Complex number fundamentals

3Blue1Brown

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Intro to the geometry complex numbers. Full playlist: https://www.youtube.com/playlist?list...Home page: https://www.3blue1brown.comBrought to you by you: https://3b1b.co/ldm-thanksBeautiful pictorial summary by @ThuyNganVu:https://twitter.com/ThuyNganVu/status...Errors: - On the first sketch of a complex plane, there is a "2i" written instead of "-2i". - At the end, in writing the angle sum identity, the last term should be sin(beta) instead of sin(alpha). - During Q9, the terms in parentheses should include an i, (1/2 + sqrt(3)/2 i) ------------------ The live question setup with stats on-screen is powered by Itempool.https://itempool.com/The graphing calculator used here is Desmos.https://www.desmos.com/The "Complex slide rule" came from Geogebra, via Ben Sparks.https://www.geogebra.org/m/mbhbdvkrCurious about other animations?https://www.3blue1brown.com/faq#manimMusic by Vincent Rubinetti. Download the music on Bandcamp:https://vincerubinetti.bandcamp.com/a...Stream the music on Spotify:https://open.spotify.com/album/1dVyjw...If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ Video Timeline (thanks to user "Just TIEriffic" - W3 Result - W4 Promp - Ask What would you call 'imaginary numbers' - Startingpoint & assumption - W4 Result - Q1 Promp - Q1 Proces - RotatingCoordinate - Q1 Resul - Q - Q3 Promp - Q3 Result - RotationAnimatio - 3 facts about Multiplicatio - Q4 Promp - Ask imaginary I vs physics i& - Q4 Resul - GeoGebraDem - Q5 Promp - Q5 Result - Q5 Solutio - RotatingImages Exampl - PythonExampl - PythonImage Rotation Exampl - Ask Vectors & Matrices for rotatio - Q6 Promp - Q6 Result - Q6 Solutio - RedefiningAngle Additio - Q7 Promp - Ask Can we do without complex numbers - Q7 Result - Q7 Solutio - Q8 Promp - Ask sum/difference of angle - Q8 Result - Q8 Solutio - DesmosExampl - Bringing it all togethe - The "cis" shorthand explaine - Q9 Promp - Q9 Result - ClosingRemarks ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribeVarious social media stuffs: Website: https://www.3blue1brown.comTwitter: https://twitter.com/3blue1brownReddit: https://www.reddit.com/r/3blue1brownInstagram: https://www.instagram.com/3blue1brown...Patreon: https://patreon.com/3blue1brownFacebook: https://www.facebook.com/3blue1brown

