16 - What do Imaginary & Complex Roots of Equations Mean? - Free Educational videos for Students in K-12

16 - What do Imaginary & Complex Roots of Equations Mean? - By Math and Science

Transcript


00:00	Hello . Welcome back to algebra . In this lesson
00:02	. I'm going to teach you something what I find
00:04	to be very , very interesting and I almost didn't
00:07	teach this lesson because it's a little bit advanced for
00:09	its quite a bit advanced actually for a typical algebra
00:12	class . However , I'm confident I can break it
00:14	down into easy to understand terms . I've got a
00:17	computer demo right at the midway point of this lesson
00:19	. That's gonna really , I think knock your socks
00:21	off as far as demonstrating what I'm talking about .
00:23	And this is the concept of these complex roots that
00:26	we've been dealing with all the time in algebra ,
00:28	we complete the square on a polynomial and solve the
00:32	polynomial equal to zero . Oftentimes we get imaginary or
00:35	complex answers right ? Sometimes we get real , sometimes
00:37	you get complex , we use the quadratic formula half
00:40	the time you're getting real answers in half the time
00:42	you're getting imaginary or complex answers . And when I
00:45	first learned algebra the very first time I asked my
00:48	teacher what do these imaginary things mean ? Like I
00:51	I kind of understand what a real root right might
00:54	mean , that's the crossing point when the graph crosses
00:56	the X axis . But in the case of imaginary
00:59	, what does that mean ? And the teacher just
01:00	didn't know and there's nothing wrong with that because we
01:03	often we don't know everything . Of course nobody knows
01:06	everything , but it kind of nod at me and
01:08	and and so on . And until I finally got
01:10	higher up in math many years later and I finally
01:12	understood why those roots pop up . So because we
01:16	learn these things in algebra , complex roots of polynomial
01:19	, I want to take the time to really explain
01:21	what they are . So keep in mind this is
01:23	kind of a more advanced concept . You won't typically
01:25	learn this stuff for years down the road , You
01:27	could probably skip this lesson . However , let me
01:29	tell you , I wouldn't spend hours of my time
01:31	writing this down and writing a computer program if I
01:35	really didn't think it was cool . And you know
01:37	with my students that watch these lessons , you know
01:39	, I often have dreams and hopes that some of
01:41	you guys or girls are going to go on to
01:43	invent the next big thing in space travel or medical
01:46	devices or power generation or whatever . So I want
01:49	to give you every bit of knowledge that I have
01:51	so that some of you can go on and do
01:52	those amazing things . All right . So the bottom
01:55	line is everything we're gonna talk about in this lesson
01:57	applies to all polynomial when you set them equal to
01:59	zero and find the roots . You know , quadratic
02:02	have two routes , cubic equations have three roots when
02:05	you set it equal to zero , you know and
02:07	on and on and on . If you have 1/4
02:08	order polynomial , you get four routes and so on
02:11	and so forth . But you all know by now
02:12	that sometimes you get these imaginary or complex roots ,
02:15	what do they mean ? So let's crawl before we
02:17	can walk . Let's take something extremely simple so that
02:21	I can explain and build up your knowledge so that
02:23	you're not gonna hit you over the head too much
02:25	with this stuff . Let's take a very simple equation
02:27	that everyone can visualize X squared minus one . How
02:31	can we visualize this ? Well because if you take
02:33	away the one here it's just F of X .
02:35	Is equal to X squared . And you all know
02:37	that that's a parabola , right ? So it has
02:39	a parabola shape , The minus one just means that
02:42	we grab the parabola and we shift it down by
02:44	one unit because whatever X squared gives us , we
02:48	just take away one from it and that's what the
02:49	output is . So , if I were going to
02:52	draw a picture of that and I am going to
02:53	draw a picture of that , it would look something
02:55	like this . If I draw a little quick little
02:57	sketch over here , X and F of X .
03:02	All right , And if I were going to draw
03:03	this guy , the regular Parabola would go down and
03:06	touch at zero and then go back up . But
03:08	this is a regular parabola minus one . So ,
03:10	basically down here at minus one , we have a
03:13	tick mark . And this parable is gonna come down
03:15	to this point because basically you take the regular parabola
03:18	and you're taking and subtracting one from every single point
03:21	. So it goes down like this . So ,
03:23	the reason I chose this one is because it's easy
03:25	to understand and you can see that there are two
03:27	real crossing points where this function cuts into the X
03:30	axis . So uh you can you can probably guess
03:34	what these values are . But if you don't remember
03:36	, let's go and find out what the roots of
03:38	this thing is . If you take x squared minus
03:41	one , set it equal to zero . You could
03:42	run this through the quadratic formula . Of course you
03:44	could . A is one , there is no X
03:47	. Term , so be would be zero and C
03:49	is negative one . You could do that . But
03:50	this is even easier because you don't even need the
03:52	quadratic formula . Just move the one over there like
03:55	this . And then you say , well then X
03:58	solving for X is going to be plus or minus
03:59	the square root of one because you just take the
04:01	square to both sides . And so then X is
04:04	going to be plus or -1 . So when we
04:06	set this thing equal to zero , the values of
04:08	the X point or plus and minus one . So
04:10	here is a root at X is equal to positive
04:13	one . And here's the other route at X is
04:16	equal to negative one . So this is what I've
04:19	been telling you up until now , uh many ,
04:21	many times over when I've had the opportunity , as
04:24	I've shown you , hey , when you solve roots
04:26	of equations in this case , we're talking about quadratic
04:29	but it works for any polynomial with fourth or fifth
04:32	order . However many wiggles the thing has , wherever
04:35	it crosses the X axis , we call it a
04:37	route . And what that really means is these are
04:40	the two values that when you plug them back into
04:43	the equation gives you zero . Now you can see
04:46	it graphically right here , because you can stick the
04:48	negative one in here and it gives you zero and
04:51	you can stick the positive one in here and it
04:53	gives you zero . That's what the graphs representing .
