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Operation with Complex Numbers: Addition, Division, And More
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Feb 19, 2025
in our last session, we discussed imaginary numbers. Now we'll learn how to perform operations with complex numbers. Chapters: 00:00 Definition of a Complex number 01:41 How To Add & Subtract Complex Numbers 04:15 How To equate complex numbers 08:20 Dividing complex numbers ( The conjugate expression)
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0:00
want all right so first we need to
0:03
understand what is a complex number by
0:05
definition right the formal definition
0:08
is this a complex number is in any
0:11
number that can be written in the form a
0:14
plus b i right anytime you can write a
0:17
number in this format like a plus b i
0:21
that number is called a complex number
0:24
now this number has two parts right the
0:26
first one is the real real number part
0:28
like a and B is the imaginary portion of
0:33
it all right example 5 + 3 I is called a
0:38
complex number because you have five
0:40
which is the real number and then 3 I
0:43
which is the imaginary portion of this
0:46
complex number you also have 3 - I where
0:50
three is the real number and then
0:52
negative I is the imaginary portion of
0:56
this complex number you also have 2 + 5
0:58
I so all those are what you call complex
1:02
numbers complex number because of that
1:04
right so we have real portion and we
1:07
have a imaginary portion so a plus b i
1:10
is a general standard form of any
1:14
complex number all right so those are
1:16
complex numbers now knowing about
1:19
complex number is fine but how do you
1:21
operate complex numbers right how do you
1:24
add these numbers we're going to learn
1:25
how to add them we're going to learn how
1:27
to multiply them and we also going to
1:28
learn how to to divide them by finding
1:31
what you call the conate expression I
1:34
hope to get to that today if not we just
1:37
going to bump it into tomorrow right so
1:39
now if I have these two numbers here
1:41
right I have two complex numbers here I
1:42
have 5 - 7 I + 2 + 4 I and I want to
1:49
condense this I want to simplify this
1:51
right this is actually a very simple
1:54
process because you have to just like we
1:57
do with like variables you have to
1:59
combine what like terms right so here I
2:04
have 5 - 7 I + 2 + 4 I so I'm going to
2:08
combine the five and the two and I'm
2:11
also going to combine the -7 I and the 4
2:15
I and I'm going to respect the sign that
2:19
are next to them right so if I was to
2:21
rewrite this expression in the right
2:23
proper way that's going to be 5 + 2
2:27
right plus 4 4 i - 7 I right because I
2:33
want to rewrite this in the right way I
2:35
want to make sure I'm putting the right
2:37
terms together okay so now 5 + 2
2:40
obviously gives me what s and then 4 i -
2:44
7 I3 I so this number now gives me s- I
2:49
- 3 I and this is your complex number
2:52
again this is what you do so you simply
2:54
just combining terms that are looking
2:56
alike it's just like doing that with
2:58
variables right and now we're going to
3:00
try to do the same thing here I have 4 -
3:02
8 I and I'm trying to subtract it from 3
3:05
- 6 I so my first step is to always get
3:10
rid of my parentheses right okay so I'm
3:12
going to have 4 - 8 i - 3 now + 6 I
3:18
right because I have the negative and
3:19
the negative again I'm going to do the
3:21
same process here I'm going to rewrite
3:23
this okay so I'm going to have 4 - 3
3:27
right and then - 8 I + 6 I so now that
3:32
gives me what I have these two together
3:34
and I have these two together so 4 - 3
3:36
is 1 8 I + 6 i - 2 i - 2 I so this is
3:42
your complex number okay so this is
3:46
really simple stuff it's not that
3:48
complicated people just get yeah and
3:50
that like the end of like doing that is
3:52
just adding up yeah just add them up and
3:54
comb put like that's it like you are not
3:56
simplify this complex number to get a
3:59
more uh simplified version of it this is
4:01
simplifying it right now next we're
4:04
going to learn how to equate complex
4:06
numbers right so what do I mean by that
4:08
let's say I give you this giant unit
4:12
here right I have 3x - 5 + y - 3 I is =
4:17
7 + 6 I right and then the question is
4:21
asking find the values of X and Y so
4:27
that this expression is true
4:30
or this equation is true right I want to
4:33
find X and Y so that this whole
4:35
expression holds right this whole