SHOW LESS

00:00

today we are going to talk about one of

00:01

my absolute all-time favorite pieces of

00:03

math it's incredibly fundamental to

00:05

engineering to mathematics itself to

00:08

quantum mechanics but it's something

00:10

that it has a terrible terrible name we

00:12

call them complex numbers and worse than

00:15

that the things that bring about complex

00:17

numbers we call imaginary numbers and

00:19

before we get into any of it what I want

00:21

to do is start with kind of a poll just

00:23

to pull the audience on seeing what you

00:26

guys to consider to be well real what do

00:28

you consider to exist when it comes to

00:30

numbers so we've already been doing a

00:32

couple polls in the warm-up animations

00:34

but as a a serious poll of so it's one

00:37

that's actually going to help me see

00:38

where you're coming from before we begin

00:41

the lesson here I just want to ask you a

00:43

very simple question okay so let's go

00:45

ahead and pull it up here pull it on up

00:47

and oh for whatever reason it seems like

00:53

we're having trouble pulling this one up

00:56

hmm okay there we go

00:58

took a little delay there among the

01:01

values - square root of 2 square root of

01:05

negative 1 and infinity which would you

01:07

personally consider to really exist

01:10

whatever really existing means to you so

01:13

in theory if you guys go to three b-1b

01:16

co / live you should be able to answer

01:18

this and then the statistics based on

01:20

your answers are going to start

01:21

populating the screen

01:22

we won't know what those answers refer

01:24

to at the moment all we know is that you

01:26

know there's someone who has a lead

01:27

another one that's pretty close behind

01:29

it and as your answers come in and as

01:32

the server is kind of digest them we'll

01:33

start to see some of the stats here if

01:36

you go to that page by the way 3 B 1 B 2

01:38

slash live which redirects to item pool

01:41

com what you're going to find is at the

01:44

very top you can ask a question on

01:46

Twitter and all that's going to do is

01:48

basically open up a tweet that's going

01:49

to have the hashtag in it locked down

01:51

math and that's the way that we're gonna

01:53

be doing questions to this instead of a

01:54

live chat anytime you have a question or

01:56

a comment that you want to you know

01:58

insert into the lesson and let that be

01:59

part of the discussion it's going to be

02:01

pulled up here and it looks like we

02:04

already have one this is from Yash Dave

02:05

who asks if you could rename the complex

02:08

numbers and the so-called imaginary

02:10

numbers

02:11

• something more intuitive - a name that

02:13

conveyed the fact that they have

02:14

numerous applications in the real world

02:16

what would you name them I couldn't be

02:18

happier that you asked - so I have one

02:21

friend who seems very passionate about

02:22

the fact that we should call them sneaky

02:23

numbers personally I'm very fond of

02:26

trying to connote spinning and rotation

02:28

and this is one of the things we'll talk

02:29

about in today's lesson is the fact that

02:31

what we call complex numbers what we

02:33

call imaginary numbers some of the main

02:35

uses that they have come from very

02:37

elegant descriptions of how to rotate

02:40

stuff and I hope you'll kind of see what

02:42

I mean as we as we proceed with that so

02:44

on our poll asking you guys which

02:46

numbers you consider to really exist

02:48

which is of course a subjective question

02:50

there's no right or wrong answer here I

02:52

am genuinely curious actually how this

02:54

breaks down because it's not at all a

02:56

strong consensus in one direction it

02:58

seems like we've got three top

03:00

contenders and then three that are

03:03

following pretty evenly behind that so

03:06

let's go ahead and take a look what do

03:08

you consider to really exist when it

03:11

comes to numbers now I can imagine which

03:15

ones might be the top two but I'm very

03:16

curious about the fact that there's

03:18

three all kind of coinciding with each

03:20

other there and it looks like I I'm

03:23

getting a little bit of a delay before

03:25

the reveal so there's kind of this nice

03:26

dramatic pause I'll tell you for me

03:29

personally I feel like it's very silly

03:31

to answer anything that's either it's

03:33

not all of them are none of them I can

03:36

maybe understand if someone wants to

03:37

treat infinity as something different

03:38

because it's ill-defined there's lots of

03:40

different things that that might mean

03:42

but insofar as numbers exist at all if

03:45

you consider what a number is to be a

03:47

real thing then it would oh man I I

03:51

can't believe that we're stalling out on

03:53

this one we had fixed it by livestream -

03:54

but I guess there's going to be an

03:56

oscillation between when it works and

03:57

when it doesn't but for me personally

04:00

basically anytime that you have a

04:02

numerical construct that's helpful in

04:05

the real world you know I'd consider

04:07

that real I would say that if it's

04:08

something that's actually useful in an

04:09

application then it is as real as words

04:13

are right you're never going to run into

04:14

an abstract word like happiness out

04:16

there but it has a kind of reality in

04:18

our mind

04:18

and things like the square root of two

04:21

which you can't express as a fraction or

04:23

things like the square root of negative

04:24

one that don't show up among real normal

04:27

numbers you know even if they might seem

04:30

a little bit different oh this is such a

04:32

shame I'm genuinely curious to see what

04:34

your answers are but it's not showing up

04:35

for me which I suppose means we'll have

04:38

to move on with a lesson but this will

04:41

presumably begin working by the end and

04:43

we can maybe pull things up again so let

04:47

me go ahead and take that away what I'd

04:51

like to do for you today basically is

04:53

show you the sense in which imaginary

04:55

numbers are useful the complex numbers

04:57

are useful and from there maybe try to

04:59

imbue them with a little more reality I

05:01

won't assume that you know what they are

05:02

yet it's meant to be a basic primer but

05:05

let's just dive right in okay the end by

05:08

the way the very end here I want to talk

05:11

about two different trigonometric

05:13

functions and this is kind of a thing

05:14

that we're going to build to two

05:15

identities from trigonometry and I

05:18

understand that maybe oh these

05:21

complicated identities from trigonometry

05:22

is not going to be the best way to lure

05:24

some people into the understanding oh

05:26

yeah complex numbers they're really

05:27

useful you're really going to love them

05:28

but I do think it's interesting that you

05:30

can have a fact that has nothing to do

05:32

with complex numbers of the square root

05:33

of negative 1 it's just trigonometry

05:35

it's everything we were talking about

05:36

last time and you can have facts that

05:38

are pretty hard to remember I remember

05:39

when I was in school and we learned

05:41

these addition formulas that if you want

05:43

to know the cosine of the sum of two

05:44

different angles you know it's this kind

05:46

of long thing in terms of cosines and

05:48

sines of the original two angles there's

05:51

this minus sign that would always trip

05:52

people up if you do the same for the

05:54

sine it looks similar but there's a plus

05:56

sign and instead of having Coast coast

05:59

you have cosine it's something that's

06:00

very error-prone if you're just trying

06:02

to memorize it as it is however if you

06:04

come at it with complex numbers this is

06:07

not only much less error-prone it has a

06:08

very beautiful meaning and it just falls

06:10

right out so even if you don't

06:12

necessarily believe in the reality of

06:14

the square root of negative one you at

06:16

the very least have to admit that it's

06:17

interesting that it can make other

06:19

pieces of math useful that other pieces

06:21

of math a little bit more understandable

06:23

too and trigonometry is just the tip of

06:27

the iceberg if you talk to anybody who's

06:28

in engineering anybody who's going into

06:30

serious math they'll tell you the

06:32

complex

06:32

numbers are as weel a part of their work

06:35

in their life as real numbers are but

06:37

the starting point looks very strange

06:40

okay when you start introducing this the

06:43

very first thing you do is to say assume

06:46

that there's some number I so that I

06:48

squared is equal to negative one and I

06:51

think through a lot of students there's

06:52

maybe one of two possible reactions that

06:54

you can have here one is no there isn't

06:57

anytime I square a number even if it's

06:59

negative if I take negative five for

07:01

example and I square it well the

07:03

negative times a negative is a positive

07:05

so I get 25 any number that you squares

07:08

if it's positive

07:10

well that just stays positive so it

07:12

seems like no matter what when I'm

07:13

squaring numbers I always get a positive

07:15

number I'm never going to get anything

07:17

negative so this does not exist no such

07:20

number however if a mathematician comes

07:24

and says oh no it exists we've defined

07:26

it so that that's the case i think the

07:28

other reaction someone can have is hang

07:30

on a second you can do that when you

07:33

have a problem that you can't solve you

07:34

can just say oh I've defined things so

07:36

that we now magically have a solution

07:37

okay next time I'm having trouble with

07:40

my homework and I don't know what the

07:41

answer to X is I will be like let X be

07:43

the value define to be the answer to

07:45

this question so if you're uncomfortable

07:47

with this you're definitely not alone in

07:49

fact Rene Descartes coined the term

07:52

imaginary for these numbers as a

07:54

derogatory it was meant to make fun of

07:56

the fact that obviously there is no such

07:58

answer and it shouldn't be taken as

08:00

serious math and then we stuck with that

08:02

as a convention and we still call them

08:03

imaginary numbers which is genuinely

08:05

absurd but that's not the only weird

08:07

assumption that we make the second weird

08:09

thing that you do when you start talking

08:11

about complex numbers is to say there's

08:13

not just such a number I but we're going

08:15

to give it a home instead of the real

08:17

number line which you know all of these

08:20

numbers we know when we square them you

08:21

can't get a negative what we do is say I

08:24

lives in a different dimension I lives

08:26

perpendicularly there's one above and

08:29

then there's one below negative I and

08:32

you can have negative 2i you scale it

08:34

however you want

08:35

essentially it's proposing that numbers

08:37

be two-dimensional and that I has a very

08:39

specific home one unit perpendicular

08:43

perpendicularly above the real number

08:45

line

08:45

and okay if we want to extend our number

08:49

system I get it maybe it's useful to put

08:52

some kind of number up there

08:53

but why I right why not say infinity is

08:57

the number that sits one unit above zero

08:58

or one divided by zero or any other

09:02

problem that you couldn't solve before

09:03

and you make up an answer to why should

09:05

that live there what on earth does the

09:07

idea of a point one unit above the real

09:10

number line in a separate dimension have

09:12

to do with squaring two negative one so

09:15

I hope to answer this for you at the

09:17

very beginning let's just talk about how

09:19

if you're adding numbers that are

09:21

two-dimensional like this the rules are

09:23

pretty straightforward and it operates

09:25

essentially the same as vectors for any

09:27

of you who might be familiar with

09:28

vectors so let's say hypothetically I

09:31

have a number oh I don't know let me

09:34

draw one here that's going to be 4 plus

09:36

I ok and then I'm going to take a second

09:40

number and it's helpful to draw them as

09:42

vectors kind of an arrow from the number

09:45

zero and this one is going to end up at

09:47

negative 2 plus 2i so what I'm saying is

09:51

you take the real number negative 2 and

09:53

then you move in that perpendicular

09:55

direction into the extension of our

09:57

number system which again kind of asking

09:59

the students to take a lot on faith here

10:01

that you're ok to do that that you're

10:03

allowed to just pretend that the numbers

10:04

extend in this direction if you take

10:06

that on faith and you follow hopefully

10:08

the fact that it becomes useful helps to

10:10

justify why we're doing any of this so

10:12

my question for you is simply what

10:13

happens when we add these two numbers

10:15

now assuming that our question system

10:18

has not broken down I should be able to

10:20

do this as a proper Pole and let me go

10:22

ahead I guess we can first check the

10:24

previous Pole ok things seem to be

10:26

working so we can take a little step

10:28

back in the lesson so I'm just genuinely

10:29

curious I want to know how you guys

10:30

