18 - Dividing Complex Numbers - Part 1 - Free Educational videos for Students in K-12

18 - Dividing Complex Numbers - Part 1 - By Math and Science

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00:00	Hello . Welcome back . We're learning how to in
00:02	this lesson , divide complex numbers , but in the
00:05	process of learning how to take one complex number and
00:08	divide it by another complex number . What we really
00:10	need to understand is the idea behind a complex conjugate
00:14	. Now . In the last few lessons I've introduced
00:15	what a complex conjugate is now . We're finally going
00:18	to actually use it for something . So let's go
00:20	take a trip down memory lane for a second and
00:22	talk a little bit about congregates because we need to
00:24	really understand that in order to divide these numbers .
00:28	So the conjugate , we call it the con jug
00:32	it of forget about complex numbers , the conjugate of
00:37	the number or the expression one plus the square root
00:41	of seven . We've used this kind of thing before
00:44	we say the conduct of this is one minus the
00:48	square root of seven . And we use this before
00:50	because when we had radical expressions with a term like
00:54	this in the denominator of a fraction , we basically
00:56	don't want any radicals in the denominator of any fractions
01:00	. So what we figured out is that if we
01:02	take this term and we multiply it by this term
01:05	with all the cross multiplication , you can see what's
01:07	going to happen . The interior term is going to
01:08	be square root of seven . The outside terms will
01:11	be negative square root of seven . And then so
01:14	they're going to add to zero . And then when
01:15	you multiply the last terms , you see , the
01:17	square root of seven squared is going to kill the
01:20	radical . So by multiplying by the conjugate , we
01:24	eliminate the radical . So we use the conjugate to
01:26	get rid of any radicals in the denominator of fractions
01:29	. We've done that in the past . So now
01:31	we need to move on to working to working on
01:35	today . So we say what we call the complex
01:40	, can you get of And we'll just take a
01:46	few , A few little examples the complex continent of
01:50	the complex number 1000 , I Is just simply 1
01:56	-3 . I all you do is you take a
01:57	copy of the exact same number , but just switch
02:00	the side plus becomes minus . Let's take a look
02:03	at the next complex number , negative one minus two
02:06	. I we say it's complex conjugate is negative one
02:09	plus two . I because it's exactly the same thing
02:12	, negative one and then the two I we just
02:14	switch the sign of the interior guy . One more
02:17	example . Doesn't matter if you have fractions here or
02:19	not , if you have one half minus one half
02:21	times I . The complex conjugate of that is one
02:24	half plus one half . I like a copy of
02:28	it . You switch the sign so positive becomes negative
02:30	, negative becomes positive . So we say that this
02:33	is the complex conjugate of this and we say that
02:35	this is the complex conjugate of this . So these
02:38	conflicts happen , these complex contour , it's happened in
02:40	pairs and notice that this is still a complex number
02:43	because we still have eyes and this is a complex
02:45	number , we have eyes involved . But what ends
02:48	up happening is when we multiply a complex number times
02:52	its conjugate . What ends up happening is that all
02:55	of the imaginary numbers disappear because of cancellations and such
02:59	. So we're going to use the complex conjugate to
03:01	get rid of any imaginary numbers in the bottom of
03:04	fractions . So we said before , we don't want
03:07	any radicals in the bottom of a fraction . We've
03:09	learned about that in the past . Well , since
03:11	the imaginary number I is actually a radical , it's
03:14	the square root of negative one , right , then
03:17	, because we don't want any radicals in the bottom
03:19	of fractions , we also don't want any imaginary numbers
03:22	in the bottom of fractions . So what we want
03:25	to do when we simplify these things , all of
03:27	these problems essentially is going to boil down to getting
03:29	rid of any imaginary numbers in the bottom . So
03:32	for instance 5/3 plus four eyes , this is the
03:37	division going on with a complex number . This is
03:40	a complex number . Of course it's just got a
03:41	real part of five and an imaginary part on the
03:44	top here of zero , but it's still complex ,
03:47	right ? And you're dividing it by a complex number
03:49	with a real and the imaginary part . How do
03:51	you actually do that division ? Do we do we
03:53	ride a division symbol , you know , like in
03:55	basic math and start trying to divide it . No
03:58	. What we really do is we say , well
04:00	we first try to simplify anything in the top and
04:02	the bottom that we can , there's nothing we can
04:03	do . Once you get down to that point ,
04:06	all you try to do at the end is try
04:08	to eliminate Any imaginary numbers in the bottom here .
04:13	So what we have to do is in order to
04:15	do this , just like we cleared any radicals from
04:18	the denominator . We need to multiply by the conjugate
04:21	. But since this is a complex number , we
04:22	multiply by the complex conjugate . So we multiply by
04:26	3 -4 . I and then of course we have
04:28	to do it on the top because we want to
04:31	be multiplied by one . So this is just a
04:32	number one , multiplied by what we started with .