4:38
equation hold so what am I going to do
4:40
here right right after I'm looking at
4:41
I'm like all right I know my definition
4:43
of complex numbers right I know that a
4:46
complex number is made of two portion
4:48
the real portion and the imaginary
4:50
portion right so if I look at this
4:52
number here I know that this is going to
4:54
be my real portion and I know that this
4:58
is going to be my imaginary portion
5:00
and I'm trying to find X and Y so that
5:03
this whole equation is true so the way
5:06
you solve it is this you're going to
5:08
have two equation right we're going to
5:10
set 3x - 5 is going to be equal to what
5:14
7 because this is the real portion of
5:17
this other number right and we also
5:20
going to make sure that y - 3 is equal
5:22
to what six and then we're going to
5:26
solve for x and y so that this is true
5:29
so so this is basically what you do this
5:31
is called equating complex numbers
5:33
you're trying to find a way to solve a
5:35
problem so that X and Y are found so
5:39
that this whole equation here is true
5:41
right so now I have 3x - 5 is 7 I can
5:44
solve for x this is just simple
5:47
arithmetic right so 3x is = 12 and
5:51
obviously if you divide by 3 x gives you
5:54
four right and then here Y is simply 9
5:57
so now why is going to be equal to 9 now
6:00
if you if you pay ATT tou if you replace
6:03
3x 4 here 12 - 5 will give you
6:06
what 7 and if you replace 9 by y 9 - 2
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will give you six which holds I have
6:14
seven and six so basically all you're
6:17
doing is finding X and Y so that this
6:20
whole equation is true
6:24
right right so that this whole equation
6:26
is true all right so that's basically
6:29
what we doing here now the very next
6:31
thing that we're going to talk about is
6:33
dividing complex numbers now that one is
6:37
a New Concept yeah how do you write how
6:40
would you write
6:42
this for homework for homework well for
6:44
homework all I want you to find is X and
6:46
Y so if x is 4 and Y is 9 you're just
6:49
finding the values of X and Y so that
6:52
this whole equation holds right you are
6:54
solving for x and y now what I'm doing
6:56
is this I'm replacing X by 9 x 4 and a 9
7:01
y by 9 to make sure what I have on the
7:03
left and what I have on the right are
7:05
the same it's just double checking you
7:07
don't have to double check if you're
7:08
sure that your algebra is right right so
7:12
but to solve it you have to make sure
7:13
you equate this one to the seven and the
7:17
imaginary unit to the imaginary number
7:19
on the side here okay and then you can
7:21
solve the actual equation now we're
7:23
going to talk about dividing complex
7:26
numbers now the dividing complex numbers
7:28
is the one that
7:30
people get hung up on I don't understand
7:32
this but it's easy Once you understand
7:34
the concept it's e really easy so we're
7:36
going to talk about dividing complex
7:43
numbers dividing complex numbers right
7:46
so say I have this
7:49
here we're going to go through a series
7:51
of steps so I have 4 I right divided by
7:57
um
7:59
no 4+ I let's say 4 + I / 5 I right and
8:05
I want to solve um I want to divide this
8:08
number so the first thing I want to do
8:10
is this I want to find what you call the
8:13
conjugate expression okay conjugate
8:15
expression so let me write that down
8:17
it's called
8:20
Uh
8:25
conjugate expression right so that we
8:28
can simplify this
8:30
okay because your goal is to make
8:33
sure when you have a complex number you
8:35
can never have an i or an imaginary unit
8:39
at the bottom you have to get rid of
8:41
that right this is why we need what we
8:43
call the conjugate expression so that we
8:45
can
8:46
simplify this we need to simplify so we
8:50
have to find a conjugate expression in
8:52
order to get rid of the imaginary
8:53
portion of it at the bottom right so the
8:56
way we do it is this so I have 4 + I
9:02
/ 5 I so the conjugate expression yes
9:07
sir okay so the conate expression of
9:11
this is 5 I right so I don't need to
9:14
worry about the five I'm just woring
9:15
about the I portion of it right so to
9:17
find a conjug expression I want to turn
9:19
this into a real number right so to turn
9:22
this into a real number I have to
9:23
multiply I by
9:25
what I I thank you by I because I sare
9:29
is equal to
9:31
itive one so I have to multiply this one
9:33
by I thank you Nicholas that deserve
9:36
extra credit that was
9:39
brilliant one brilliant thought of the