answered on this one it looks like

10:33

there's a there's a back and forth

10:35

between answers F and D so f is all of

10:38

them saying that all of these should be

10:40

considered real and interesting D is the

10:43

one that says you should consider to

10:44

square root of 2 and negative 1 but not

10:46

infinity so there's a good contingent of

10:48

you out there who would just reject

10:50

infinity as being considered real but

10:51

are very comfortable with the square

10:52

root of negative 1 that's awesome and

10:54

then after that it looks like C people

10:56

who reject the square root of negative

10:58

fascinating I actually would have

11:00

thought that none of them would have

11:01

come higher than that none of them is

11:03

much lower at a okay so it looks like

11:06

we've got a cohort of people who are

11:07

comfortable with negative one a large

11:09

cohort are uncomfortable with infinity

11:11

that's a topic for another day don't

11:13

worry about it and then a number of

11:15

people who are kind of in that middle

11:16

ground of maybe not being super

11:18

comfortable with the idea that negative

11:19

one might be real let's see if we can

11:21

convince you of the difference of that

11:22

so for our first much more mathematical

11:25

question that's kind of a warm up I just

11:28

want to ask you to add these two before

11:29

I've taught you how to add them make a

11:31

guess at how it might work and I hope

11:35

that it feels pretty straightforward

11:37

addition is actually the least

11:38

interesting part of this but it is it's

11:41

a good thing to know when you're

11:42

learning about complex numbers it's

11:43

definitely one of those operations that

11:45

you are going to need to know

11:46

unfortunately and you can tell by the

11:49

fact that I'm stalling and what I'm

11:50

saying here

11:51

it looks like the question is still not

11:53

loading completely correctly so I'm

11:54

going to have a stern word with Cam and

11:57

either behind the scenes who have

11:59

otherwise built such a beautiful

12:00

beautiful interface that's helpful for

12:02

this kind of back and forth between you

12:05

guys and me I'm gonna have a stern word

12:08

with them behind the scenes but in the

12:09

meantime let's go ahead and move forward

12:11

with a lesson here so I guess I can pull

12:14

it up on the just on the piece of paper

12:16

and you can follow along at home see

12:18

what the addition might be it turns out

12:20

to be relatively straightforward if

12:23

you're moving four units to the right

12:24

and then one unit up and you want to add

12:27

the idea of moving two units to the left

12:28

and then two units up will you just do

12:31

each of those one at a time I'll go

12:33

ahead and pull out black here the real

12:35

part is going to be those four to the

12:37

right then minus two to the left okay

12:39

straightforward enough and then the

12:41

imaginary part is going to be this one

12:43

unit up and then these two units up one

12:46

plus two times I so is that one I plus

12:49

two I and then when you work that out

12:51

four minus two is two one plus two is

12:53

three a nice simple introduction here

12:57

addition doesn't really have anything

12:59

complicated going on which is great that

13:01

means that it's one fewer thing for us

13:03

to worry about what is so complex about

13:05

complex numbers after all well where

13:07

everything becomes interesting is when

13:10

you try to multiply these numbers

13:11

together

13:12

so with victors there's not really any

13:14

notion of multiplying them to get two

13:16

vectors back at least when we're in the

13:17

2d plane you have some notions like

13:19

cross products and dot products that in

13:21

three dimensions can get you something

13:23

like it but the rules end up being very

13:24

different from that in the number system

13:26

you can't really do algebra you can't do

13:28

things like assume that if two numbers

13:30

multiply to make zero then one of them

13:31

has to be zero but complex numbers are

13:34

going to end up behaving much like the

13:35

real numbers so rules from algebra can

13:37

carry over but to understand what that

13:39

rotation rule is I know I'm giving

13:41

things away what that multiplication

13:43

rule is I just want to ask you a simple

13:46

question which is basically suppose I

13:48

have the point mmm

13:51

3 2 okay we're not even going to think

13:54

of it as a complex member per se if I

13:56

just have some sort of coordinate grid

13:58

and I go to the point with x-coordinate

14:00

3 and y coordinate 2 what is the 90

14:04

degree rotation of this if I rotate it

14:06

90 degrees and let's say counter

14:08

clockwise counter down geez writing is

14:15

difficult counter clockwise

14:23

okay now what's lovely about this is we

14:28

can basically just turn our paper to

14:29

figure it out you say okay if it started

14:31

at three two and then I rotate 90

14:33

degrees counterclockwise I can just read

14:34

that off now as being negative 2 in the

14:37

X direction and then 3 in the Y

14:40

direction if I had rotated the whole

14:42

plane like that so what we've done here

14:44

is we've taken 3/2 and then we convert

14:48

it to negative 2/3 something which may

14:52

be in our original system you know it

14:56

looks like this negative 2 and then 3

14:57

that's going to be the 90 degree

14:59

rotation and what's nice here is that

15:01

that rule is very simple and it applies

15:03

to any pair that we might have if I took

15:05

a pair of numbers a comma B okay and

15:09

then I said where is that going to

15:10

rotate to if I rotate it 90 degrees it's

15:13

going to end up by swapping the

15:14

coordinates B a and then making that

15:17

first one negative that's a 90 degree

15:19

rotation and a nice gut check here is to

15:22

say what happens when we do that twice

15:23

what if we do that same very mechanistic

15:25

operation again twice and I'm gonna go

15:28

and take this I swapped the two

15:30

coordinates we get a negative B but then

15:33

that first one becomes negative so that

15:36

was another 90 degree rotation well

15:38

what's happened here is we've just made

15:40

both of the coordinates negative and

15:42

that's reassuring because if I take some

15:44

point sitting at a B and then I rotate

15:48

it 90 degrees so this will be my initial

15:51

90 degree rotation and then another 90

15:53

degrees that's the same as 180 degree

15:56

worked no I've done that wrong that will

16:00

be the same as a 180 degree rotation

16:02

which should look like this ignore the

16:05

other vector that I drew which is just

16:08

taking both of the coordinates and

16:09

making them negative negative a negative

16:12

B okay so that's reassuring this

16:16

operation that does a 90 degree rotation

16:17

actually behaves like you would expect

16:19

it to now why am I asking you this well

16:22

I'm being told that supposedly I'm

16:24

allowed to ask you questions again so

16:26

I'm gonna have you do your very first

16:28

complex product oh look a lot of people

16:30

did submit answers very good okay let's

16:34

let's grade the complex addition

16:35

actually let's let's see if it is

16:37

and straightforward to process as I was

16:39

hoping it was see how much explanation

16:41

is demanded okay so it looks like a

16:43

majority of you did get the correct

16:45

answer which is 2 plus 3i very good very

16:48

good

16:48

52 of you answered simply 2 which would

16:51

have been the real part of the answer so

16:53

maybe just the fact that there's some

16:54

vertical component and you need to still

16:56

add those vertical components or maybe

16:58

those of you who answer to reject the

17:00

reality of imaginary numbers so you just

17:02

don't even acknowledge that vertical

17:04

component some of you answer negative 2

17:06

3 which I guess is just making a vestra

17:11

swapping up whether you're taking 4

17:13

minus 2 or 2 minus 4 so that's

17:15

completely understandable we've got 2

17:18

plus 3 which is maybe just dropping off

17:21

the I so I think maybe a lot of like

17:22

simple errors and entry and you know

17:24

that happens to all of us especially on

17:26

tests is sometimes you know what the

17:28

right answer is but then you you forget

17:29

a symbol or you swap too so that's all

17:31

very good let's go ahead and try our

17:33

very first product though like I said so

17:36

here because I already talked through

17:38

one of the questions we're going to go

17:39

ahead and skip ahead of it we know how

17:41

to rotate something like 3 comma 2 so

17:44

I'm not even going to give you time to

17:46

do that and properly grade it stall

17:50

stall words words you know they tell me

17:53

that it's working and yet it's very slow

17:55

for me to progress forward so you know

17:58

if I'm not gonna have a stern word with

17:59

and you guys can go at them on Twitter

18:01

too under the same place that we asked

18:03

questions and just say hey Cameron eater

18:06

can't you make the live questions work a

18:08

little bit better for us ok I think

18:11

we're finally there everybody ready

18:14

aha wonderful very simple question I

18:18

want you to take the number I and I want

18:20

you to multiply it by 3 plus 2i and even

18:23

though I haven't really talked about the

18:25

rules for multiplication what I can say

18:27

is pretend like it operates just like it

18:29

does for normal numbers you've got

18:30

things like the distributive property

18:31

where you can distribute this throughout

18:33

and then the defining feature of I is

18:36

this idea that I squared is negative 1

18:38

that's the only special thing you need

18:40

to know about that other than that just

18:41

treat it like it's a normal number ok

18:44

and then proceed forward with the

18:45

product wonderful ok so it looks like

18:48

we've got quite a few of you coming in

18:49

to answer which is always lovely

18:51

super exciting for me by the way just

18:53

how many people are enthusiastic about

18:55

coming and like getting back to the

18:56

fundamentals of math in this lockdown

18:58

and just you know we're gonna sit back

19:00

for an hour and we're gonna learn about

19:01

complex numbers and we're actually going

19:04

to participant we're actually going to

19:05

answer questions as you do it rather

19:06

than sitting and passively watching this

19:08

is genuinely delightful to me okay this

19:12

is this isn't necessarily question I was

19:14

expecting to divide the audience

19:16

necessarily so unsurprisingly it looks

19:19

like we have a very strong majority in

19:21

one direction hopefully in the correct

19:24

direction but if not that would that

19:26

would heavily inform where the lesson

19:27

should go so that would be quite useful

19:29

and I think that's probably enough time

19:32

so I'm gonna go ahead and lock the

19:35

answers into this one okay and it looks

19:38

like the majority of you answered

19:39

negative two plus 3i which is absolutely

19:42

correct absolutely correct

19:44

so there's two ways to think about this

19:46

okay one of them is to walk forward with

19:49

the algebra and just do it a little bit

19:50

mechanistically okay so if we pull

19:54

ourselves up our sheet if we take I

19:58

times 3 plus 2 I 3 plus 2 I

20:03

it just distributes I times 3 is going

20:06

to be 3i I times 2i is going to be 2

20:09

times I squared by definition I squared

20:14

is negative 1

20:15

which means that our final answer is

20:17

going to look like negative 2 plus 3i

20:21

ok and like I said it looks like a

20:23

majority of you correctly did that

20:25

product now it's one thing to just walk

20:27

through it mechanistically it's another

20:28

to step back and say what just happened

20:31

geometrically right because what we just

20:34

talked through was the fact that if you

20:36

want to rotate numbers 90 degrees the

20:38

rule is to swap the two coordinates and

20:39

then multiply that first one by negative

20:41

2 well look it's what's happened here

20:43

we've got 3 & 2 those coordinates have

20:46

been swapped 2 is now the real part 3 is

20:49

the imaginary part but that 2 got

20:51

multiplied by a negative 1 because I has

20:53

this defining feature of squaring to

20:55

become negative 1 so that should give

20:57

you some indication that okay

20:59

multiplying by I has this action of

21:01

rotating things by 90 degrees may

21:04

that means that it's not a crazy thing

21:06

to do to geometrically position I at a

21:09

90 degree angle with the real number

21:12

line okay

21:13

now to really see why that ends up being

21:15

useful it turns out that if you have

21:17

that if you have a number that behaves

21:19

this way it gives you a computational

21:22

mechanism for all of the other types of

21:24

rotations that you might want to do that

21:25

might not necessarily be ninety degrees

21:28

and to show you why this works I'm going

21:30

to go ahead and pull up an animation so

21:32

let's say we have any number Z and in

21:35

this case Z is going to be let's see

21:37

where do I have it C is going to be a 2

21:40

plus I great and let's say I want to

21:43

understand what is multiplying by Z due

21:46