04:54	So in general , I tell you the roots ,
04:57	is basically telling you where the graph intersects the X
04:59	axis . We've talked about that many times , but
05:02	because I'm gonna generalize this in a second , what
05:05	is , what the roots really mean is we can
05:08	take these values of the route and we can sub
05:11	back into the equation . So what is basically telling
05:14	us is that if we take the value of one
05:16	and substitute it back into the equation that we have
05:19	, we know it's gonna give a zero from the
05:21	graph . So it's kind of redundant . But if
05:23	you take that one here and you say , well
05:24	it's one squared minus one and we know this is
05:26	one minus one which is zero . So a route
05:29	is a value that I can substitute into the equation
05:32	and gives me zero and I can of course do
05:34	the same thing with negative one . If I take
05:36	this and take the negative one and square it ,
05:38	I'm gonna get exactly the same thing . So it's
05:40	gonna be one when you square this minus one which
05:43	is zero . So we've proven about two different ways
05:45	by looking at the shape of the graph , the
05:47	intersection points here , and also by direct substitution that
05:50	these are the special values , the roots of the
05:53	special special values . So that when you substitute them
05:56	back into your function you get zero . And that's
05:58	what we've learned for functions we understand very well like
06:01	this one . Now let's change this function very slightly
06:05	so that it no longer has any intersection points here
06:08	. And let's discuss what happens there . So let
06:11	me kind of draw a little dividing line here .
06:12	We're gonna do a little comparing contrasting above here .
06:16	Let's take another related function F . Of X is
06:19	equal to X squared exactly the same shape , but
06:22	instead of a minus one , we're going to add
06:24	one . So what this means is that the graph
06:27	is still a parabola , it's still the same exact
06:29	shape . But instead of shifting it down by subtracting
06:32	one , we're actually going to shift the thing up
06:33	by one unit . So now we all in our
06:36	mind know exactly what the thing is going to look
06:37	like . It's still gonna look like a parabola .
06:40	So let's go over here and let's draw that shape
06:43	. So here's X . Here's F . Of X
06:46	. And we'll try to do our best to pick
06:48	the same color . And so what's gonna happen here
06:50	, if you do a table of values here ,
06:52	it's going to shift up . This is a one
06:54	unit up . And that parable is going to go
06:56	and touch or at the very lowest value at this
06:59	at this point here , because every point on this
07:02	parable a graph , you just add one to it
07:04	, that's what the output is . So the whole
07:06	thing is shifted up . The regular curve of course
07:08	comes down to the origin right here . Now the
07:11	interesting thing about this thing is that it appears to
07:14	have no intersection values with the X . Axis .
07:17	There are no places where this function cuts into the
07:21	X axis . So you really think there should be
07:23	no zeros of dysfunction because there are no values that
07:26	make the thing equal to zero . So we see
07:29	that this is the case from the graph . Let's
07:30	go and calculate it the same way we did over
07:33	here . We're going to take this exact thing .
07:35	And we're gonna say , let's find all values where
07:38	where this graph here , X squared plus one is
07:41	actually equal to zero from the graph . We see
07:43	that there are no values were equal zero , but
07:46	let's go calculated . So what we're going to get
07:48	is X squared is negative one . All right .
07:50	And then we're gonna get that . When we solve
07:52	for X , it's going to be equal to plus
07:54	or minus . We have to insert that square root
07:57	of negative one . You all know what the squared
07:59	of negative one is now it's plus or minus I
08:03	. So the interesting thing about this and we've seen
08:05	this many times when we solve equations or use the
08:07	quadratic formula or whatever . Sometimes we get routes that
08:11	are imaginary . This case it's just plus I and
08:14	-1 . But oftentimes we get even weirder roots like
08:17	one plus two I or square root of two plus
08:20	three I . Or square root of two minus three
08:23	. I it's these weird complex numbers that pop up
08:26	all the time when we're trying to find the roots
08:28	of an equation , but no one ever tells you
08:30	, especially in algebra class at a college level ,
08:33	even algebra class , like what they actually mean ,
08:35	because it's really beyond the scope of a typical class
08:38	, but that's what we're gonna talk about today .
08:40	What do these things mean ? Okay , well there's
08:43	a couple ways to interpret it and we're gonna do
08:45	a computer demo to show you exactly visually what they
08:47	are . Um But what we have to go back
08:51	to is our original definition . It looks like there
08:54	is no values where this graph crosses the axis .
08:57	So how can there be any roots and why are
08:59	they imaginary ? Remember from the original problem we said
09:03	a route is a value . When we substitute it
09:06	back into the equation it gives zero . So if
09:09	we've done our math right then we can substitute in
09:12	these values . The ones that we got here F
09:16	evaluated at I . And it should equal zero .
09:19	So the original equation is X squared plus one .
09:21	So if we put in the value of the imaginary
09:23	number i it'll be I squared Plus one . I
09:26	know it looks a little weird putting an imaginary number
09:28	in there . But what you're gonna get is you
09:30	know that I squared is always equal to negative one
09:32	plus one . And so what you get is zero
09:35	. Which it checks out the check here a root
09:38	of a equation is a value that when you put
09:41	it in their drives the function to zero . And
09:44	then we can show the exact same thing is true
09:46	for negative . I because that's the other route ,
09:48	right ? So if you're putting negative i in squaring
09:50	it , you'll have negative I and you're squaring it
09:53	like this , you have to be it's not negative
09:55	ones are negative . I hear you have to be
09:57	a little careful about negative I squared . So let's
09:59	be really careful about it . This is negative .