9:42
day that m m
9:47
all right let me let me I give
9:50
nothing I give him nothing who's
9:54
Niche there's no h no who's nit
9:59
it doesn't have
10:01
it's hey hold on a second now have to
10:04
multiply this
10:06
right four * I right
10:10
look four * I is what four I right and
10:15
then I * I is 2 I which is I Square
10:20
let's go step by step right and then 5 I
10:22
* I is 5 i s right we have to go step by
10:26
step now that we're beginning to learn
10:27
this process so now I'm going to turn
10:29
this into 4 I right minus -1 right over
10:36
5 * -1 so now that gives me 4 I + 1
10:42
over5 and then you can stop here because
10:44
now you have your real number at the
10:46
bottom right where'd you get the X what
10:50
x times that's plus I
10:54
plus come on I'm about to say right come
10:57
on come on so this is the first type of
11:00
conjugate expression we have two
11:01
different types so now we're going to
11:03
talk about the second type
11:05
right
11:08
now how did it become
11:11
minus 5 * 1 i 4 I become 4 IUS i s i
11:18
multiply the top and the Bottom by I
11:20
whatever you do at the bottom you also
11:21
have to do it at the top okay yes but
11:23
where does where does the minus come in
11:25
when it was originally a oh
11:30
oh my God it's 4 I plus I
11:33
squ oh
11:36
somebody she let me in
11:54
ER drowned out by
12:01
Three 6 now look here hey hey hey hey
12:05
stop so now let's say we have this here
12:09
right 3 + uh 2 I 3 + 6 I now this is
12:13
where you going to have to just believe
12:14
me and accept what I say right now if I
12:17
have this here and I want to find the
12:19
conjugate expression so the conjugate
12:22
expression of any term let's say if I
12:24
have a plus b i right the conjugate
12:27
expression of a plus b i is is a - b i
12:32
and then the con expression of aus b i
12:35
is A + B I so basically I'm trying to do
12:39
the same thing right I'm trying to get
12:41
rid of I'm trying to make sure that the
12:44
number that is under the um div the
12:47
dividing bar is a real number right so
12:51
to find the conjugate expression of this
12:53
this is always going to be multiplying
12:56
this by 3 - 6 I and I have to do the
12:58
same
12:59
up here now I pause what now pause I say
13:04
I pause for a second right why I want to
13:07
tell you something wait what is here
13:09
wait wait wait wait wait wait wait don't
13:11
don't don't L right so here I want to
13:14
show you something here in this in the
13:17
first instance it was just five eye and
13:20
to get rid of my eye I just needed to
13:22
multiply by what I right which you say
13:25
like brilliantly earlier right now here
13:29
to turn this into a real number I have
13:32
to multiply 3 + 6 I by 3 - 6 I because
13:35
that's the only way you have to multiply
13:37
an expression by its
13:39
conjugate in
13:41
order can you use theate is the
13:44
conjugate just the opposite can you just
13:45
do I kind of like the opposite what
13:48
about if you do2 I for which one
13:52
it's no because remember remember wait
13:55
what remember remember for the con
13:58
expression I never want to have a i i a
14:03
number of the imaginary number at the
14:05
denominator on the denominator I only
14:06
want I don't it doesn't matter if it's
14:08
in the numerator it can't be in the
14:09
denominator that's the the end point you
14:11
can't have an ey in the uh in the
14:13
denominator yeah but don't you already
14:15
have one
14:16
question what is it where did you get
14:18
the negative one from conjugate
14:21
expression I didn't get a negative one
14:22
what you because I S is equal to I
14:24
Square is1 that's established all right
14:27
right yes why don't we just multiply by
14:30
I again which well you can't because
14:32
this is 3 + 6 I here you just have 5 I
14:35
right yeah well can I just multiply that
14:37
one by I well if you do that then if you
14:39
multiply by I you're still going to have
14:40
a 3 I in the denominator which you don't
14:42
want oh yeah you see it you're going to
14:44
have 3 I + 6 I Square so that doesn't
14:46
help so this is why you have to use the
14:48
conjugator because if you multiply this
14:49
by 3 - 6 I then that's going to happen
14:52
right you just have to accept that this
14:54
is it I can't tell you why you do this
14:59
going use like I don't care I'm just
15:01
fing it right right now look here now
15:04
look at how I'm going to multiply this
15:06
right I'm going to use my what my uh
15:08
distributive law 2 I * 3 is what 6 I all