to every other possible complex number

21:48

what we can go one by one the very first

21:51

one that's relatively simple as if I ask

21:53

what is Z times one where does it take

21:55

the number one

21:56

well Z times one is going to be Z so

21:59

we're going to take the number one which

22:01

I might draw with a little yellow arrow

22:02

and we're going to rotate and stretch

22:04

that arrow up to the point where Z is

22:07

great a kind of trivial fact even though

22:10

it's trivial I'm actually going to take

22:11

a moment to write that down just so that

22:13

we can oh no no that's for the best for

22:16

later that is Randy don't you guys worry

22:17

about him he'll be coming in in just a

22:19

moment so I just want to write down

22:22

three crucial facts that are getting an

22:24

influence rotation three facts I'll call

22:29

it three facts about multiplication the

22:36

first two are going to look simple the

22:37

third one is going to look innocuous but

22:39

it ends up being extremely constraining

22:41

so the first one is that if I have any

22:43

number Z and I multiply it by one well I

22:46

get the number Z back okay

22:48

geometrically that means with that we

22:49

stretch and rotate that vector one to

22:51

get what we want the second one is that

22:54

if I take Z times I from what we just

22:56

talked about that is the 90-degree

22:58

rotation what I might write is like

23:00

rotate by 90 degrees the number Z okay

23:04

so often our animation what does that

23:06

look like let's see we've already

23:07

multiplied by one we take I and then

23:10

it's going to move to whatever the 90

23:12

degree rotation point for Z itself is

23:14

okay

23:16

two down infinitely many to go and we

23:20

know what it does to one we know what it

23:21

does to I let's see if we can understand

23:23

what Z does to any other possible number

23:25

well it turns out those two is really

23:27

all we need to work with if we have the

23:29

distributive property so the third fact

23:31

that's going to look kind of innocuous

23:33

is let's say I take this Z and I

23:36

multiply it by C plus D times I where C

23:41

and D are just any two numbers okay

23:44

well this is going to distribute so Z

23:46

times C I'm actually going to write that

23:48

a little differently I'm going to write

23:49

it as C times Z plus Z times D I which

23:54

again I'm going to write in kind of a

23:55

funny order and write that as D times Z

23:59

I now the idea here is what we know

24:03

where Z is we also know where Z times I

24:06

is so if we're just scaling them up by

24:09

some other constants that completely

24:10

constrains where we need to go so let me

24:13

go ahead and write this down with an

24:14

example okay let's say that we go back

24:16

here and I want to know what multiplying

24:18

by Z does to anything I want to tell I

24:21

want to convince you that it rotates all

24:23

of the grid in a way that keeps these

24:25

lines parallel it keeps them evenly

24:27

spaced keeps them perpendicular to each

24:29

other

24:29

it applies this very constrained rule to

24:32

the whole plane and really just think

24:34

through any one particular point for

24:35

this let's say that we have two times

24:39

negative I okay so you move two units in

24:41

the positive right direction and then

24:44

negative one unit in the vertical

24:45

direction well after the product where

24:48

that's going to land has to be two times

24:51

wherever zealand's plus negative one

24:53

times wherever I lands okay and we see

24:56

that right it's two times this yellow

24:57

vector and it'll be negative one times

24:59

the green vector so here even before you

25:02

actually work out the product we could

25:03

just read off the fact that Z here which

25:06

looks like it's two plus I multiplied by

25:09

two - I must land on the number five

25:13

okay so I actually want you guys to work

25:16

this out algebraically because I do

25:18

think that it's it's edifying to see

25:20

this work in practice right the fact

25:22

that we have this geometric rule where

25:23

we were just kind of stretching and

25:25

rotating and this corresponds to an

25:27

otherwise very mechanistic process of

25:28

expanding out

25:30

some so let's pull back our quiz and I'm

25:34

going to it's a little bit silly because

25:40

I've revealed the answer but what I

25:42

actually want you guys to do here is

25:43

take a look at the question which says

25:45

what is 2 plus I times 2 minus I and if

25:49

you have notes right now if you have a

25:51

pencil and paper which I encourage you

25:52

to always come to class with I want you

25:54

to try working it out do the first

25:56

inside/outside last distribution

25:58

property just to see mechanistically

26:00

what number ends up popping out from

26:02

this and then we'll try to see how that

26:03

squares with the geometric intuition so

26:06

while you're doing that while you're

26:08

working that out hopefully on pencil and

26:09

paper it looks like we've got a question

26:11

from the audience which is is I the same

26:15

as I and J the vectors in physics great

26:18

question actually that's that's a much

26:20

more interesting question than you might

26:21

think it is the short answer is no so in

26:24

physics you often describe the rightward

26:26

direction as I in the up and down

26:29

direction as let's go get another one

26:31

here let me finish answering the

26:33

question first so in physics you often

26:36

have this situation where instead of

26:39

describing the XY plane you describe the

26:42

right vector as I sometimes I hat and

26:45

the vertical one as J hat so that's a

26:49

separate thing that's how physicists

26:51

will describe the right direction of

26:52

direction for vectors they're not

26:53

necessarily describing complex numbers

26:55

totally separate thing what's more

26:57

interesting actually is that the reason

27:00

that convention came about I believe has

27:03

to do with a number system that extends

27:05

even beyond complex numbers it's called

27:07

the quaternions so with the complex

27:08

numbers we introduced something called I

27:10

with the quaternions we introduced J and

27:12

K these two other numbers both of which

27:14

also have squares two negative one as it

27:16

happens and the guy who invented them

27:19

Hamilton he was trying to describe

27:20

three-dimensional space very elegantly

27:22

and using the purely imaginary

27:24

directions I J and K became a nice way

27:26

to do that and I think it's because of

27:29

that and his attempts to push that

27:30

forward that the conventions slipped

27:33

into normal physics to just describe

27:35

those as I and J of course that leads to

27:37

serious confusion now because if we look

27:40

at how physicist thinks of things you

27:42

know I is in the right direction

27:44

J is in the vertical direction but with

27:45

complex numbers I is of course in the

27:48

vertical direction so it just makes for

27:49

a whole potential mess now what was our

27:52

oh no it looks like they took away the

27:54

comment oh man there was comment giving

27:56

them some sass and then they got

27:57

embarrassed so they pulled it away well

27:59

it's on screen it's in there for

28:01

everyone in the future to see now I

28:03

think that's probably been enough time

28:04

to work through our a little bit of

28:06

algebra here so let's go ahead and grade

28:08

it for everyone to see and it looks like

28:11

the vast majority of you well I told you

28:13

the answer so I'm gonna assume the vast

28:14

majority of you are giving the correct

28:17

answer Oh interesting

28:18

okay so it looks like five five is

28:24

correct and that's the most common

28:25

answer but then the ones where missed

28:27

were three and four and maybe we can try

28:28

to understand why three and four would

28:31

have been common misconceptions on this

28:33

one 69 is close 0 i plus 5 is correct

28:37

but I guess just formatted in a weird

28:40

way so let's let's go ahead and work

28:43

this one out by hand ok it's actually

28:46

kind of a pleasant process to multiply

28:49

complex numbers because it's so

28:51

deterministic you know what you have to

28:53

do you know where you start you know

28:54

where you're going to end and it's

28:56

really just kind of practicing through

28:57

it so two times two that's the first

29:00

thing that we get and then you take the

29:02

inside parts I times two so that's going

29:05

to be 2i and then the outside parts

29:07

which is negative I times two so that's

29:10

negative 2i and then the last part which

29:13

is I times negative I so that'll be

29:16

negative I squared the defining feature

29:19

remember is that I squared is the same

29:21

as negative one which means that the

29:24

real component of our answer is going to

29:26

be 2 times 2 or 4 now minus negative 1

29:29

which is plus 1 so that'll be 5 and then

29:33

the imaginary parts cancel each other

29:34

out that 2i cancels out with a negative

29:36

2i so to everyone on the quiz who

29:39

answered 5 plus 0 I you know extra

29:42

credit to you because that actually is a

29:44

hundred percent correct you're even

29:45

emphasizing that we're doing this in a

29:46

complex context so there's no reason

29:48

that you should have been graded

29:49

incorrectly on that one however what's

29:53

uh what's interesting here those of you

29:55

who watch the very first lecture might

29:56

notice the same

29:57

with difference of squares and this is

29:59

effectively a difference of squares type

30:00

question and I'm going to guess that

30:02

those of you who answered three or four

30:03

it probably came down to if you answered

30:06

three it would have been because you

30:07

took this four and you subtracted one

30:09

rather than subtracting negative one

30:10

very common mistake no harm in that and

30:13

then if you answered four it comes from

30:16

forgetting the fact that I squared can

30:18

be simplified into a real number so that

30:21

you know that very much counts for the

30:23

the real component of your final answer

30:25

now what's interesting I think is that

30:27

that a process of just walking through

30:29

the algebra feels very different from

30:31

this geometric idea that we had before

30:33

of saying multiplying by Z has the

30:36

action of rotating and stretching things

30:38

and how it rotates and stretches things

30:40

is in the way that's necessary to get

30:42

the number one to sit on the number Z

30:44

and in fact I have a have a good friend

30:47

Ben Sparks who is a frequent number file

30:50

contributor and as some of you might

30:52

recognize and I asked him to whip up a

30:55

little geogebra thing because he's just

30:57

a geogebra whiz and he actually created

30:59

basically the complex number equivalent

31:01

of a slide rule that we can play with

31:03

here so the way that this is gonna work

31:05

is that I can take the number one and

31:07

sort of drag it around the plane and as

31:10

I do

31:10

the rest of the grid moves in the way

31:12

that it has to so that the origin stays

31:14

fixed in place and then everything else

31:16

kind of stays rigid with the exception

31:18

of being able to stretch so the way that

31:20

you might use this is to say let's say

31:22

we want to know what Z times W is well

31:25

I'm gonna stretch and rotate the number

31:27

one so that it lands on Z and then I'm

31:30

going to follow what happens to W which

31:31

is now drawn as a a orange vector follow

31:35

what happens to W and as it lands there

31:37

we can just read off that it ends up at

31:39

five

31:39

now even though in practice

31:40

computationally you would be working it

31:43

out with the algebra like we just showed

31:44

having in the back of your mind that

31:46

this is how they operate helps draw the

31:48

connection between complex numbers and

31:50

geometry and also trigonometry which

31:52

we'll get to in a moment

31:54

and that turns out to be a much more

31:55

helpful connection than you might expect

31:57

so let me go ahead and ask you guys a

32:00

second question here which is do to

32:03

bring back our helpful little friendly

32:06

quiz

32:09

given what I just told you about how

32:12

complex numbers act I want you to guess

32:16

for me it's okay if you don't know it's

32:17

okay if you just want to hazard a guess

32:18

here

32:19

what is the complex number Z so that

32:22

multiplying by Z has the effect of

32:24

rotating 30 degrees or pi/6 radians

32:27

counterclockwise okay so you want to

32:30

number Z so that when it's being applied

32:32

as an action and you're saying what do

32:34

you do to all the other possible complex

32:35

numbers the corresponding action is to

32:38

rotate by pi/6 radians or 30 30 degrees

32:43

okay great we've got a lot of answers

32:45

coming in strong consensus around one of

32:47

them which is always a good sign let me

32:51

show you how I might draw this out if I

32:54

were doing things so I would take a look

32:57

at a unit circle of some kind and

33:00

luckily I have my infinite supply of

33:02

unit circles sitting right here and I

33:05

would say okay I know that the action

33:06

that I want is going to be whatever