10:01	I multiply by negative . That's what negative I square
10:04	it is plus one . And so negative times negative
10:08	is positive , which means you have a positive thing
10:11	and then I times eyes I squared plus one .
10:14	This turns out to be exactly what we got before
10:17	. We know this is gonna be a negative one
10:18	plus a positive one . Which is going to give
10:20	you a zero . So it checks out as well
10:23	. So what we have found here is actually a
10:26	little more profound than it seems on the surface .
10:28	What we have found is that the concept of roots
10:31	of an equation are basically just finding the values so
10:35	that when we substitute them in they give us zero
10:38	. Now when we graph and we have real values
10:40	for these for these routes , then when we graph
10:43	it , we get these graphically illustrated these graphical points
10:47	here where the crossing point happens . And so it
10:50	makes sense That plus and -1 are the places where
10:53	it goes to zero because we can see it here
10:55	. This makes a little less sense because the graph
10:57	never actually goes to zero . So how can we
10:59	actually have these routes ? We see that when we
11:03	substitute them in we get zero . And so that's
11:06	half of the answer . But really , what do
11:08	they mean ? So here is the point where I
11:10	need to do some talking or have some notes here
11:12	and I want to talk to you and make sure
11:13	I get it all down . And we're gonna go
11:14	through where I'm gonna explain . We're gonna do a
11:17	little more board work here where I can get you
11:20	all prepared and then we're going to do you have
11:22	a computer program that I wrote to show you graphically
11:25	what these things mean . But in order to understand
11:27	it , you really need to understand some background .
11:29	So I have to talk a little bit . I'm
11:30	sorry about that , but that's just the way it
11:32	is . What do these roots mean ? Okay ,
11:35	there's no intersection point of the second function with the
11:38	X axis . So how can there possibly be a
11:40	route ? The quadratic formula frequently gives us these fruits
11:44	, We talked about that you get imaginary or complex
11:46	roots . So what do they actually mean ? The
11:49	punch line here is for every function you've ever been
11:54	exposed to in this case here is a function X
11:56	squared plus one . Here's another function X squared minus
11:58	one . But really in general for any function that
12:02	you've ever heard of ever talked about , you know
12:05	two X plus three . That's a function X squared
12:08	plus four X minus nine . That's a function of
12:11	X to the fourth plus three X squared plus two
12:14	. That's a function . I can go on and
12:16	on and less functions until the end of time .
12:18	But every function you've ever ever been exposed to .
12:20	Nobody ever told you this . But here's the punchline
12:23	. All of those functions . Normally we take numbers
12:26	and we stick them in the input and then we
12:28	get numbers in the output . That's what we typically
12:30	do in algebra class . But all of those functions
12:32	can actually take as an input to the function .
12:35	Not just numbers but complex numbers , they can all
12:38	take as inputs complex numbers . And then of course
12:42	as the output they will spit out another complex number
12:45	and you can kind of see that as snuck it
12:46	in right here . I said , hey you've never
12:48	done this before . I don't think I've never taught
12:50	in this lesson in this class . We substituted an
12:53	imaginary number I in . And what do we do
12:55	with it ? We just did exactly what we always
12:57	do . We put numbers in . We get numbers
12:59	out , it's just we put a number in that
13:00	was imaginary . We squared it we added it ,
13:03	we got zero , we put this number in ,
13:05	we squared it , we added up , we get
13:06	zero . But the general thing applies as well .
13:09	Here I put in plus I and here I put
13:11	in minus I . But I can stick into any
13:14	function . Any complex number . I can put one
13:17	plus two . I that's a complex number . I
13:19	can put that in as an input to any function
13:22	and then calculate the output , which will be another
13:25	complex number . So I'm broadening your horizon right all
13:29	the way up until now your entire exposure and math
13:32	has been these things called functions . You put a
13:34	number in like two or three . You get a
13:36	number not a number out like five or seven .
13:39	Right ? But I'm expanding that for you And telling
13:41	you that all functions are really a bigger thing called
13:45	complex functions . Actually when you get in the higher
13:47	math , there's this thing called complex function theory is
13:50	really what this actually is . So we're kind of
13:52	what I said , it's kind of advanced but you
13:53	can definitely understand it because what I'm telling you is
13:56	complex numbers four plus five , I that's the most
14:00	general kind of number . All of these functions can
14:02	take as an input a complex number and then you
14:05	crank through and calculated and you get an answer out
14:07	that's a complex number . So I want to show
14:10	you that explicitly . You know , when we talk
14:12	about these functions of X , like X squared plus
14:15	one . Typically we we know we're gonna be putting
14:17	numbers in for X . But when you talk about
14:19	substituting complex numbers in , usually don't use the letter
14:22	X . You usually use the numbers E . It
14:25	doesn't really matter so much , but I can change
14:27	this function . For instance , instead of F of
14:29	X is X squared plus one , I can make
14:31	it F of Z , is z squared plus one
14:34	. All this means when you see a Z .
