11 - Simplify Expressions with Imaginary Numbers - Part 1 - Free Educational videos for Students in K-12

11 - Simplify Expressions with Imaginary Numbers - Part 1 - By Math and Science

Transcript


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

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