33:08

takes the number 1 to something that's

33:10

the 30-degree rotation of the number 1

33:12

it's it's up here whatever complex

33:14

number Z this is is going to have the

33:17

appropriate action and then from there

33:19

it's all of the trigonometry that you

33:21

and I were working through last time so

33:23

let's look at our answers and let's go

33:24

ahead and lock them into place now

33:27

before we talk through the explanation

33:29

and the correct answer in this context

33:32

is d which is cosine of pi/6 plus I sine

33:36

of pi/6 congratulations to the majority

33:39

of you who correctly answer that very

33:41

interesting to me actually is that the

33:43

second most common answer was a which is

33:47

1 plus PI sixths and I think that's

33:49

that's an utterly reasonable guess

33:51

actually and let's take a look at why so

33:55

if we go back to our friendly unit

33:56

circle the number we were going for is

34:02

like I said sitting pi/6 radians around

34:05

and from everything we talked about last

34:07

time the X component of that is the

34:10

cosine of PI sixths this is sort of how

34:13

cosine is defined and then the Y

34:16

component of that is going to be sine of

34:18

PI sixths

34:21

which means that that number Z that

34:23

we're dealing with yeah he's going to

34:27

have an X component of cosine of PI

34:29

sixths and then an imaginary component

34:33

what's in the Y direction of sine of PI

34:36

sixths great now that second most common

34:42

answer like I said that's very

34:43

interesting um if we pull it up it was a

34:45

which was 1 plus PI 6 over I let's take

34:48

a look at where that sits with a I'll

34:50

bring in a second cleaner unit circle

34:52

here like I said I've got an infinite

34:54

supply they were guessing that it's 1

34:56

and that instead of walking PI sixths

34:58

around that you walked a straight up PI

35:00

sixths okay which is actually quite

35:05

close right especially for small angles

35:07

if you're just walking straight up

35:09

that's not too different from walking

35:10

around the circle so if you were to try

35:12

to use that number to have the action of

35:15

rotating by 30 degrees it would have an

35:17

action of rotating by something that's

35:19

slightly less than 30 degrees and then

35:21

they would stretch things out by

35:22

something that's slightly more than a

35:23

factor of 1 but it's actually close

35:25

which is cool because usually the the

35:27

misconceptions might be like very wrong

35:29

can computationally but the idea that

35:31

the the misconception here was like

35:33

computationally quite close indicates

35:35

that maybe the geometric reasoning was

35:37

lining up with truth which i think is

35:39

awesome so to give you an illustration

35:40

of how this might actually be used okay

35:43

we have a number that has this action of

35:45

a 30 degree rotation I want to bring in

35:47

of course a friendly PI creature for us

35:50

to all think about Mandy so let's say

35:54

that I have a PI creature and I'm I

35:55

don't know animating a video and at some

35:58

point for whatever reason I want that PI

36:00

creature to rotate 30 degrees okay I

36:03

just want them to rotate there's lots of

36:04

ways you can do this one of the most

36:06

common ones in computer graphics is to

36:08

use matrices which might be a whole

36:10

topic for another day but a tactic you

36:12

could use is to use complex numbers and

36:15

the way this would work is to think of

36:16

our PI creature as living in the complex

36:19

plane and then in any kind of computer

36:21

graphics it's always defined with

36:23

various points often these control

36:24

points around it and each one of those

36:26

control points I would give complex

36:28

coordinates to so for example let's say

36:31

the the lower right foot here or I guess

36:35

Stage Left our right looks like it has

36:37

coordinates about one negative three so

36:40

that would be given the number one minus

36:42

3i and then from what you guys just told

36:44

me in answering the question if we want

36:46

to take that point and then figure out

36:48

where it gets when we rotate 30 degrees

36:50

about the origin what I should be

36:52

multiplying it by is cosine of 30

36:54

degrees by sine of 30 degrees and if

36:57

ever you wanted to be convinced that

36:58

complex numbers are just as real as real

37:00

numbers I think one factor to consider

37:02

is that many programming languages have

37:04

built into them

37:05

complex numbers they are considered as

37:07

real as real numbers so what I might do

37:10

I can pull up a little Python terminal

37:12

for us to play with and complex numbers

37:15

if I write something like 3 negative 1

37:17

are considered a valid type in this case

37:19

they use J I think basically because in

37:23

programming parlance I is often used as

37:25

an indexing variable so they just go

37:27

with J instead that actually matches

37:29

with electrical engineering convention

37:31

to where they instead of using I for the

37:34

imaginary number they use J because

37:35

again I was kind of already used so 3

37:38

minus 1 J you can just read that as

37:40

being the same as 3 minus 1 I and I can

37:43

take this number and I can multiply it

37:44

by other complex values I don't know

37:46

like five for two if I multiply that

37:50

it's going to apply the rules of complex

37:53

multiplication which basically is just

37:54

foiling it out you can think of this as

37:57

being the real part is 3 times 4 or 12

38:00

minus negative 1 times 2 because

38:04

remember that I squared gives us that

38:05

negative so that's where the 14 comes

38:07

from and then the imaginary part is

38:09

going to be that negative 1 times 4

38:11

which is negative 4 plus 3 times 2 which

38:17

indeed works out to 2 so that's kind of

38:18

cool Python seems to be working all of

38:20

this out for us and what that means

38:22

remember is let's say that we wanted to

38:24

rotate our pie creature and let's say

38:26

I'm just going to focus on one point

38:27

like his foot sitting here which is that

38:29

one negative 3 so let's say I define the

38:32

foot to be sitting at real part 1

38:36

imaginary part negative 3 the rotating

38:39

number that I want Z is what you guys

38:41

figured out for me we're going to need a

38:42

little math for this so let's import

38:44

math always a sign that you're about to

38:46

have some fun

38:47

so we're gonna call Z a complex number

38:49

whose real part is the cosine

38:52

well I can't write 30 degrees because

38:53

this will all be in radians so I'll

38:55

write it as PI sixths that's the same

38:57

thing as 30 degrees and then the Y is

38:59

going to be the sine of that sine of

39:02

math PI over 6

39:05

great so if we look at its components

39:08

the real part is around 0.866

39:11

and the imaginary part should be exactly

39:13

1/2 but you know often you get a little

39:15

numerical error because computers they

39:17

can't get all the real numbers perfectly

39:18

there's just not enough information in

39:20

the universe so they have to have a

39:21

little bit of an error there some of you

39:23

might recognize that that first part is

39:26

the square root of 3 divided by 2 this

39:29

is everything we were working out last

39:31

lecture for computing things like the

39:32

cosine of PI sixths now what this means

39:34

for us is if we take Z times foot right

39:37

or remember foot was that one negative

39:40

3z times foot should tell us where it

39:42

goes when you just go through the

39:43

mechanistic operation of complex

39:45

multiplication this point should tell us

39:47

where it lands and violently it's

39:49

supposed to land at around 2.30

39:51

real-part

39:52

and then negative 2.09 as the imaginary

39:55

part and indeed when we play that out

39:57

and we follow our little foot point here

39:59

where does it land now all right it

40:02

looks like the real part is a little

40:03

above 2 and the imaginary part is a

40:05

little less than negative 2 and any

40:08

deviation there is probably because the

40:09

foot wasn't exactly at that point 1

40:11

negative 3 like I said but you can see

40:13

what I mean in principle you tell the

40:15

computer to do this for every single

40:16

point in a given image that you want and

40:18

it will tell you where the outputs

40:20

should be it'll tell you where the

40:22

rotated versions should be so complex

40:24

numbers are certainly as real as real

40:26

numbers in that respect it looks like

40:29

we've got a question from the audience

40:30

so let's go ahead and pull that up seal

40:32

programmer asks what is the biggest

40:35

reason that people use vectors and

40:37

matrices to represent rotation instead

40:38

of complex numbers that is actually a

40:41

very interesting question

40:42

I don't know if there's a great reason

40:45

actually because if all you're gonna do

40:47

is rotate and then potentially scale

40:48

things you could have everything defined

40:50

in terms of complex numbers in 2d

40:53

graphics in 3d graphics that actually

40:56

wouldn't work

40:56

complex numbers only describe rotations

40:58

in two dimensions

40:59

however if you want to use a number

41:01

system to describe rotations in three

41:03

dimensions instead of using matrices you

41:05

could use these things called

41:06

quaternions which are basically the

41:08

complex numbers on steroids and in fact

41:11

they expose you to fewer errors like

41:13

there's errors that come about when you

41:15

try to use matrices for 3d rotation this

41:17

thing called gimbal lock talk to any

41:19

robot assist or any computer graphics

41:21

programmers they'll tell you that gimbal

41:22

lock is a real pain and they're

41:24

quaternion this sort of complex numbers

41:26

on steroids helps to solve that problem

41:28

there's a whole other contingent of

41:30

people who feel passionate about

41:31

something called geometric algebra that

41:34

is a whole other topic potentially for

41:35

another day but that's that's a really

41:37

valid question you've got these two

41:38

perspectives on one one possible

41:41

operation but if we're just coming at it

41:44

from the idea that complex numbers play

41:46

a role in other parts of math and in

41:48

other parts of engineering I think

41:49

thinking of the rotation implications

41:52

really helps us see that and this is

41:55

particularly visceral for me when I

41:58

think in terms of trig identities so if

42:00

you remember at the very beginning I

42:01

opened up by talking about one

42:03

particular trig identity and without

42:05

pulling it up again I don't want to go

42:06

back and just read off of it I'm going

42:08

to ask you one more question which this

42:10

is going to be a tricky question this

42:12

one will actually take some time to work

42:13

through words that I want you to

42:16

calculate for me the cosine of 75

42:20

degrees okay and let's go ahead and pull

42:24

this up as a quiz question so let's pull

42:26

up our quiz let's move on to the next

42:29

question let's cross our fingers and

42:32

hope the cam and either have not broken

42:33

anything again and that's all it's

42:37

asking what is the cosine of 75 degrees

42:39

okay now let me get you started here on

42:41

how you might think about it 75 degrees

42:44

it's not gonna be based on any triangles

42:48

that we might have seen maybe you've

42:50

seen a 75 degree triangle and you happen

42:52

to remember what its side lengths were

42:53

but most people won't have however what

42:55

you can do is note that 75 is the same

43:00

as taking 45 degrees and then walking

43:02

another 30 degrees around okay 45 plus

43:06

30 degrees and if I

43:09

question is asking us for the cosine of

43:10

75 one way to think of it is if you

43:13

first walk 45 degrees around and then

43:16

you walk another 30 degrees around what

43:19

is the x coordinate of that

43:20

corresponding point okay so this is

43:22

going to have some coordinates XY the

43:25

cosine of 75 degrees is asking for the x

43:27

coordinate of that now like I said I

43:30

think this is this is not easy okay so

43:32

it's gonna take some time to work out

43:33

and while you guys are working this out

43:36

I want you to actually pull out some

43:37

pencil and paper and think it through

43:38

I'm just gonna pull up a little pause

43:41

and ponder music and give us I don't

43:43

know I might give us a full 60 seconds

43:45

or so to like noodle this one out and

43:47

think it through and in the meantime if

43:50

you want to ask any questions you know

43:52

how to do so through Twitter it's with a

43:53

hashtag lock down math and I might go

43:56

ahead and take a couple questions while

43:57

people try working this out by hand

43:59

which again I heavily encourage you to

44:01

do even if you're not watching this live

44:03

even if you're watching this at a later

44:04

date this is the time to say okay I'm

44:07

gonna pull out my paper I'm going to try

44:08

to think this through and and now I'll

44:10

stop talking I'll let you just think

44:30

certainly don't feel rushed by the way

44:32

even if there are some answers rolling

44:34