14:35	There is it just generally means that I'm probably gonna
14:38	be substituting complex numbers in and getting complex numbers out
14:41	. So as a simple example , we've already substituted
14:45	in plus or minus . I . Let's substitute in
14:47	something a little more complicated . Let's substitute the complex
14:50	number one plus I it's just a number , it's
14:52	just a different kind of number than you had to
14:54	deal with and we can stick it right into here
14:56	as we always do . So when we square the
14:59	entire complex number has to be squared and then whatever
15:02	we get as an answer , we have to add
15:04	one to it . So normally in all the past
15:06	, you know , you can stick real numbers in
15:08	like 3456 put them in here and get answers .
15:11	I'm just telling you that you can put for any
15:13	of these functions . Any complex number you can think
15:15	of here . I'm just picking one as an example
15:17	. So how do you do this ? What you
15:18	have to do foil on this ? Right ? Because
15:20	it's one plus I times one plus I that's what
15:24	this is what's not equals there's a plus one here
15:28	. So I have to do foil . So it's
15:29	more work , right ? But the first term is
15:31	gonna be one times one is one . Inside term
15:34	is one times I . Which means just I outside
15:37	term is again one plus I . Which is I
15:39	last term is I squared . And then I have
15:42	a plus one . So here I have one plus
15:46	I plus eyes to I plus I squared . Which
15:49	you all know . You can always substitute negative one
15:51	for that . Plus one . So the negative one
15:53	plus the one is gonna disappear . We're gonna add
15:55	to zero . So what you're gonna get is one
15:57	plus two . I So what you basically found out
16:01	that when you substitute the value I into this equation
16:05	you get another complex number out . So I'm broadening
16:07	your horizon to a really complex topic that you learn
16:10	down the road . But it's really not that hard
16:12	to understand . How do you know what a complex
16:14	number is ? All of these functions that we have
16:16	? Every function you've ever heard of ? X squared
16:19	plus 10 X plus five . You can of course
16:21	put real numbers in and get real numbers out .
16:23	That's what we've been doing all along . But you
16:26	can also put complex numbers in and in general you
16:29	will always get a complex number out some sort of
16:31	way . Now when you actually do the calculation ,
16:34	there's more work involved because you have to do foil
16:36	on the on the complex number often . But or
16:38	some other expansion or some other multiplication . But in
16:41	the end of the day you get a number which
16:43	is a complex number . Now that goes in and
16:45	then a complex number that comes out the other end
16:48	. All right . So when you really broaden your
16:51	horizons , when you go back to the beginning of
16:53	time and you talk about the concept of function ,
16:54	all we did was plug in these values of X
16:57	. But these are all just real numbers . So
17:00	I put in negative four and I get an answer
17:02	and I put negative three and I get an answer
17:04	and I put in negative one here and I got
17:06	an answer of zero . And here I put zero
17:08	in for X and I got negative one and I
17:10	put Positive one , I get zero . I've put
17:12	to you see all the numbers I was substituting in
17:15	in the olden days when you first learned this is
17:17	just the real numbers on this single line . But
17:20	now I'm telling you that a whole new universe exists
17:23	where you can substitute any complex number you want into
17:26	a function and you will get another complex number out
17:29	. But remember , complex numbers are not represented on
17:32	a single line because they have two parts , they
17:34	have a real part in an imaginary part . So
17:36	we have to represent them on a complex plane .
17:39	So let's quickly sketch that complex plane . And I
17:43	believe me , I wouldn't go through this if I
17:44	didn't think it was critical for you to understand it
17:47	. And I didn't think it was easily understandable by
17:49	everybody here . I'm confident that everybody here is going
17:52	to understand there's no problem . So here we have
17:55	not an xy plane , This is not an xy
17:57	plane , this is a real axis and this is
18:00	the imaginary axis . And we talked about the real
18:03	the complex playing in the past . Um you know
18:07	, several lessons ago . So when you look on
18:10	this axis , these tick marks are like 1234 And
18:13	this on this side is negative one , negative two
18:15	, negative three , negative four . But these tick
18:17	marks are I two I . Three I . Four
18:20	I negative I negative to negative three negative four I
18:24	. Right , So I'll just write that down ,
18:25	here's I here's to I . And so on three
18:28	I . And four , here's negative I negative two
18:31	I . And then these numbers are just the regular
18:33	numbers that you know , I'm not gonna put negatives
18:35	because it'll clutter it up 123 negative 123 So then
18:39	if you want to represent a number , let's say
18:42	one plus I . The number we stuck into this
18:44	thing into this plane . It's gonna live right here
18:47	at the intersection point right here . Why ? Because
18:49	the real part is one and the imaginary part is
18:52	I . So it lives as a point in this
18:54	plane right here where the real part is one and
18:57	the imaginary part is I right ? And here's the
19:02	part where you have to kind of really stretch your
19:04	imagination a little bit at every single point in this
19:07	plane , no matter where you are here , I
19:09	picked one called one plus I here's another point over
19:12	here . It's four plus four high . Here's another
19:15	point over here . It's it's negative three comma negative
19:18	two . I every single point in this plane has
19:22	a real and imaginary part of its complex number ,
19:25	right ? But at every point in this plane I
19:28	can substitute that point into this thing , crank through
19:31	the math and get the output . So at this
19:33	point this input , right . This input point was
19:38	one plus I as an input . Mhm . Right
19:44	. At one plus side this is the input point
19:46	. But the output when you put it through that
19:48	function is a totally another complex number one plus two
19:53	. I which lives somewhere else on this plane .
19:56	So you see , whereas a regular xy plane ,
19:59	you put numbers in for X and you get numbers
20:02	out for why ? It's a mapping we always talked
20:05	about with functions . It's a mapping . A number
20:07	goes in . A number goes out . It's a
20:08	mapping . It's a relationship between two numbers . Stick
20:11	a number in . Get a number out here .
