Up next in 10
Complex Numbers Part 1: Operation With Imaginary Numbers, Negative Square Roots, And equations
1 views
Feb 19, 2025
Section 4.4 We've already learned how to simplify square roots, in this lesson we're going to learn how to perform operations with pure imaginary numbers. Chapters: 00:00 Introduction 02:55 The imaginary unit 04:27 How to find square root of negative numbers 09:47 How to multiply imaginary numbers 12:08 How to solve equations with imaginary numbers
View Video Transcript
0:00
all right so it's still there chapter
0:04
4.4 we're going to talk about complex
0:07
numbers complex numbers are fun I love
0:10
working with complex numbers I'm not
0:12
even I'm like serious I love complex
0:16
numbers right so we have been talking
0:20
about about the realm of the integers
0:23
right in real numbers most people are
0:25
use like real numbers negative and
0:27
positive and stuff like that but we also
0:29
have some numbers called complex numbers
0:32
right now for example I drew an equation
0:35
uh a graph here right this is a second
0:37
degree or we also call it quadratic
0:40
equation right quadratic function and if
0:43
you look at this quadratic function it
0:45
does not cross the x axis because it
0:49
does not cross the x axis this does not
0:53
have any real Roots meaning you cannot
0:56
solve this right you won't have any
0:58
solution in the realm of real numbers
1:01
but this can be solved if you are going
1:04
to use complex numbers and that's why
1:08
they come in handy right so we must use
1:10
them we must use them to solve problems
1:13
that cannot be solved using real numbers
1:16
all right for example if I have
1:20
x² =
1:23
-5 why would you say why would most
1:25
people say x² = 5 is that possible no
1:29
why
1:31
it's not real you leave it excuse me
1:33
what it's negative right you can't have
1:37
a number Square in the realm of real
1:39
numbers becoming a negative number it's
1:41
impossible right no real number can
1:43
become negative if you square it
1:46
absolutely none what if it's x² equals a
1:50
number that can't be perfect square like
1:53
square rooted well you can still do it
1:55
is there a square root of five yeah
1:57
square root of five is like 2 point
1:58
something something something a it's a
2:01
it's a irrational number but it is is a
2:04
real number but it's just irrational you
2:06
can use it right but you cannot have the
2:10
square root of a negative number if you
2:11
did one that's what Mrs hman told you x²
2:14
= -5 you cannot solve them in the realm
2:17
of real numbers you can't yeah this is
2:22
like how would you use this
2:24
because I don't see any circumstance be
2:27
like you'd have to use
2:31
not not on this level right not on this
2:34
level this is why they call them complex
2:36
numbers right
2:42
now imag walk up to a kindergarten and
2:46
you really want to ruin his day you show
2:48
him this now look at this right so if
2:51
you go so we're going to call those
2:54
numbers imaginary numbers right
2:56
imaginary numbers basically you
2:57
imagining that right so the imaginary
3:01
unit is called I and is defined by i²
3:05
isal to1 so now today you've learned
3:08
that what in the complex numbers
3:12
i² is equal
3:14
to1 this is called the imaginary unit
3:18
this is what we're going to use to solve
3:20
i² = 1 I is called the imaginary unit
3:24
okay so i² by definition is equal to 1
3:30
in complex numbers when you deal with
3:33
complex numbers I sare is1 and I is
3:36
called the imaginary unit so example
3:40
numbers like 6
3:42
I -2 I I root of three are all called
3:48
Pure imaginary numbers right any number
3:52
in the form like 2 I 3 i 4 I 6 I 7 I 8 I
3:57
9 I 20 I that's called imaginary number
4:00
a pure imaginary number right so now why
4:05
are they useful
4:07
example let me say I want to find a
4:10
square root
4:11
of9 if you put that in prealgebra
4:15