15:13
right 6 I 2 I * -6
15:16
I -2 I 2 right now here what 3 * 3 9 3 *
15:24
6
15:25
I I 6 I * 3
15:30
18 I and 6 I * 6
15:34
I6 i s now what what's going to happen
15:36
here right now I already know I can get
15:38
rid of these two right they're gone cuz
15:41
89 + 9 that's gone hey knock it off cuz
15:46
if she not paying attention I can't
15:47
teach you that later on I didn't get it
15:49
stop doing it come on you you guys are
15:51
like going up so 6 I we got 6 I right
15:56
now here I have -12 * what
15:59
1 right CU i s
16:01
is1 and then I have in the denominator 9
16:05
right - 36 *
16:09
-1 okay because 12 i² is 12 * 1 now I'm
16:14
left what 6 I + 12 over 9 + 36 okay 9 +
16:23
36 and then that gives me 6 I + 12 over
16:29
45 now I don't leave it like this I can
16:32
still simplify this right because each
16:34
of these number is divisible by what 12
16:37
no 12 by three three goes into six three
16:39
goes into 12 three goes into 45 so I'm
16:42
left with what 2
16:44
i+ 6 no + 4 over 15 now I have to write
16:50
it the right way because all complex
16:52
numbers are supposed to be written how a
16:55
plus b i so I have to write this the
16:57
proper way so it should be really 4
17:01
right + 2 I over 15 because you have to
17:05
put it in the right order it's always
17:06
the real portion first and then the
17:08
imaginary portion next to it all right
17:12
so again if this this is to abstract for
17:15
you I'm sorry we just have to learn
17:18
right so now case number two yes sir
17:21
more are we doing like uh problems no
17:25
today tomorrow so now let me ask you a
17:27
question what if I have said right over
17:30
1 - 2 I and I wanted you to find the
17:34
expression what would you do yes yeah
17:38
what would you do here what would be the
17:39
conjugate expression there and how would
17:40
you simplify this we're doing which one
17:43
we're doing seven s over 1 - 2 I so how
17:46
you that thank you multiply everything
17:49
by I not by I by what 1 + 2 I 1 + 2 I oh
17:54
yeah right because here if you do I
17:56
again you going to still end up with I
17:57
in the denominator so 1 + 2 I so
18:00
basically you just switching the sign
18:01
here and multiplying that by the
18:03
opposite right does that make sense so
18:05
this will be inside of that be 1 + 2 I
18:08
but don't if you multiply by I becomes 2
18:10
I 2 which is just yeah but then you
18:12
still going to have 1 * I and that's
18:14
gives you another I in the denominator
18:16
which really defs the purpose but that
18:17
would be
18:19
1- all right let's try it so say you say
18:21
you want to multiply by I right let me
18:23
show you what I'm talking about so let's
18:24
sayy this by I right and do the I here
18:27
oh I see now yeah I and I still have i -
18:30
2 I 2 right so this is why we have to
18:33
use this and we have to multiply this by
18:35
the conjugate which is basically 1 + 2 I
18:39
and then the same thing here 1 + 2 I and
18:41
then notice something that you're going
18:42
to have a recurring thing here right on
18:44
top we're going to end up with what 7 +
18:47
14 14 I and the denominator you going to
18:50
see that every time the two middle
18:52
numbers are going to cancel for let me
18:53
let's go step by
18:55
step so this is it right 1 * 1 is 1
18:59
1 and 2
19:00
I 2 I right -2 I and 1 - 2 I and then -2
19:06
I * 2 I is 2 I 2 I 2
19:13
right 4 i
19:15
s I need some more
19:19
zinc right so you see the middle one are
19:21
always doing what that's canceling
19:24
constantly canceling so you going to get
19:25
to the point where you won't even have
19:26
to compute this once you get the hang
19:28
off it's going to be like what my
19:30
favorite term it walk in it
19:36
holy right and then 1 - 4 * -1 which is
19:41
basically 7 + 14 I over 1 + 4 all right
19:47
so 7 + 14 I over 5 right and this is the
19:52
conjugate expression so what I want to
19:54
do tomorrow is I want to spend some time
19:56
working on some problems to
19:59
WRA yourself all right so it's not it's
20:02
not that hard if you understand when do
20:05
I use the I for example what if I gave
20:07
you this 3 over 2 I and I say find
20:11
simplify this you multiply this by what
20:13
I I that's it I and I that's all you're
20:16
doing right but then if I have 3 1 - I
20:21
then you do what conjugate which is 1 I
20:24
1 + I 1 plus I so that's itj
20:28
expression the cont is the opposite of
20:30
it right kind of like the opposite so
20:32
this is it all right it's easy stuff
20:35
this is this is like a