20:13	I'm putting a complex number in . I'm getting a
20:15	complex number out . But the thing is you have
20:18	to imagine that there's a function kind of hovering over
20:21	this plane . You can't see it because I can't
20:23	draw in three dimensions . But in this plane is
20:26	all the possible numbers that are that are complex possibilities
20:29	. That could go into this function . And in
20:31	every one of these points like here , I can
20:33	hover my finger and I could put a little dot
20:35	that says the output is one plus two . I
20:37	here . If I put a number over here ,
20:39	I could say there's a number output of one plus
20:41	of whatever output of the function . I could calculate
20:44	their at every single one of these infinite number of
20:47	points . I could calculate the output of the function
20:50	which would be a new complex number . All right
20:54	. That's what I want to say there . And
20:57	at some very special locations in this plane , if
21:01	I substitute into this function , it will drive that
21:04	function not to a complex output , like some other
21:06	complex number , but it'll drive it to zero .
21:08	Those are the roots of the function . So if
21:11	I had to kind of give you a punch line
21:13	, I'm trying to give you a punch line several
21:14	times through the lesson . When you get complex roots
21:17	of a polynomial and you say , well you usually
21:19	we just flush them and say , well , I
21:20	don't care what they mean , I don't have any
21:22	idea what they mean . Is that all of these
21:24	functions we plot are really functions in the complex plane
21:28	. We can put infinite amount of complex numbers into
21:31	them and we can get complex numbers out . But
21:34	in the complex plane there are certain special places where
21:37	when you put those complex number into the function ,
21:39	the function goes to zero . Those are special points
21:42	. We call roots for this point . Um over
21:45	this function over here , X squared plus one .
21:47	The very special points where plus or minus I .
21:50	Those are the very special points we could put into
21:52	the function and get zero out . But if you
21:54	can use your imagination and say , here's the function
21:57	here is the complex plane that every single one of
21:59	these points and a complex function , a complex number
22:02	pops out of this function . But at very special
22:05	points which are actually plus and -1 . For this
22:08	example , the function is driven to zero . And
22:12	that is why we get complex uh an imaginary roots
22:16	of functions , even though there is no intersection point
22:19	here . The reason there's no intersection point here is
22:21	because there are no real numbers that drive this function
22:24	to zero . The closest I can get to zero
22:27	is actually here , but it's still never gets to
22:28	zero . But that just means there's no pure numbers
22:31	that I can put in and calculate and get zero
22:33	. But there are other numbers called imaginary or complex
22:36	numbers that when you put them in actually drive the
22:39	function to zero . So I want to make sure
22:41	I've said everything and then we're gonna go do the
22:43	computer demo because I think we're gonna get um a
22:48	lot more um Yeah , I think I've said everything
22:51	here . So what I want to do before we
22:53	do that is I want you I want you to
22:54	imagine this complex playing here full of complex numbers .
22:57	You can plug those complex numbers into the function and
23:00	then out spits another answer . So one more thing
23:03	I want to write down because I'm going to have
23:05	it in the computer demo is I want to we
23:08	want to let's make a quick chart of this .
23:10	Let's make a grid . We can construct the table
23:12	. In other words , we can make a table
23:13	of values . You know how when we plot functions
23:15	, regular ffx functions , we make a table of
23:18	values . We say here's the X values for input
23:21	. We calculate the answer , we put the outputs
23:24	for F of X . So we plot X .
23:26	F of X F F . Put those points in
23:28	the plane . That's how we grab functions . That's
23:29	what they are here . We're going to construct something
23:31	similar . But because it's a little more complicated ,
23:34	we have to make a table . So the function
23:36	F of Z is Z squared plus one . Now
23:40	I'm using Z because I'm gonna be using complex numbers
23:43	here . So let me get the table on the
23:45	board and then you'll understand it . We have a
23:47	real part of all these things And we have an
23:50	imaginary part . So the real part is going to
23:52	be zero 1 2 . I'm just gonna go out
23:56	to to negative one negative two . And then we
24:01	have I'm gonna go sideways here , we have the
24:04	imaginary we have the imaginary parts here and I'm gonna
24:10	switch colors here . Let's go ahead and make this
24:13	let's say this is um zero and we're going to
24:17	say this is negative I . And negative two .
24:20	I and this is positive I . And this is
24:23	too I . So one eye to eye negative one
24:25	negative two . I . So in other words because
24:27	we have real and imaginary parts , we can't just
24:30	make a simple table of values like we did for
24:32	regular functions , we have to actually make a chart
24:34	. So we know we've calculated one point at the
24:37	value one plus I we calculate that one plus two
24:41	eye pops out . So for the real part of
24:44	one and the imaginary part of I we go down
24:48	in the table at this point real part of one
24:50	imaginary part of I one plus I in other words
24:53	and the value that we get out of that is
24:56	one plus two . I So we're constructing a table
24:59	of values when we put an input of one plus
25:02	I we get an output of one plus two .
25:04	I right here and then we have let's see here
25:11	, we can fill in some other values here at
25:13	a value of negative two with no imaginary part negative
25:17	two . We put in a value and we get
25:19	a five -1 . We put in a value of
25:22	that , we get an output of two , we
25:24	put a value of zero , we get a value
25:26	of one , we'll put a value of 12 and
25:30	then we have a five here . You see the
25:31	symmetry here . This line right here in the table
25:34	is exactly what we plot . When we normally plot
25:38	dysfunction , we're putting in values of X which are
25:40	just real numbers . And we're getting outputs and we
25:43	plot them . If you can ignore the rest of
25:45	this entire chart . And just look here , see
25:47	the imaginary part was setting equal to zero . When
25:50	the real part is negative two , we get five
25:52	real parts negative one . We get to we plug
25:54	in a zero , we get a one , we
25:55	plug in the one , we get it to we
25:57	plug into two . We get a five . All
25:58	we're doing is putting it in here . Two squared
26:00	plus one is +51 squared plus one is +20 square
26:04	plus one is one . This line right here is
26:07	the table of values you would set up if you
26:10	were just plotting a regular function with real numbers .