in right now that is by no means in the

44:37

indication that you should have finished

44:38

or anything like that

45:04

so interestingly we do have a strong

45:07

lead but it's not at all a strong second

45:10

place so it seems like the the way that

45:13

the assuming that the lead is correct

45:14

the way that the error errors have

45:16

happened is not in a consistent way

45:18

which is interesting and that'll be

45:20

that'll be fun to dissect once we reveal

45:23

what the answers are

45:38

let's them let me check if there's any

45:41

new questions that have rolled in no ah

45:44

which suggests stew me that the two

45:47

folks that I have behind the scenes who

45:48

were supposed to feed me questions are

45:50

probably desperately wrangling with

45:53

something item pool related

46:01

those of you who have already submitted

46:03

an answer or if we think you already

46:05

know the answer by the way I want you to

46:07

take a moment to seriously guess what

46:10

you think the second most common answer

46:12

is going to be and where exactly the

46:14

misconception could have taken place or

46:16

to double check your work you very well

46:19

might be one of those sitting in the the

46:22

misconception territory which you know

46:24

it's it's a lot to work out there's a

46:27

lot of different failure points along

46:28

the way

46:40

all right so with that I'm gonna go

46:43

ahead and lock in the answers and if you

46:46

wanted to still work it through even if

46:48

the answer isn't going to be recorded

46:49

that's totally fine we're going to walk

46:50

through it in a moment

46:51

but I'm gonna lock them in just so that

46:53

we can see what the correct answer turns

46:55

out to be and the correct answer is D

46:58

which it looks like a majority of you

46:59

got so that's awesome the second most

47:01

common was C which really only differs

47:04

by a minus sign and I can I can actually

47:06

guess why that's true already so that's

47:07

that's interesting

47:09

after that is B where there's a square

47:11

root of 3 instead of a square root of 2

47:12

okay so we have we actually have a

47:14

pretty good distribution on this I think

47:16

that this shows that this is not at all

47:18

an obvious question and can take take a

47:21

little bit of noodling to figure out so

47:23

if I saw this on a test okay let's say

47:25

that someone threw like a high school

47:27

math test at me right now and it said

47:30

cosine of 75 you know I would know that

47:33

in principle I'm supposed to be thinking

47:34

of this as in terms of friendlier angles

47:39

I know that 45 degrees is a friendly

47:40

angle for trigonometry and I know the 30

47:42

degrees is what I don't have memorized

47:44

though are the addition formulas I

47:47

learned them once slipped out of my head

47:49

definitely don't have them memorized but

47:51

I do know how to read arrive them on the

47:53

fly basically the idea is that we take

47:55

the complex number associated with a 45

47:57

degree rotation which is going to be

47:59

cosine of 45 degrees plus I times the

48:05

sine of 45 degrees I take that complex

48:09

number and I'm going to rotate the

48:11

complex number corresponding to a 30

48:13

degree rotation so I might even just

48:15

pull up yet another unit circle here and

48:17

say the one that sits 30 degrees away I

48:23

want to multiply it by this number well

48:25

what is that number its x-component is

48:27

the cosine of that so the real component

48:29

is the cosine of 30 degrees and then the

48:32

imaginary component

48:34

okay the vertical component here is the

48:36

sine of 30 degrees which looks like it's

48:38

you know the smaller leg of a 30-60-90

48:40

triangle which is going to be one half

48:43

but we'll work out the numbers in just a

48:45

moment

48:45

sine of 30 degrees times i and by the

48:49

magic of complex multiplication

48:50

which I hope isn't magical based on what

48:52

we walked through a little bit earlier

48:53

the effect that this will have is to

48:56

rotate by one and then rotate by the

48:57

other so what are these values cosine of

49:01

45 degrees is the square root of 2

49:03

divided by 2 sine of 45 degrees is the

49:06

square root of 2 divided by 2 and if you

49:09

want to if you don't remember that and

49:10

you want to work it out the way to think

49:12

about it is that a 45 degree angle is

49:18

going to create an isosceles right

49:21

triangle so if we don't know what this

49:24

side length is we at least know that

49:25

it's the same as the other side length

49:27

so if the hypotenuse is 1 we know that x

49:29

squared plus x squared is supposed to

49:31

equal 1 that's the Pythagorean theorem

49:33

that tells us the 2x squared equals 1 so

49:37

x squared is the square root of 1/2 and

49:40

then that's actually the same as writing

49:43

the square root of 2/2 you could also

49:47

write it as 1 divided by the square root

49:49

of 2 if you want but these are these are

49:50

equivalent forms so if you forget you

49:52

just draw the triangle it's no big deal

49:53

cosine of 30 degrees well if we have our

49:57

30-60-90 triangle the long side of that

50:01

from everything we talked through in the

50:02

last lecture is square root of 3 over 2

50:04

square root of 3 over 2 and then the

50:07

sine of 30 degrees is 1/2 and now we

50:11

just carry out the complex product it's

50:13

not exactly a fun prospect maybe but

50:15

it's really just for operations so it's

50:17

not going to be too bad the real part is

50:19

going to be our cosine times our cosine

50:21

okay so that's going to end up being our

50:23

square root of 2 over 2 multiplied by

50:28

square root of 3 over 2

50:31

okay I'm also going to package in when

50:34

we multiply both imaginary parts okay

50:36

I guess I was a little inconsistent I

50:38

wrote I times sine and then sine times I

50:40

but it's the same thing so I'm gonna

50:42

include that square root of 2 over 2

50:46

multiplied by the 1/2 but because each

50:50

of them has an I we also multiply it by

50:52

I squared which by definition is

50:54

negative 1 so we get this negative sign

50:56

in here and that's going to be the real

50:59

part okay which is actually all we need

51:02

we just need the real part and that

51:03

gives us

51:04

the that gives us the x coordinate of

51:07

the where they drop before here we go

51:09

if we rotate 45 degrees then we rotated

51:11

another 30 the X component of that the

51:14

real part is going to be the final

51:16

answer so we'll do the sine part in a

51:18

moment too when we make it more general

51:19

but just for the sake of answering the

51:21

question this ends up being root 2 times

51:23

root 3 which is root 6 divided by 2

51:27

times 2 so root 6 over 4 minus and that

51:30

minus sign came about because of the I

51:33

squared minus square root of 2 over 4 ok

51:38

so all in all that ends up being root 6

51:41

minus root 2 divided by 4 and if we pull

51:45

up the question I think it's very

51:47

interesting that the second most common

51:49

answer the the most common mistake you

51:51

might say is only different because of

51:53

that plus sign and if we look at the

51:55

math where that came from as it happens

51:57

is the idea that this was a minus sign

52:01

which came about from the idea that

52:03

there was an I sitting in front of each

52:04

sign so the most common misconception

52:07

came from maybe forgetting to apply the

52:08

fact that I squared is equal to negative

52:10

1 at least if you were going about it

52:12

using complex numbers which from the

52:14

whole context of the lecture of course

52:16

we want to so this actually corresponds

52:20

to a much more general formula and I

52:22

think it's edifying to write out that

52:23

general formula for ourselves because

52:25

anytime that I've forgotten what the sum

52:27

of two angles in trigonometry should be

52:29

it's totally ok because you can read

52:32

arrive it relatively quickly

52:34

if you're comfortable with complex

52:36

multiplication and the more math that

52:37

you do because it shows up all over the

52:39

place you actually get very comfortable

52:41

with complex multiplication even if you

52:43

like me become very rusty on things like

52:46

trigonometric identities but with that

52:49

comfort you can very quickly read arrive

52:51

what the angles should be or what the

52:54

sum some identities should be sign of

52:58

data writing and talking at the same

53:00

time so remember the way to think about

53:02

this is that this first number is what

53:05

we get by finding a complex number

53:08

that's basically just rotating alpha

53:10

around the unit circle if you rotate an

53:13

angle of alpha around that's what this

53:15

first number is the second number

53:17

tells us what you would get if you

53:19

rotate an angle beta around so let's say

53:21

there's some other angle which I might

53:24

write down here the whole angle from the

53:25

x-axis would be how should I write it

53:28

I'll write it as beta that doesn't

53:31

really connote that it's the fall angle

53:33

just thing I'll just I'll just do it as

53:36

a separate a separate friend over here

53:41

beta so I think like when I when I see

53:44

these terms cosine alpha plus I sine

53:46

alpha I think in my head very visually

53:48

of a point on the unit circle in the

53:50

complex plane similarly I think of a

53:52

point on the unit circle in the complex

53:53

plane and I know that multiplying them

53:56

together means that the action of one of

53:58

them is rotating the other so the final

54:00

result we know is going to be the point

54:04

that sits alpha plus beta around alpha

54:06

plus beta plus I times the sine of alpha

54:11

plus beta wonderful but on the other

54:15

hand if you go through the mechanistic

54:17

foil operation you will get an alternate

54:19

expression for the answer and by having

54:21

an alternate expression you have an

54:23

identity which i think is pretty cool

54:25

it's a sign that when you have a

54:26

non-obvious relationship like the

54:28

algebra of complex multiplication and

54:30

the geometry of complex multiplication

54:31

you can leverage that non-obvious

54:33

relationship to get yourself real

54:35

identities okay so how does this work

54:37

the real part is going to consist of

54:39

multiplying those cosines so we have

54:42

cosine alpha times cosine of beta and

54:46

then it also consists of the two

54:49

imaginary parts multiplied together

54:50

because that I squared becomes real I

54:52

times sine I'd say I won't write

54:56

anything here it'll be that sine of

54:58

alpha and then the sine of beta and

55:03

because we have an I squared we can

55:06

consider that a negative one okay so

55:08

it's cosine alpha cosine beta minus it

55:11

sort of accidentally started writing a

55:13

plus so we're just going to live with

55:14

that minus sine of alpha plus sine of

55:17

theta sine of alpha times sine of beta

55:19

okay obviously it's a lot of algebra but

55:23

it is satisfying the idea that you don't

55:25

need to memorize this and again when you

55:27

do this regularly when you do complex

55:28

multiplication regularly you can kind of

55:30

see

55:31

this pop out we're like oh yeah it's the

55:33

the real put the two real parts - the

55:35

two imaginary parts that as a pattern is

55:38

something you start to recognize and

55:39

that when you want the imaginary part of

55:41

the product that you take the other two

55:43

like in this case it's gonna be I times

55:46

sine times cosine so the sine of alpha

55:49

times the cosine of beta plus the cosine

55:55

of alpha times the sine of theta the

55:57

cosine of alpha times the sine of beta

56:00

long complicated formula all of that

56:04

times I this long complicated formula is

56:07

giving you the angle on some identities

56:08

and sometimes you can write this out in

56:10

a much more compact form this this is

56:13

not a commonly used notation and I'll

56:15

explain why in a moment but sometimes

56:17

cosine plus I sine of something comes

56:21

about commonly enough it has this

56:22

meaning on the unit circle that it's

56:24

written as C is okay in this case C is

56:27

of alpha so everything that we just

56:29

wrote down could be written more

56:31

compactly as CIS of alpha times CIS of

56:36

beta is equal to C is of alpha plus beta

56:42

okay this is communicating the very

56:45

non-trivial fact that complex

56:48

multiplication lets us you know lets us

56:51

work out rotations in the way that we

56:52

want to again that's not standard

56:55

notation and I'll explain why there's a

56:57

much more standard way to write these

56:58

but it's weird so I don't want to just

57:00

kind of throw it down there but before

57:02

we do that I want to illustrate how you

57:04

can do a little bit of magic when you

57:06

have this proper intuition for complex

57:08

numbers and and how they multiply so let

57:11

me pull up our quiz again and let's move

57:13

on to another question and this question

57:19

asks which of the following values of Z

57:22

satisfies Z squared equals I okay so

57:28

there's two ways that you can think

57:29

about this I'll talk about them both and

57:31

