13 - Add and Multiply Imaginary Numbers - Part 1 - Free Educational videos for Students in K-12

13 - Add and Multiply Imaginary Numbers - Part 1 - By Math and Science

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

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