26:13	But we have all of this beauty around it where
26:17	we have other values in the complex plane . So
26:19	we had when we calculated before one plus I we
26:22	got this value now . What about the value ?
26:24	I'm just I'm not gonna feel the whole chart up
26:26	but one minus I would live up here and it's
26:28	gonna be one minus two . Y you see there's
26:31	a lot of symmetry here and the value of real
26:33	value of one minus I we get this one plus
26:36	I would get this and I'm gonna fill out a
26:38	couple of more . Just so you can get an
26:39	idea four minus four . I lives over here and
26:43	four plus four I lives over here . That means
26:46	if I put in a value of two minus I
26:49	into this equation and calculate the answer . I'm gonna
26:51	get this . If I put a value of two
26:54	plus I as a number into this equation calculated I'm
26:57	gonna get this value . And so the rest of
27:00	the chart I'm not going to fill out . But
27:02	you can imagine that in every possible location here -1
27:06	-2 . I I could put that in as a
27:07	complex number , get an output , stick the value
27:09	here and so on on the computer . I'm actually
27:12	going to fill this entire chart out so you can
27:14	see it in all its glory . But the point
27:16	is , is that every value of the complex plane
27:19	? Really ? What I've done is I've constructed a
27:20	complex plane here and I'm calculating output values at every
27:24	point in the complex plane , right ? But for
27:27	this particular example , we know that when we put
27:30	I in we get a zero and we put a
27:32	negative I in , we also get a zero .
27:34	Because we calculated that . So when we look at
27:37	the complex plane , I'm gonna do this in red
27:39	to make it stand out . When we put a
27:42	value of zero minus I . Which just means pure
27:45	negative . I we get an output of zero and
27:48	when we put a value here of positive , I
27:51	we also get a value of zero . These points
27:53	here are the roots of this equation . So the
27:56	reason and I'm trying to show different ways and the
27:58	reason in a nutshell why you get complex roots ,
28:01	even though there's no intersection points , is because the
28:03	graphs of these functions that we were using all the
28:06	way up until now to show you what the function
28:08	look like really are incomplete . It looks like it
28:11	doesn't make any sense that this function has any routes
28:14	because it doesn't ever cross the X axis . But
28:17	I'm trying to show you that the graph of those
28:19	functions that we learned a long time ago are totally
28:21	incomplete . Those graphs are only putting in real numbers
28:24	and getting real numbers out . There's a whole richness
28:28	to this where I can put complex numbers in and
28:30	get complex numbers out . And it is true that
28:32	if I set the imaginary inputs to zero , I'm
28:35	gonna get only real numbers out . So there's no
28:37	routes along the real axis . That is true .
28:39	There's no intersection points on the real access . However
28:42	, there are two special numbers that don't even live
28:44	on the real axis , negative and positive , I
28:47	that do drive the function to zero . So when
28:50	you're solving a polynomial equation of any order and you
28:52	get complex roots . Those are just the numbers in
28:56	imaginary or complex form . That when you plug them
28:58	into the function they drive to zero . And so
29:01	you can't see them when you draw a regular plot
29:03	of a function . Because the regular plots of these
29:05	functions the old way that we do it , they
29:07	only involve real numbers . They're not showing you everything
29:09	else out there . So you have to imagine this
29:12	grid of reality of complex numbers that in some special
29:15	cases drive the function to zero . And now what
29:17	I wanna do is I want to go to the
29:19	computer and show you graphically . I can show you
29:21	what these functions look like . We can play around
29:23	with it and I can show you how the roots
29:24	actually pop out of these equations . Hello , welcome
29:28	back here . We have the computer demo here .
29:30	We're going to talk about what do complex roots of
29:32	a quadratic mean ? We're gonna do this uh this
29:34	demo here . So here we have the equation dialed
29:37	in that we've actually talked about in the lesson ,
29:39	X squared minus one . We see it cuts through
29:41	the X axis and two locations and the roots are
29:43	calculated for this thing . And I'll be able to
29:45	change this equation and show you , you know that
29:47	as we for instance , shift the thing up here
29:50	, the roots change right ? So here we have
29:52	uh imaginary roots here is we don't have any intersection
29:56	points on the real axis . But then as we
29:58	go deep below , we start to get real roots
29:59	and so on . So let me go set it
30:01	back to the place that it was before X square
30:03	. If I can get it there X squared minus
30:05	one like this . And so here are the roots
30:08	, negative positive one . So let's go down and
30:10	scroll down and take a look at what this table
30:11	of values looks like in the complex plane . So
30:15	if you were just plotting this , like we like
30:17	we do like we did right here , like we
30:19	just plotted it right here at table of values then
30:21	really what you need to look at in terms of
30:23	real numbers is just this line right here . This
30:26	line right here represents what happens here we have I
30:29	guess I should explain the column , the row along
30:31	the top is the real part of the input to
30:34	the function . And the over here on the side
30:37	is the imaginary part of the input . So if
30:39	we're just plotting this function in terms of its the
30:42	real numbers , plotting what we have up here ,
30:44	then we're gonna set the imaginary part equal to zero
30:46	, which is this value right here , set it
30:48	equal to zero . And we're really only going to
30:49	read this line of the chart . So you can
30:51	see here that at zero you get a value of
30:54	negative one at negative one , you get a value
30:57	of zero and a positive when you get a value
30:59	of zero . And you have all of these other
31:00	values here , these values 15 , 83 and so
31:03	on . These are the values that you plot in
31:05	this curve up above . And you can see that
31:07	the zeros of this graph are at negative one ,
31:09	X is equal to negative one , and X is
31:12	equal to positive one here , the value of the
31:14	imaginary zero . So these are the roots here ,
31:16	this is a route and this is a root .