algebra one most of them will tell you I
4:17
can't solve this because you cannot have
4:19
a square root of
4:21
a you can't have a square root of a
4:23
negative number it's just impossible
4:24
right and then when we were in like
4:27
lower classes we go W can't solve no no
4:29
solution you go like this no solution
4:32
right and we go yay and I got the full
4:35
grade now we complex numbers now that we
4:38
talking about complex numbers this can
4:39
be solved this can be solved right so
4:43
now show you how to solve this if I have
4:45
root
4:46
of9 I have to break it
4:49
down right I have to break down this
4:52
number how we'll find out so Square Ro
4:56
of9 is basically me saying I'm doing
4:58
what square root of one time what 9
5:03
right -1 * 9 is9 do you all agree with
5:06
me on that yeah yes right are we all in
5:10
agreement right now the the great thing
5:14
about the square root is a property that
5:16
can be split right so of 9 is of1 * 9
5:22
and I can split this into whatk of1 * <
5:27
TK of 9 I can do
5:31
that right I can do
5:34
that because the square root property
5:37
can be split like this okay now I know
5:40
one thing we were introduced to I squ we
5:43
say that I square is equal to what one
5:46
so therefore we're going to substitute 1
5:48
by i s
5:52
right I sare * < TK of 9 now the
5:56
beautiful thing is this Ro of I sare is
6:00
I because I square is a perfect square
6:02
so square of I square is
6:04
automatically I
6:07
right and what is sare root of 9 three
6:11
so therefore this is I * 3 which is what
6:13
3 I 3 I right 3 so you see what we do
6:17
right so this is the breakdown this is
6:19
the basics now without even blinking an
6:22
eye what would what do you think Square
6:24
of4 will be 4 I no 2 I 2 I yeah two I
6:29
right because we have what because we
6:32
havek of1 * 4 and 4 is 2 so therefore
6:37
this will be what 2 I 2 I so do you do
6:41
need to do all that or can you just do
6:42
it in your head you can you can just do
6:44
this once you know the principle you can
6:46
do this real quick right but now it's
6:48
not so easy with this because 27 is not
6:51
a a perfect square so now we're going to
6:54
have to do what work out the number to
6:56
get something right we're going to have
6:57
to work with it yeah I know hat it but
7:01
that's FY awful right so now let's look
7:03
at this here we're going to work with 27
7:06
right we can break down - 27 that can be
7:09
written as what -9 right time what 3
7:14
right or this can be broken as what of 9
7:19
* < TK of
7:21
3
7:23
right did we just work with 9 what is
7:28
it 3 3 I right we already know that of9
7:33
is what 3 I because it's negative n
7:38
inside a perfect square so this is 3 I
7:40
so therefore this will give you what 3
7:43
I sare root of three right this is the
7:47
simplified form of this number okay is
7:51
that making
7:53
sense yes now how do you think we will
7:57
do this guy right here squ of8 we can
8:01
break down 18 into
8:02
what 92 9 and2 so be of 9 right time
8:08
root of two whatare root of9 3 3 I so
8:13
this is what 3 I square root of
8:18
those
8:20
right press
8:26
e all right now let's work with
8:30
more more
8:33
more
8:35
huh of the day yeah what's today I
8:40
lovex even the ones
8:44
that let's work with 125 right watch
8:48
here now we have St of25 it's 125 the
8:52
perfect square no right so we have to
8:55
try to see if what we can do with 125 I
8:58
know that 125 is what it's 25 and 5 25
9:01
and 5 so I'm going to go this isun of -
9:04
255 Right Time s otk of sinkle right
9:11
now what is the sare of -25 who knows
9:15
five what I 5
9:18
I because it's a negative right so it's
9:21
5 I because it's a negative is 5 I so
9:24
this is 5 I and what otk 5 Okay so you
9:29
see once you understand the gist of it
9:31
it becomes what a walk in the park A
9:34
Walk in Park walk in the
9:37
park walk walk in the park right
9:41