I'm just going to give you a moment to

57:33

kind of think this one through

57:47

and while you work that out let me go

57:50

ahead and take a question from the

57:51

audience

57:52

Achmed Osama asks what is an application

57:55

that is impossible to achieve without

57:57

complex numbers convince me that they

57:59

are needed they're fun to work with

58:01

maybe but they made things easier but

58:03

can we do without them I would say I

58:07

believe yeah you can absolutely work

58:09

without them because you can basically

58:12

use rotation matrices in place of

58:15

complex numbers so it's nice about

58:17

complex numbers is that when you're

58:19

adding things and when you're

58:20

multiplying things you're dealing with

58:21

the same type they're both complex

58:23

numbers but if you wanted to describe

58:25

this all with matrices and vectors and

58:27

such when you're adding stuff you'd be

58:28

adding vectors but then when you want to

58:30

do rotation type actions and stretching

58:32

type actions you'd be transferring to

58:33

instead dealing with matrices so that's

58:36

fine you just have to keep that jumbled

58:37

and use them differently I'd be willing

58:39

to bet pretty much anything you can do

58:40

with complex numbers you could do there

58:42

it'll become much less elegant and I

58:46

guess one of the one of the main things

58:48

that would be just really awkward is the

58:51

fundamental theorem of algebra which

58:52

we'll talk about in a moment but it

58:54

basically says anytime you have a

58:55

polynomial of degree n so the highest

58:57

term you see is something like X to the

58:59

N among the complex numbers it has n

59:02

different solutions so that's very

59:04

elegant it's very nice to work with it

59:06

definitely makes the lives easier of

59:08

anyone who's working with things where

59:10

polynomials come about to include

59:12

matrices and things like I don't want to

59:14

like throw around terminology but when

59:15

you're computing eigenvalues and such

59:17

you find yourself with polynomials often

59:19

of high degree and it's nice to have

59:21

solutions to them I can conceive of

59:23

constructing some kind of way to work

59:26

with the numbers where you have like

59:27

solutions to all of these where the

59:29

solutions are expressed in terms of

59:30

vectors and matrices and so maybe the

59:34

strict answer to your question is you

59:35

could do without them you definitely

59:37

would not want to life for

59:38

mathematicians and quantum physicists

59:41

and really anyone who works with like

59:42

linear algebra and polynomials would

59:44

just become incredibly awkward so in in

59:47

some very real sense they are a natural

59:49

construct so much so that even if you

59:52

could do without them conceivably you

59:55

never would want to it would really make

59:57

things quite the pain

59:58

so coming back to our quiz here it looks

60:01

like again we've got a lot of strong

60:03

consensus which is good and let me go

60:06

ahead and create this alright so the

60:09

correct answer is a it's root 2 over 2

60:13

plus root 2 over 2 I it looks like the

60:16

majority of you got that and ok so very

60:19

interesting is that the second-most

60:21

common answer was D that no solution

60:23

exists without extending beyond the

60:25

complex numbers because the question

60:27

we're asking in this context is

60:28

basically to take a square root of I so

60:30

if you think of oh I is defined to be

60:32

the square root of negative 1 I think a

60:34

common question is to say you know how

60:36

deep does this rabbit hole go okay if we

60:38

had to make up an answer to the question

60:39

square root of negative 1 do we have to

60:41

make up another one for the square root

60:42

of I and the square root of that and

60:44

just kind of keep extending and

60:45

extending and it's interesting and

60:47

surprising that once you have I you have

60:49

pretty much everything else that you

60:51

want at least when it comes to solving

60:53

anything that looks like a polynomial ok

60:55

so the answer here there's two ways to

60:57

think about it one of them is the brute

60:59

force way and then one of them is the

61:00

more elegant way so the brute force way

61:02

looks like this you're reading through

61:04

the multiple choices and you just kind

61:05

of try them out and in this case when

61:07

you try out the square root of 2 over 2

61:11

plus square root of 2 over 2 times I and

61:15

you square it so we're going to multiply

61:17

it by itself I'm going to actually do

61:19

this just to illustrate that it's

61:21

possible and that it's not super fun

61:25

taking root 2 times root 2 for that

61:28

first real part is going to give us 2

61:30

divided by 2 times 2 so 2 over 4 the

61:36

real part will also include the products

61:38

of the imaginary parts because of that I

61:40

squared so we're going to have root 2

61:42

over 2 times root 2 over 2 which like we

61:44

just saw well excuse me that was

61:47

supposed to be 2 over 4 you can tell

61:50

them a little all over the place today

61:51

which you know just because sometimes

61:54

complex numbers get me excited root 2

61:56

over 2 times root 2 over 2 is going to

61:58

be another 2 over 4 but this time it's

62:01

times I times I so we're going to have a

62:03

negative in there yeah that's going to

62:06

be the full real part and then the

62:09

imaginary part is going to come from the

62:11

cross terms

62:12

something where there was only one I

62:13

included so to over to I times root 2

62:16

over 2 it's the same product we're doing

62:18

over and over it's always 2 over 4 and

62:21

then we're gonna add 2 root 2 over 2

62:24

times root 2 over 2 I so we're gonna add

62:25

the same thing and I'm not writing the

62:28

eyes here because I'm gonna factor it

62:30

out okay so the real part involves

62:32

subtracting these the imaginary part

62:34

involves adding them and indeed when we

62:36

subtract them that real part is 0 and

62:38

when we add them that imaginary part is

62:41

1 so we just get 0 plus I or I and okay

62:45

it works but imagine that you looked at

62:49

this question and it wasn't multiple

62:50

choice right someone asks you to find a

62:52

square root of I if the only option you

62:54

have is this terrible brute forcing it

62:57

seems like a very painful process right

62:59

to try every number try squaring it if

63:01

this is what you have to do but in an

63:03

insane trial and error it would seem

63:04

like magic if someone just pulled that

63:06

answer out of a hat

63:07

but if you're thinking about it

63:08

geometrically it's actually a pretty

63:11

straightforward question to ask because

63:13

what you're saying is we want a number

63:15

such that Z squared is sitting where I

63:19

is right it's sitting one unit above

63:20

well if we know the rotation corresponds

63:23

to complex multiplication and that means

63:26

what we want is a number that rotates by

63:28

45 degrees because if you have a number

63:32

that rotates by 45 degrees then when you

63:35

multiply by it twice that's the same as

63:37

rotating 90 degrees so it gets you up to

63:40

I all right well what number is that we

63:44

could think of and in terms of sines and

63:46

cosines if we want you know this is

63:48

going to be cosine of 45 degrees this

63:51

vertical part is going to be the sine of

63:53

45 degrees and it's root 2 over 2 for

63:58

each of them this is something we were

64:00

working out just by looking at the the

64:02

isosceles right triangle each one of

64:05

these is equal to root 2 over 2 so

64:10

that's very fun simply by thinking of

64:12

the geometry you could pull out of your

64:14

hat the fact that the square root of I

64:16

where I should say a square root of I is

64:18

root 2 over 2 plus root 2 over 2 I

64:21

another answer would have been what

64:23

happens when you rotate not 45 degrees

64:25

but if you were to rotate 40 45 plus

64:29

another 180 degrees okay so 45 plus 180

64:34

actually gets us 225 degrees all the way

64:36

around that's another number where if

64:38

you multiply by it twice it'll have the

64:40

effect of rotating you a grand total of

64:44

360 plus 90 degrees it'll kind of rotate

64:47

you all the way around and then an extra

64:49

90 so in the same way that most positive

64:52

numbers have two square roots you know

64:55

square root of 25 is 5 but it's also

64:57

negative 5 and in the same way that even

65:00

negative numbers like negative one has

65:02

two square roots not only is it I but

65:05

negative I also satisfies this property

65:07

that when you square it you get negative

65:09

one even i itself has two square roots

65:11

and in general everything except for

65:13

zero has two square roots zero you might

65:16

consider to have like a double root but

65:18

again that's sort of a story for another

65:19

time so that's pretty magical and now

65:22

let me show you another bit of magic

65:24

okay something where again you can come

65:26

at a question that seems like it's going

65:27

to involve just insane trial and error

65:29

but if you're thinking about the

65:31

geometry of it you're able to come to a

65:33

pretty a pretty meaningful answer

65:35

relatively quickly alright what's your

65:40

question here the equation X cubed

65:43

equals one has one real number solution

65:46

x equals one makes sense you keep one

65:49

you get one among the complex numbers

65:51

there are two more solutions which of

65:54

the following is one of them okay

65:56

so this time it's actually asking us to

65:58

cube some kind of complex number or to

66:01

verify that when you cube it you're

66:03

going to get one we'll give you some

66:07

some time to sort of think about that

66:08

and in the meantime let me take a sip of

66:14

water and then take the question from

66:16

the audience

66:26

what do we have you showed that in order

66:30

to find the cosine of a sum of two

66:32

angles you can multiply two complex

66:33

numbers does that mean that if I need

66:36

the cosine of a subtraction of two

66:38

angles I need to divide two complex

66:39

numbers wonderful question

66:41

yes yes it does mean that and dividing

66:44

by a complex number is actually the same

66:46

as multiplying by one in the same way

66:48

that dividing by two is the same as

66:51

multiplying by 1/2

66:52

you basically say 2 has the action of

66:54

stretching the number line by 2 1/2 as a

66:58

multiplier has the action of squishing

67:00

everything down by a factor of 2 that's

67:02

kind of the inverse action when it comes

67:04

to complex numbers 2 divided by it you

67:06

ask what's the inverse action so an

67:08

example here might be let's say let's

67:12

pull up something here let's say I

67:14

wanted to rotate you know negative alpha

67:19

that's what I want to do so I'm going to

67:21

do like cosine of beta minus alpha or

67:24

something like that that number could be

67:29

considered the inverse the

67:31

multiplicative inverse of what you get

67:33

by rotating alpha the other way so up

67:35

here we have I'm just going to write it

67:37

as CIS

67:38

4 cosine plus I times the sine of alpha

67:41

the inverse number over here is CIS of

67:44

negative alpha and that number is 1

67:48

divided by this I could even write it

67:50

out over here to say C is the number

67:54

that's alpha the way a long and unit

67:56

circle is equal to 1 divided by CIS of

68:00

negative alpha and vice versa if I took

68:03

one divided by CIS of alpha this is

68:07

equal to C is of negative alpha now

68:11

everyone who knows Euler's formula is

68:12

kind of screaming at me that there's an

68:14

easy way to see this in terms of

68:16

Exponential's which is of course where

68:17

we're getting with this but because it's

68:19

weird this connection between complex

68:21

numbers and Exponential's I don't want

68:22

to just throw that down right away I do

68:24

want to emphasize that this is this

68:25

should all make sense before anything

68:27

like Oilers formula enters the picture

68:29

if you think of CIS as being what

68:31

basically a function to ask where do you

68:34

end up if you walk

68:35

alpha units around the unit circle in

68:38

the complex plane

68:39

it's the Opera it's the number that you

68:41

can use for for rotating things by alpha

68:44

degrees so dividing by it will also get

68:46

you rotating the other way but the

68:49

overall answer there is the division is

68:51

the same as multiplication so that's

68:52

absolutely the right instinct you can

68:54

derive for yourself the angle what would

68:57

you call them the trigonometric

68:58

difference formulas if you wanted to

69:01

great so I think that's probably enough

69:03

time if we punch back to our quiz and

69:06

let's tone down the pause and ponder

69:07

music here it seems like answers have

69:10

sort of stopped rolling in so this is as

69:12

good a time as any to grade things so

69:16

the correct answer here of the four

69:19

options that we were given ends up being

69:21

drumroll please negative one half plus

69:25

the root three over two times I now I

69:27

just want you to imagine for a moment

69:29

that if you didn't know any of the