31:18	It lines up exactly with what is calculated up here
31:21	above , right ? But I want to show you
31:23	a little more uh the grandeur of this down below
31:27	. So here this is a plot of the function
31:29	with all of the value . See this plot up
31:31	here is just showing the real values of the plots
31:34	. Here is the table of values in the complex
31:37	plane . Like I showed you on the board with
31:38	the whole entire thing filled out . You can see
31:41	complex numbers everywhere , you know , here's a complex
31:43	number at a value of negative two minus three .
31:45	I this would be the output right there if you
31:47	stick negative two minus three I into the equation and
31:50	so on . But there are two very special values
31:52	where the function is driven to zero . And you
31:54	can see this when we draw in three dimensions .
31:57	When you look down below here , these little bump
31:59	outs here and I know it's hard to read all
32:01	this stuff , but right here and right here is
32:04	where the function , these little kind of these ,
32:06	these points down here , that's where the function is
32:08	driven all the way down to zero . And you
32:11	can see and again I know it's hard to imagine
32:13	visualize it , but I'm drawing a three dimensional plot
32:15	. The real axis is down here , this axis
32:17	here and the imaginary axis is over here . So
32:21	when you kind of like turn it edge on here
32:23	, you can see the imaginary axis being zero is
32:26	lining up exactly what these little dimple points right here
32:29	. And then when you look at , if I
32:31	get a good angle here , the real part here
32:34	is negative one and here is over here , a
32:36	positive one . So the dimples here in this function
32:38	occur exactly where the chart says they should occur ,
32:41	negative one and positive one on the real axis .
32:43	Uh here because the imaginary part zero , where these
32:46	dimples are gonna shifted over here , so you can
32:48	see the imaginary part zero and those dimples are basically
32:51	right there . So as this curve changes as I
32:55	drag it farther and farther down , X squared minus
32:58	two , X squared minus three . Let's go to
33:00	x squared minus four . The roots are now at
33:03	negative two and positive two . And again in this
33:07	chart I see the roots again are now at negative
33:09	two and positive to know imaginary parts uh needed for
33:13	this because it's the roots are actually on the real
33:16	axis . And then you can see that these routes
33:18	are now farther apart , they're occurring at here .
33:21	I know again , it's hard to read . The
33:22	real part is negative two , that's this and positive
33:24	too . And if you kind of line it up
33:26	, you can see that these dimples at the bottom
33:28	occur at that location . And again , if you
33:30	shift it this way , you can see that it's
33:32	right with the imaginary part is zero . So basically
33:35	as as this function shifts around right here , you
33:39	can make these routes farther and farther apart . If
33:42	I go down here they're going to be even farther
33:44	apart , Right ? And then of course if I
33:47	go the other direction that can get closer and closer
33:49	together , which is where we kind of started ,
33:51	let me go right here , let's start right there
33:54	. So you can see that they're getting closer together
33:55	. Now let's make them a little bit closer together
33:57	still . So we'll go X squared minus one ,
34:01	which is where we started . This lesson will go
34:03	and see where x squared minus one is . So
34:04	the um the roots are occurring at negative one and
34:09	positive one on the real axis , negative one here
34:12	, positive one here on the real axis . That's
34:14	where the roots are basically driving this thing to zero
34:16	. Now if we take this quadratic and stick it
34:19	right on the axis where the equation is X squared
34:23	, Then we only have one dimple because they got
34:25	so close together . Remember they were getting closer and
34:28	closer and closer together . Finally they there's only one
34:31	spot which now the chart is updated to reflect only
34:33	one location here where the functions driven is zero .
34:36	It was plus or -1 . Now it's right at
34:38	zero because we only have one point right here where
34:40	it's touching and that dimple in the complex plane is
34:43	right in the middle of the whole entire thing ,
34:46	you can see it right there and so that's where
34:48	the function is driven to zero . So we can
34:52	go and drag this thing down and we have real
34:55	roots that are separated . They get closer and closer
34:58	and closer and closer and closer together . Finally they're
35:00	right on top of each other , but then we
35:02	go this direction and now we don't have any real
35:04	roots anymore . Let me see if I can pick
35:06	this one here , this is the other equation ,
35:08	we get X squared plus one . We know that
35:10	the roots are plus or minus , I it's listed
35:12	right here in the table . Now we see that
35:15	this is a route and this is a route .
