nine plus one is gonna
04:54	give me 10 and we add these imaginary parts to
04:57	minus seven is gonna give me negative five but it's
05:00	not negative five , it's negative five . I because
05:02	that's what I'm adding together and this is actually the
05:04	final answer . So adding complex numbers is really ,
05:07	really simple . You just add the imaginary parts ,
05:09	you add the real parts and you're done what if
05:11	you have nine plus two ? I Yeah right .
05:15	And you're gonna add to that just a purely imaginary
05:18	number three . I some students look at that have
05:20	no idea what to do , but then you realize
05:22	this is just a complex number , but the real
05:25	part of it is just zero because it's purely imaginary
05:28	. So you add the real part of this to
05:30	this but it's nine plus zero . So you get
05:32	nine And then you add the imaginary parts , you
05:34	get the five I back nine Plus 5 by that's
05:38	the final answer to that problem . And similarly you
05:42	can have nine plus two I as a complex number
05:46	and I can add instead of adding a purely imaginary
05:48	number , I can just add up your real number
05:50	to it . So I add the real parts together
05:52	. Nine plus six is 15 . I add the
05:55	imaginary part to the imaginary part here but it's the
05:58	imaginary part of this number zero . So I just
06:00	keep it as two I because I'm just adding zero
06:02	I to it so I get 15 plus two .
06:04	I so it's really really very simple . You just
06:08	treat it as if it's any other algebraic expression and
06:10	go from there . What if I have five minus
06:13	seven ? I and I'm gonna subtract eight plus two
06:17	. I Again if this was just an expression you
06:19	would know what you have to do . You have
06:21	to distribute the negative end to both of those terms
06:23	. And that's what you have to do in this
06:25	situation too . So you can take the princess away
06:27	here 5 -7 . I the negative goes in making
06:31	negative eight . The negative goes in making negative two
06:33	II . And then you basically collect like terms ,
06:37	the real part is going to be five minus 85
06:39	minus eight gives me negative three . And then what
06:43	you have left over just for completeness what you have
06:45	left over is the negative seven . I and the
06:47	negative to I and I can now add these together
06:50	negative three . What's negative seven minus two is negative
06:53	nine . I So you get negative three minus nine
06:57	. I All right . We just have a few
06:59	more problems to kind of increase the complexity . But
07:02	none of these are going to be really difficult at
07:04	all . Now we change it a little bit and
07:09	we say what if we have three times the complex
07:11	number , negative two plus I . And then we
07:14	subtract four times the complex number . Three minus two
07:18	. I how would you handle this if this wasn't
07:20	a complex number ? Well you would distribute the three
07:23	in . You distribute this negative four N . And
07:26	then you collect like terms and that's all you're gonna
07:28	do here . So you have three times negative two
07:30	is negative 63 times I is three I This is
07:34	negative four times three is negative 12 , negative four
07:37	times negative two is positive eight . I And then
07:41	you collect terms . So the real part is negative
07:44	6 -12 gives you negative 18 . But I still
07:47	have these imaginary things . I'm going to keep them
07:49	along for the next step and then finally have what's
07:53	three plus eight is 11 I . So you have
07:57	negative 18 plus 11 . I . And that's the
08:00	final answer there . Okay . All right . We
08:04	only have a couple more problems now . Here is
08:06	where a lot of students get sort of start getting
08:08	confused and I understand why because it's not obvious until
08:12	you deal with this stuff a lot . But what
08:13	if you have I times three plus forearm ? Sometimes
08:17	students will get really confused . Like , well ,
08:19	I have I have an eye out here . What
08:21	do I do ? I've never seen a problem where
08:22	I have an eye out there . You follow the
08:24	rules of algebra . This i is a number .
08:26	It's just not a real number . So it has
08:28	to distribute into the three . And it also has
08:30	to distribute in here . So , when you take
08:32	I times three , uh What you're gonna get is
08:35	three . I And I times four . I is
08:39	going to give you four . But the items I
08:41	give you I squared , but you now know that
08:43	I squared is always negative one . So , anywhere
08:46	you see that you have to replace the I .
08:48	Square with negative one . So you have three I
08:51	minus four . And we always want to flip it
08:54	around . So the real part is first negative four
08:56	plus three . I do . You have the real
08:58	part first . The imaginary part next . And this
09:00	is the final answer . So if you ever have
09:02	to distribute things in , you do exactly the same
09:04	as you always do . But any time you have
09:06	I squared , you got a substitute for a negative
09:08	one and go from there . All right . Last
09:15	two problems on the outside . What if I have
09:17	negative four times I . And on the inside negative
09:21	two plus I Well , I have to distribute this
09:24	in just as I always would . So the negative
09:27	four I times this . The negative times negative gives
09:29	me a positive four times two is eight . And
09:32	then you have the eye there as well . So
09:34	it's going to be positive eight . I but then
09:36	when I do it here , negative times positive means
09:39	I'm gonna have a negative four times one means four
09:42	and I times I is I square . Don't try
09:45	to do too many things at once . Right ?