04:35	So we're not changing anything . And so what we
04:39	end up having here is we have to multiply the
04:41	five times this . Now . Here's the here's the
04:43	thing I want you to think about . I'm not
04:45	gonna do it for every problem . But any time
04:47	you have these complex numbers on the numerator in the
04:49	bottom , I want you to imagine invisible parentheses are
04:52	surrounding them like this . And also over here you
04:55	could draw it around the five if you want .
04:57	But it's kind of not terribly helpful . We can
04:59	do that if we want to no big deal ,
05:00	you can go ahead and envision them there because then
05:03	it helps you think when the five is multiplied by
05:06	this , it gets distributed into each term . So
05:09	the five times the three minus four I five times
05:12	3 becomes 15 . Five times negative four I becomes
05:16	negative 20 because five times four is 20 that's the
05:20	numerator . The denominator . We have to multiply these
05:24	complex numbers together and now we know how to do
05:26	that . So three times three is nine . Mhm
05:29	. Inside terms three times four is 12 . But
05:31	we have an eye . So it's 12 I three
05:33	times negative four is negative 12 . I notice we
05:36	have a nice cancellation going on here , negative times
05:39	positive gives me negative four times four is 16 .
05:42	I times I don't forget . That is I squared
05:45	. So for the next step , what's going to
05:47	happen here is we're going to have 15 -20 times
05:51	I . And in the bottom and we're gonna have
05:53	the nine . All of this is going to subtract
05:57	away but we have this minus sign 16 and then
06:00	the I . Square we need to substitute for negative
06:02	one . So you see it already looks a lot
06:04	simpler Moving along and the numerator will have 15 -20
06:09	times I . And the denominator will have nine .
06:12	This becomes a positive 16 And so then we will
06:15	have 15 -20 I in the denominator . What is
06:19	this ? nine um plus 16 is gonna give me
06:23	25 . So you could basically stop there if you
06:26	wanted to 15 -20 I over 25 . However ,
06:30	you notice this is divisible by five , this is
06:33	divisible by five and this is divisible by five .
06:35	So if you do a math problem and you give
06:36	me an answer of 5/15 as the answer 5/15 or
06:41	2/8 or 5/10 . I mean yeah that's right .
06:44	But you could have simplified all of those fractions ,
06:46	right ? So this is just a little more complicated
06:49	because we know these are all divisible by five but
06:52	it's a complex number on the top . So often
06:54	in the beginning you don't know what to do .
06:55	So here's what you want to do . you want
06:57	to cancel the common factor of five from the top
06:59	and the bottom here is the cleanest way to do
07:01	it . What I want you to do on the
07:03	top is just factor out of five . So we
07:07	can write the top . We know this is divisible
07:08	by five and this is divisible by five . So
07:11	we can factor out of 55 times three is 15
07:14	. We have a minus sign from here , five
07:16	times four is 20 , but we have the eye
07:18	here . So this you should convince yourself when you
07:21	multiply back gives you the top on the bottom ,
07:24	you have the 25 . Yeah . All right .
07:28	Now , now that we have a five kind of
07:30	pulled out and then we have a 25 on the
07:33	bottom , you can say five divided by five is
07:34	1 , 25 , divided by five is five .
07:38	And so then for the final answer , well ,
07:40	it's not exactly the final answer , but closer to
07:42	the final answer , it's going to be three minus
07:44	the four i on the top . But on the
07:46	bottom , all I have left is five . And
07:48	again I could simplify , that is I could circle
07:51	, that is my final answer . But when we
07:53	divide complex numbers , we want the number as a
07:55	complex number , right ? So when we divide 20
07:58	divided by four , we want to get a number
08:00	back . Uh a complete self contained number . What
08:04	we have here is a complex number divided by five
08:07	. So what you need to do is is work
08:09	backwards and I'm gonna do it once . It's going
08:11	to seem weird . But then as we solve more
08:13	problems will become common sense or it will become more
08:16	familiar to you . This has two terms on the
08:18	top and we have one term on the bottom .
08:20	This is we can break this up as follows .
08:23	3/5 experiences minus sign from here . Four , I
08:28	over five or a better way to say it .