69:31

geometry of complex numbers someone asks

69:33

you this question and if you were to

69:34

able to spit it back decently quickly it

69:36

makes you look like a machine

69:38

right because you would imagine that the

69:41

only way to answer this is to look at

69:43

each number and then cube it that you

69:45

have to go through and do the whole

69:46

first inside/outside last but twice

69:48

because you're cubing it for each one of

69:50

those and then deduce that answer B

69:53

turns out to have this property that we

69:54

want geometrically there's a very nice

69:57

way to see it so let's go ahead and pull

69:58

it up what we want is an action so that

70:06

when you do it three times you end up

70:08

back at the back at one basically I want

70:11

if I were to multiply by this number

70:13

three times I end up back at once let me

70:15

clear for myself nice new piece of paper

70:18

here so just remind ourselves what we're

70:23

looking for which is X cubed is equal to

70:27

one okay so one such action is the

70:30

do-nothing action right if you multiply

70:32

by one which does nothing ya do that

70:34

three times you end up back at one that

70:36

is our real solution the other ones

70:38

would be to rotate a third of the way

70:40

around the circle okay to rotate 120

70:44

degrees or you could rotate 120 degrees

70:48

the

70:49

which is equivalent to thinking of

70:51

rotating 240 degrees around those are

70:54

other actions where if you do them three

70:55

times it's the same as doing nothing if

70:57

I rotate 120 degrees then do that three

71:00

times I cube it that's the same as doing

71:02

nothing that's how you can read this

71:04

equation which I think is very nice and

71:05

elegant rather than just trying to plug

71:07

in numbers and see what comes out so

71:09

using the trig that we learned about

71:11

last time this again comes from a

71:13

30-60-90 triangle you can see that the

71:15

real part is negative one half and then

71:18

the imaginary part is the square root of

71:20

three over two so even before working it

71:22

out I can know with confidence that if I

71:25

were to go through the painstaking

71:26

process of taking negative one half plus

71:29

the square root of three over two and

71:31

cubing that I'm going to get back one I

71:34

don't have to go through that process to

71:36

know that that's going to happen which

71:37

is pretty magical this also answers for

71:41

us a mystery that we had in the last

71:42

lecture if you guys remember what we did

71:45

is we pulled up decimals so let's go on

71:48

back let's go on over to Chrome let's

71:51

open up a desmos tab and let's get rid

71:53

of whatever else was going on in here

71:55

and what we played with was the idea

71:58

that if you take a cosine graph and then

72:00

if you square it the shapes look

72:02

remarkably similar you have this bizarre

72:04

relationship we're squaring cosine looks

72:07

like squishing it and more specifically

72:09

we were able to play around and say if

72:11

you double the frequency of that cosine

72:13

wave and then you manipulate it in some

72:15

other ways we're going to shift it up

72:16

and then we divide it by two this is all

72:18

something we did last time we got an

72:20

identity between them but this raised an

72:22

interesting question what on earth does

72:24

cosine of 2x have to do with cosine of x

72:27

squared well now with complex numbers we

72:30

can actually know the answer we can work

72:31

this out we know that if you rotate

72:34

around by theta degrees okay if we call

72:40

that number Z then when we rotate around

72:42

another theta degrees this new number is

72:46

Z squared right and Z is what I was

72:50

calling CIS of theta which we could

72:53

expand out as saying cosine of theta

72:56

plus I sine of theta

73:00

and we can work out that square nearly

73:04

algebraically we take cosine of theta

73:06

plus I sine of theta the real part of

73:12

that is going to come from two products

73:14

of the real part so it's going to be

73:15

cosine squared of theta but then also

73:20

two parts of the imaginary and because

73:22

of that I it becomes negative one so

73:25

we'll end up with negative sine squared

73:27

of theta and then the imaginary part is

73:31

going to come from yeah I might as well

73:32

just write this out in full cosine theta

73:37

plus I sine I realize I'm kind of

73:41

running against the edge which is why I

73:42

was hesitant to do this but basically

73:44

that cosine squared from comes from

73:46

taking the cosine times the cosine the

73:48

negative sine squared comes from taking

73:50

I sine times negative I sine our

73:54

remaining terms are going to be that

73:56

cosine plus I sine so cosine theta times

74:02

sine of theta and then there's going to

74:05

be an I I'll just write it out times I

74:08

and then we get exactly the same thing

74:11

sine of theta times cosine of theta

74:12

times I so sine of theta times cosine of

74:16

theta times I which we could factor out

74:22

this whole last part ends up just being

74:25

two times the cosine of theta times the

74:30

sine of theta all right all of that

74:35

times I so what we've done here is

74:37

basically said there's two ways to think

74:40

about cosine theta plus I sine theta

74:41

squared one of them is oh of course it

74:44

doubles the angle and so that's going to

74:46

be cosine of two theta on the other hand

74:48

algebraically it's equal to deep breath

74:50

cosine squared theta minus sine squared

74:52

theta plus two cosine theta sine theta I

74:55

now that might seem like a big pain but

74:58

the big pain is showing us an identity

75:00

it's showing us the double angle

75:02

identity so what we found here just to

75:04

summarize on the real part is that the

75:07

cosine of two theta the real part of

75:11

what happens when we rotate by that

75:13

is equal to cosine squared of theta

75:16

minus sine squared of theta now from

75:22

another trig identity that we were

75:24

working with last time sine squared of

75:27

theta is actually the same as taking 1

75:31

minus cosine squared this was the

75:35

alternate way of expressing the

75:36

Pythagorean theorem so we can get

75:38

everything written in terms of cosines

75:40

and what we have is now 2 times cosine

75:45

pardon the messiness of the handwriting

75:48

here by the way 2 times cosine of theta

75:49

all minus one that is evidently equal to

75:54

the cosine of 2 theta and we could

75:57

rearrange that further and see what we

75:59

were playing around with in desmos that

76:01

this cosine of 2x expression over here

76:06

this cosine of 2x expression corresponds

76:08

with squaring the cosine this was a

76:10

reflection of the fact the cosine

76:12

relates two complex numbers which i

76:14

think is pretty cool yeah and is the

76:18

very last thing that I want to mention I

76:19

want to talk about why we didn't why

76:23

nobody really uses the CIS notation and

76:26

I was saying that CIS of alpha is not

76:30

really standard notation when we're

76:32

talking about rotating alpha around the

76:34

unit circle as a reminder this was my

76:37

shorthand for cosine plus I sine the

76:41

reason is because what you're always

76:43

going to see instead through electrical

76:46

engineering through quantum mechanics

76:47

through math through all sorts of things

76:49

is instead the way that this is written

76:51

is actually before I show you how it's

76:54

written no no I'll just I'll just show

76:56

you how it's written it's it's it's

76:58

weird and we're going to talk about it

76:59

at length the next time but you imagine

77:02

that you're raising something to a power

77:04

okay e to the I times alpha now this

77:08

obviously makes a lot of people very

77:09

uncomfortable first of all if you don't

77:11

know about e what is e and then secondly

77:13

what on earth does it mean to raise

77:16

something to an imaginary power so that

77:18

is a full lectures worth of material I

77:19

do want to talk about it next time but

77:22

just to to give an indication of why

77:24

this might not be a totally crazy thing

77:26

okay I want you to remember the core

77:29

property that we wrote down earlier that

77:31

when you rotate alpha around the unit

77:33

circle and then when you rotate another

77:36

beta around the unit circle this is the

77:39

same thing as alpha plus beta which

77:43

means our CIS function has this kind of

77:45

interesting property that adding the

77:47

inputs is the same as multiplying the

77:48

outputs so with that I want to ask one

77:51

final quiz question to try to relate

77:54

this to other functions that you might

77:55

be familiar with at this point it might

77:57

be halfway obvious where we're going

77:59

with this but is the very final question

78:01

to cue you up for where we're going to

78:03

go next time and maybe appreciate the

78:05

relationship at play here and why it's

78:06

not that crazy I want you to answer

78:09

which of the following functions

78:11

satisfies F of a plus B is equal to F of

78:15

a times F of B okay

78:32

I'm being told by the way that um when I

78:36

advanced the questions too quickly it

78:37

kind of reveals the answer of the next

78:39

one which happened all throughout last

78:41

lecture and you guys certainly told me

78:42

about it again this is one of those

78:44

things we're like this is what happens

78:46

with live production we thought we had

78:47

fixed that usually you know this is a

78:50

very weird process for me because

78:51

usually when I make videos you catch

78:53

some kind of error so you edit out the

78:54

error then you put out the video this is

78:56

an experience for me is much more like

78:57

alright world here it is raw here are

79:00

the errors as they come I hope you enjoy

79:02

watching them happen because there's no

79:04

editing it does not get clipped out this

79:06

is math presentation raw all right so as

79:10

we do this it looks like pretty pretty

79:13

strong consensus I'll give you a little

79:16

bit more time to submit the answers all

79:28

right and with that let's go ahead and

79:31

take a look so the question is what

79:33

function has this property we're adding

79:34

the inputs looks like multiplying by the

79:36

outputs so 2700 of you correctly

79:40

answered the two to the X an exponential

79:42

function behaves this way looks like the

79:44

most common second answer was a that

79:48

logarithms which makes sense there's an

79:50

identity in logarithms that looks very

79:52

similar it's basically the opposite so

79:54

if we were to you know write this out I

79:57

might say 2 to the a times 2 to the B is

80:02

equal to 2 to the a plus B and when

80:06

we're thinking in terms of repeated

80:07

multiplication this makes a lot of sense

80:09

because one of these is saying multiply

80:11

in a copies of two then multiply in B

80:14

copies of 2 and the result should be all

80:16

a plus B copies of two the rule with

80:19

logarithms is exactly the opposite and

80:22

it's because logarithms are the inverse

80:24

of multiplication on that front

80:26

it looks like taking the log of a plus B

80:29

ends up being the log of a sorry the log

80:34

of a times B multiplied log of a plus

80:38

log of B

80:39

logarithms is another thing I think

80:41

we're going to talk about in a later

80:42

lecture of this series so if you don't

80:44

know about them don't worry but to those

80:46

of you who answered that totally

80:47

understandable it's because it's a very

80:49

similar property but it's just getting

80:51

mixed up whether the outputs are being

80:52

added or the inputs are now you'll

80:55

notice is how strange this factor that

80:58

comes about because 2 to the X is

81:01

defined as repeated multiplication looks

81:03

very similar to the rule that we found

81:05

for complex numbers that sit on the unit

81:08

circle that multiplying the outputs is

81:10

the same as adding the inputs and that

81:12

should be very suggestive that there

81:13

might be some kind of unifying theory

81:15

between them which of course there is it

81:17

has everything to do with why we write

81:19

things as e to the I theta which I

81:22

actually think is kind of a bad

81:23

convention the good thing about it is

81:26

that it does make it pretty obvious when

81:28

you take e to the I theta times e to the

81:30

I beta sorry e to the I alpha that

81:34

you're allowed to just add them it

81:35

really highlights that fact which

81:37

otherwise you kind of have to think

81:39

through but maybe that's a good thing

81:41

that you kind of have to think it

81:42

through and you're not just blindly

81:43

applying rules but I'm gonna talk all

81:46

about that next time we're gonna dig

81:47

deep into Oilers formula a way that you

81:50

can try to understand what is this

81:52

really saying like computationally it's

81:54

not talking about repeated

81:55

multiplication so what could you do if

81:57

you are on a desert island with a bunch

81:59

of paper to just work out by hand the

82:01

fact that this is a reasonable

82:02

expression all of that is to come next

82:05

time so thank you for joining apologies

82:08

for being mildly scattered throughout