35:17	So if you look at only the real numbers of
35:20	this graph , which is this guy here , the
35:22	imaginary parts set to zero . The real parts go
35:25	from negative four to positive for every one of these
35:28	values that shaded here is just a number , none
35:30	of them hit zero , and that's because the function
35:33	never gets down to zero . These are the values
35:35	that we would use to plot this function but it
35:37	never gets to zero . There are no real roots
35:39	. However , there are other routes here . Zero
35:42	, I'm sorry , located at negative I . And
35:44	positive I . Negative and positive I . With a
35:47	real part of zero . These are the values in
35:49	the complex plane that are driven , driving this function
35:51	to zero . Even though there's no intersection point here
35:54	. That's just because this graph originally only has real
35:57	numbers . It's a very thin slice of the actual
36:01	full blown curve which is now down below . So
36:04	now you can see two dimples whereas before they were
36:07	on the real axis . But now these dimples are
36:10	actually on the imaginary axis and you can see it
36:12	if I flip this thing around now see now the
36:15	imaginary axis is down here and this graph , the
36:18	two dimples are at negative I and positive . I
36:20	know it says negative two and negative four . This
36:22	is the imaginary axis . So it's negative two ,
36:24	I negative four I positive to positive four . The
36:27	dimples are occurring now in the imaginary part of the
36:30	plane . And then as I shift this thing farther
36:32	up , the same thing is going to happen when
36:34	I get farther and farther away like this , then
36:36	I'm going to have now I have roots at negative
36:39	two and positive to I . And now I can
36:42	see in this chart the roots have moved farther apart
36:45	, negative two I am positive to I . And
36:47	then in this diagram the roots have again moved farther
36:50	apart , negative two , I positive to I .
36:52	But if I line them up with the real axis
36:55	, there is no real part to these routes because
36:57	here the real part is zero . So the bottom
36:59	line is as I play around with this function and
37:02	I dragged those sliders around and change what where the
37:05	roots are . These dimples will appear in different places
37:08	of the complex plane , which will correspond to these
37:11	zeros being in different parts of this little chart that
37:14	I have pulled around . So that's basically the bottom
37:17	line . When you see these imaginary roots pop up
37:20	, then what you're really saying is that the that
37:24	the complex plane has those locations that drive the function
37:28	equal to zero . Now , here I have it
37:30	set up where I have the graph first and then
37:32	I have the chart and then I have this .
37:33	But I have another version of this down below where
37:36	I can more easily drag the sliders and you can
37:38	see how these complex roots move around . So here
37:41	is X squared minus one here . And now I'm
37:45	going to drag the sliders farther up so it's going
37:49	to be x squared plus one , X squared plus
37:51	two . Here's the here's the function over here on
37:53	the left , X squared plus three . So you
37:54	see what's happening is as I drag the slider and
37:57	change the function , the dimples are getting farther and
38:00	farther and farther apart because the roots are getting farther
38:02	apart . Um uh in this case the roots are
38:06	in the imaginary axis here because if you saw this
38:10	equation X squared plus 11 is equal to zero and
38:13	move the 11 over , it'll be square root of
38:15	negative 11 . You're getting imaginary roots here and they're
38:18	farther and farther apart . As I slide this thing
38:20	around as I make it closer and closer and closer
38:23	to the axis , then what's going to happen is
38:26	the function is going to have roots that are gonna
38:29	be closer and closer and closer together . Now watch
38:31	what happens when I go right through zero here ,
38:34	here's export plus one . Now here's the special equation
38:37	why is equal or f of X is equal to
38:39	X squared . The roots are right on top of
38:41	each other . And then when I go through it
38:43	then the the roots pop into the imaginary , I'm
38:46	sorry , the real access here . So here now
38:48	the roots are getting farther and farther apart in the
38:50	real axis . So as I drag the slider down
38:53	again approaching where this graph is approaching , uh the
38:57	origin here , then the roots get closer and closer
39:00	and closer together in terms of real uh numbers .
39:02	And then they pop over into imaginary territory . And
39:05	you can see the roots are now imaginary here .
39:07	So that's what I wanted to show you . The
39:09	reason why we have these complex roots is because there
39:11	are just different values in the complex plane that drive
39:14	the function to zero . And the reason that's not
39:16	taught to you so much in algebra classes , just
39:18	because it's a little beyond the scope of what you
39:20	typically learn . Yes . Hello , Welcome back .
39:23	I hope you've enjoyed the computer demo . It took
39:25	me a while to put it together , but it's
39:26	actually I think really instructive to show what's really going
39:29	on with complex roots . So now when you saw
39:31	the complex , I'm sorry , a quadratic equation with
39:34	the quadratic formula and you get to complex roots out
39:37	like three plus four I and three minus four I
39:40	. Or something like that . Then you shouldn't so
39:43	much wonder exactly what they mean now , because the
39:45	point of this is trying to tell you what they
39:47	mean , what they are is that these functions have
39:49	a much richer nature to them than is what is
39:52	originally described . All these functions can take complex numbers
39:55	in and get complex numbers out . And all of
39:59	these functions have special places in the complex plane .
40:02	That when you substitute them into the function will drive
40:04	the function to zero . I filled out just a
40:06	portion of the chart here and then the computer ,
40:08	we saw the full chart . But you can imagine
40:10	that even that chart on the computer is incomplete .
40:12	I mean , there's millions of other infinite number of
40:14	other numbers that can be substituted in . But the
40:17	point is is all functions live in the complex plane
40:20	. So when you give complex roots , even though
40:23	the function never actually crosses the axis , there can
40:25	still be other values in the complex plane that drive
40:28	that function to zero . That is what a route
40:30	is . If the root happens to be complex ,
40:33	it's okay . It's just another number in the complex
40:36	plane . They're a pair of numbers that will drive
40:38	the function to zero . I hope you've enjoyed this
40:40	. I hope you've learn something from it . The
40:42	concepts in this lesson are beyond the scope of most
40:44	typical algebra classes until you get to the university level
40:47	. But I think especially with the computer , it's
40:49	it's easy to understand where these complex roots come from
40:52	and that's what I was hoping to accomplish today .

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