08:30	Let's just do like this before I over five first
08:34	, make sure you understand this step . If I
08:36	give you this and say how do you add these
08:39	? These are fractions . I mean yeah there's an
08:40	imaginary number but it's not really any different . You
08:43	have a common denominator of five So the answer that
08:46	you get is going to have a common denominator of
08:48	five And then you have 3 -4 I in the
08:51	numerator , 3 -4 I in the numerator . So
08:54	it's very easy for you to look at this and
08:56	know how to get here . But we don't often
08:58	in math look at this and have to break it
09:00	up , but we want to do it with complex
09:03	numbers because we always write them as a real part
09:06	in an imaginary part . So in almost all of
09:09	these problems you're going to get down to the answer
09:11	and then you're gonna have to break it apart into
09:13	the real and imaginary . So all you do is
09:15	you say well this is 3/5 minus this over five
09:18	and then this is actually totally acceptable , but a
09:20	better way to write it . I think it's 3/5
09:23	minus sign 4/5 times . I this is the only
09:28	change four I over five is exactly the same as
09:30	4/5 times . I I like to see it like
09:33	this because The real part is 3/5 , the imaginary
09:38	part is negative 4/5 I that's the imaginary part .
09:42	And this is a true complex number , You can
09:44	have the real part , you can look at it
09:46	on a number line , The imaginary part , you
09:48	can plot this in the complex plane like we've done
09:50	before and it's very very uh easy to understand .
09:54	So all of the problems will behave the same way
09:56	when we divide these things , we're gonna multiply by
09:58	conjugate , we're gonna do a lot of math to
10:00	multiply by the conjugal , we're gonna get a number
10:02	on the bottom and then we'll have to simplify .
10:04	Usually we'll have to factor , cancel , simplify and
10:07	then break it all up again at the end to
10:09	make it pretty alright next problem . This is no
10:14	harder than the last one but we're just getting practiced
10:16	. What if we have two divided by three minus
10:20	I . Well again there's not much to do in
10:23	the numerator or denominator , so now we just want
10:25	to get rid of the imaginary number in the bottom
10:27	. So we say to over three minus I .
10:29	And we have to multiply by something to get rid
10:32	of that imaginary number . And the only way to
10:34	really do it is to multiply by the conjugate three
10:36	plus I And on the top three plus odd .
10:40	And again you can imagine invisible parentheses surrounding all of
10:43	these terms to help you visualize the multiplication process .
10:47	But at the end of the day , what you're
10:50	going to have here is on the numerator side ,
10:52	you're gonna have two times this complex number . So
10:55	you distribute two times three is 62 times I is
10:58	too I This is the numerator of the answer .
11:01	The denominator is much more involved because we have to
11:04	do foil three times three is nine Inside terms is
11:07	negative three I Outside terms is positive three I negative
11:14	times positive is negative items . I as I square
11:16	. That's the final kind of intermediate answer of the
11:19	the end here . But what we have is six
11:24	plus two I on the top and on the bottom
11:26	we have a nine we have a negative three I
11:29	and a positive three I . They go away but
11:31	we still have a minus and we have an eye
11:33	square which we can substitute for minus one . So
11:36	what you really get is six plus two . I
11:39	over this is nine plus one . And so just
11:43	to make it totally clear six plus two I over
11:47	10 . Now again you could probably circle that but
11:50	it's not exactly in the right form number one .
11:53	It's not written as a complex number should be written
11:56	, it's not written as real part plus imaginary part
11:59	. So we want to fix that secondly , um
12:02	we see that this is divisible by two , this
12:04	is divisible by two and this is divisible by two
12:06	, so we think we can simplify it . So
12:08	we want to do first if you want to factor
12:10	out of two on the top , so this would
12:12	be two times three will give me six and then
12:14	two times one would give me too . Don't forget
12:16	the I here make sure that going back in gives
12:19	you what you started with and you have a 10
12:22	Now by factoring it out , you can more easily
12:24	see that we're gonna have two divided by two is
12:26	1 , 10 divided by two is five . Yeah
12:30	, so then what you're gonna have is When you
12:33	do it three plus I over five , that's the
12:37	simplified kind of fractional form of that complex number .
12:40	And then what you have to do to write it
12:42	properly is you have to break this up into real
12:44	and imaginary parts . So you say 3/5 is the
12:47	real part Plus whatever is over here , I over
12:51	five like this and you should always check yourself .
12:55	If I were given this I would say okay I
12:57	have a common denominator of five and I would just
12:59	add the numerator which would be that . So you
13:02	can circle that . I wouldn't give you full credit
13:03	but a better way to ride It is 3/5 plus
13:07	. This should be better written as 1/5 times I
13:11	Because then you can read it off the real part
13:13	is three fits the imaginary part is 1/5 . 3/5
13:16	plus 1/5 . I all right , Not too hard
13:21	. Um dividing complex numbers essentially becomes multiplication essentially .
13:26	But when you really think about it , dividing any
13:28	number really becomes multiplication . I mean if I take
13:31	you know um Two divided by let's do eight divided
13:35	by two . Eight divided by two . Right ?