now now we going to do this walk par so
9:45
now we're going to do this here right
9:46
now check this out I think you
9:48
understand the principle right so now
9:50
we're going to learn how to multiply
9:52
pure imaginary numbers knowing all the
9:54
stuff that we just learned so if I do5 I
9:58
* 3 I what do you think that's going to
10:00
give me step by
10:02
step I
10:05
what I squ now here's the thing don't I
10:08
know what i sare is what is I
10:11
S1 so this giv you what5 * 1 which is 15
10:17
15 see how easy this is fun stuff right
10:22
now what's 2 I * 4
10:23
I what's that
10:26
equal I 8 I Square which is
10:30
pretty much four no 8 I
10:33
S8 I sare is
10:36
what1 so this 8 * 1 which
10:41
is8
10:43
okay right now now we're going to have
10:46
to work with like square root of
10:48
negative numbers here so let's try to
10:49
play play with it a little bit what is
10:51
do we are we doing any of this today I'm
10:55
not I don't know yet so I have of -6 *
10:59
15 right I want to make my life easy so
11:01
I'm like ah I'm not going to use I here
11:04
cuz I have a negative and a negative
11:06
gives me what positive so that gives me
11:08
sare of square of 90 now I can break
11:11
down 90 can I yes is 9 and what 10 10 so
11:16
this isun of 9 * < TK 10 what isun of 9
11:22
three so this is 3 S root of 10 so this
11:26
is how you multiply uh pure imaginary
11:30
numbers right
11:33
so now what we're going to do is we're
11:37
going to learn how to solve equations
11:40
with pure imaginary numbers and we can
11:41
stop at that and then do some problem
11:44
okay so we're going to learn how to
11:45
solve equations so can I wipe this out
11:48
or no yeah all
11:54
right so we going to learn how to solve
11:56
equations because now
11:59
so we have equations
12:03
right equations with imaginary numbers
12:06
so suppose I have
12:08
X2 + 16 = 0 and I going to solve for x
12:13
right X2 + 16 is = to zero what am I
12:17
going to do
12:25
first
12:27
uhhuh subtract what 10
12:29
where's the 16 16 right you subtract 16
12:34
right so we got x² = 16-6 now in the
12:38
realm of real numbers if we stop here
12:41
what would we say no what because it's a
12:45
negative right but we no longer dealing
12:47
with complex numbers so we have to take
12:49
the square root right so we going to be
12:51
like X is equal to what plus or minus
12:55
right square root of what -6
13:00
right course this is what we do we take
13:02
the square plus or minus the of16 okay
13:06
now what is theot of -16 we've done it
13:10
and4 and four what I I thank you very
13:14
much right so this should be x = what
13:16
plus or minus
13:19
4
13:21
I that make sense right so first I go x²
13:26
= -16 take the square Ro I know it's
13:29
going to be positive or negative squ of
13:32
-16 and then when I take the square root
13:34
of that I get what because the negative
13:37
number plus or minus 4 I
13:42
okay uh let's do another one
13:45
here uh let's
13:47
see let's say I have
13:51
4x2 plus 100 is equal to
13:56
zero um
13:59
just um I I would say first just divide
14:02
the whole thing by four all right divide
14:04
whole thing by four because we can right
14:06
we can divide whole thing by four that's
14:08
possible right so we get x² plus what 25
14:11
25 is equal to Z and then
14:14
what subtract 25 right so we get x2 =
14:19
--25 and then
14:21
what take the square root
14:24
right so we get x = what plus orus
14:28
square root of 5 - 255 first right and
14:32
that gives us X = plus or minus 5 5 I
14:36
right 5 and this is so we can stop here
14:40
for this cuz we've done a lot we've done
14:42
uh we've talked about imaginary unit
14:45
we've done square root we've done a
14:47
multiplication the product and now we
14:48
done equation so we need to stop here
14:51
and then do some just work to kind to
14:54
get the hang off going to be a work yes
14:57
tomorrow's going to be a workday