09:47	The items eyes I squared . But in the very
09:49	next step , you know that it's going to be
09:51	negative one . You have to replace this with negative
09:54	1 to 8 . I plus four . But we
09:57	always like to flip it around . So the real
09:58	part is first four plus eight . I answer is
10:02	four plus eight . I Okay . The last problem
10:06	we have is a little more complicated , but not
10:09	too bad . What if we have 3 -1 Times
10:13	three plus I three minus seven times three Plus .
10:17	I a lot of students look at that first and
10:18	say what do I do ? But I'm trying to
10:20	tell you basically treat I as a variable . So
10:22	you're gonna do foil on this . It's not a
10:24	variable of course , but you treat it as if
10:26	it is . So we do foil on this .
10:28	The first terms multiplied together three times three gives me
10:31	nine . The inside terms give me three times negative
10:34	. I give me negative three I The outside terms
10:38	three times I give me positive three I . And
10:41	the last time you got to be careful negative times
10:43	positive , gives me negative but items I gives me
10:45	I squared like this . A couple of interesting things
10:49	happen here though because notice you have a negative three
10:52	I and a positive three I . So these actually
10:54	add together to give me zero . And also the
10:57	I squared is going to give you a negative one
10:59	. So you gotta be careful with the signs you
11:01	have nine , ignore all this because it's zero ,
11:03	the minus sign is still there . But the I
11:05	squared is replaced with negative one . Don't do too
11:09	many things at once , right ? It is nine
11:11	minus and replace this negative one . So you get
11:13	nine plus one , so you get 10 . So
11:16	the answer is actually 10 . So the interesting thing
11:18	is a couple of things I want to talk to
11:20	you about is that you can have a complex number
11:23	multiplied by another complex number . And the answer that
11:27	we got is just a well it's a complex number
11:29	because all all numbers are complex , but it's purely
11:31	real . There's no imaginary part to it . And
11:34	that's kind of one of the things I was telling
11:35	you early in the beginning , complex numbers are so
11:37	useful because a lot of times we get the answers
11:40	out of the equation and complex form , but sometimes
11:42	we'll combine equations together or multiply it by something else
11:45	and we might not get a complex answer , we
11:47	might get a purely real answer . So some students
11:50	say , well , why do I care ? Well
11:51	, it's hard to explain without you getting into more
11:53	advanced math and be showing you solutions to really advanced
11:56	equations . But this is the crux of it .
11:58	A lot of times we multiply these complex numbers together
12:00	and we get real numbers back . So even though
12:03	it seems like they're useless , they're not useless because
12:05	they often combine to give us real numbers . The
12:08	other thing I want to tell you about is I
12:11	want to point out something to you here . The
12:14	three plus I is what we call the complex conjugate
12:18	of this . So three plus I is what we
12:23	call the complex conjugated of 3 -1 in fact are
12:32	complex conjugate of each other . So a complex conjugate
12:36	is very , very similar to what we already talked
12:38	about . We already talked about the complex . I'm
12:40	sorry the conjugate of a radical . So you have
12:43	something plus something and there's a radical involved to make
12:46	it a conjugate , you just replace the sign in
12:48	the middle , flip the sign from a negative to
12:50	a positive or positive and negative . And that's called
12:52	a conjugate . We use that all the time to
12:55	clear the radicals from denominators , right ? But don't
12:57	forget I is really a radical to I . Is
13:00	the square root of -1 . So we have a
13:02	thing called a complex conjugate . So anytime you have
13:04	a real plus an imaginary part , you just flip
13:07	the sign of the inside and that's its conjugate .
13:09	Or if you start with this one , you flip
13:11	the sign of the negative . This is the conjugate
13:13	of this and this is the context of this .
13:14	The consequence of each other , basically , there's peanut
13:17	butter and jelly , they go together . But the
13:19	idea is when you multiply a complex number times its
13:22	conjugate , you will always get this cancellation in the
13:25	middle and you'll always get an I . Squared ,
13:27	so you'll always get a real number . So in
13:30	the next few lessons , we're going to use that
13:32	because we're going to use it to clear away any
13:34	imaginary numbers in the denominator of fractions by multiplying by
13:38	the conjugate , just like we multiplied by the regular
13:41	conjugate for radicals . To clear the radicals , we're
13:44	gonna end up multiplying by the complex con you get
13:46	to get rid of any eyes and the imaginary numbers
13:49	and the denominator of fractions . So this is what
13:51	a complex conjugate is . And the property of multiplying
13:54	a complex conjugate or a complex number of times its
13:57	conjugate will always give you a real number . That's
13:59	true for any conjugal , any conjugal pair like that
14:03	. So this is the idea of multiplying and adding
14:07	complex numbers in algebra . Make sure you can do
14:09	all of these . Follow me on to the next
14:10	lesson will increase the problem complexity to give you more
14:13	practice , but ultimately , we're gonna solve all the
14:15	problems the same way . So , follow me on
14:17	the next lesson , we'll do it right now .

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