13:38	That's division . That's what we do . Was division
13:39	but eight divided by two can always be thought of
13:42	as eight times one half . Right ? So you
13:45	can always turn the division dividing by two . You
13:48	can turn it into multiplication of something else . So
13:51	we're dividing these complex numbers basically by doing multiplication and
13:55	that's pretty much how you're going to divide complex numbers
13:58	. Um At least for now we're gonna revisit this
14:00	a little bit down the road and show you show
14:02	you a different technique down the road . Uh It's
14:05	a different ball of wax . But for now this
14:07	is how we're gonna do it . So last problems
14:10	we have negative one minus two times I On the
14:13	bottom we have negative one plus to lie . So
14:17	interestingly this is the conjugate of this . We have
14:20	a conjugating the numerator and the consequent in the denominator
14:22	. So it'll be interesting to see what happens here
14:25	. What do we do ? Well we can't simplify
14:27	top or bottom . So we're gonna rewrite the whole
14:29	thing -2 I . And then we'll have negative one
14:32	plus two . I . And then we have to
14:34	multiply by the conjugation And the conjugate here of the
14:37	bottom term is -1 -2 . I And we have
14:41	to do it as a fraction negative 1 -2 .
14:44	I like this . All right . So we have
14:49	a a multiplication going on on the top and on
14:52	the bottom that we have to take care of .
14:53	And it's helpful for you again to think about invisible
14:56	parentheses kind of around all of these . So in
14:58	the numerator we have foil that we have to do
15:00	. It's more complicated than the previous problem . So
15:02	we have negative one times negative one is going to
15:05	give me a positive one . The inside terms negative
15:09	one times negative two . I is gonna give me
15:11	a positive to I . The outside term negative one
15:15	times negative two . I again gives me positive to
15:17	I . The last terms negative times negative positive .
15:20	Then two times two is four and items I as
15:23	I squared . So that's the numerator . The denominator
15:26	will be very similar negative one times negative ones positive
15:30	one . Inside terms negative one times the two .
15:33	I gives me negative two . I outside terms negative
15:37	one times negative two . I gives me positive to
15:39	I . There's a cancellation showing up here negative here
15:43	, times positive two . I is gonna give me
15:45	the same term except it will be a negative here
15:48	. Uh negative two times two is four items I
15:50	as I squared . And so then we can simplify
15:55	Over here . We have a one To I plus
15:59	two , I is four . I Then we have
16:02	four times the I square which is negative . One
16:06	will handle that in the next step here we have
16:08	a one . But notice we have a cancellation ,
16:10	this guy becomes zero , so it drops away ,
16:13	we have a negative sign and then we have a
16:15	four and then the I squared again is negative one
16:18	. So notice we don't have any imaginary numbers in
16:19	the bottom , which is what we wanted to do
16:21	in the first place . So what do we have
16:24	here on the top ? We have one plus four
16:28	times I . This becomes a -4 . And on
16:31	the bottom we have a one . This becomes when
16:35	you multiply by the minus one . A positive four
16:37	. So what we have is what's one minus four
16:40	, you get negative three plus four . I over
16:44	five . When we catch up to myself and make
16:45	sure I'm okay so far negative three plus four .
16:47	I over five . That's right . So again you
16:49	want to break it up into the real and imaginary
16:51	parts ? So you say because there's there's nothing to
16:53	factor out , I can't factor and cancel 34 and
16:56	five , there's nothing else to do there . So
16:57	you have uh negative three over the five and then
17:02	plus you can do four I over five , make
17:06	sure you understand that common denominator . If I were
17:08	to add them and then this plus this is exactly
17:11	what I have . And then let me clean it
17:13	up a little bit . Let me write this as
17:14	a negative sign . 3/5 the fraction will be 3/5
17:18	plus 4/5 times the imaginary number I so we have
17:22	negative 3/5 plus 4/5 I this is what I would
17:25	want to see is the final answer . Mhm .
17:28	So I hope you've seen in this lesson that dividing
17:30	complex numbers basically always becomes multiplication . At least dividing
17:35	them . Now . At this point in your education
17:36	it turns out later down the road we're going to
17:38	represent complex numbers in a slightly different way , a
17:41	different way of thinking about them . That makes multiplication
17:44	and division even a little bit easier to do than
17:47	this . But for now it all boils down to
17:49	turning division into a multiplication . So you try to
17:52	clear out the denominator becomes multiplication and then you just
17:55	simplify the answer . So we have one more lesson
17:58	, we're going to be continuing to divide complex numbers
18:00	. These problems in the next lesson are significantly more
18:02	difficult , but the same process will apply no new
18:06	rules . Just we have to be a little bit
18:08	more careful with our with our actual work . So
18:10	follow me on to the next section and we'll wrap
18:12	it up with dividing complex numbers .

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