Classical to Quantum

Classical to Quantum | Special Relativity and Complex Numbers (blue) | Wild Egg Maths

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The true nature of Einstein's Special Relativity is much clarified by understanding an algebraic structure that plays a largely hidden but fundamental role in mathematics: the red complex number algebra. However to understand that, it is best to first appreciate the usual (blue) complex numbers in a highly innovative way which is more in keeping with physicists' understanding of spacetime, or as we prefer to call it, timespace: since for us the time axis comes before the space axis.

In this video we give a concise introduction to this familiar object; but in a mostly novel way that may be a bit disorienting, but leads to important insights.

This is part of the Classical to Quantum series of videos that will attempt to give an elementary mathematical viewpoint on lots of fundamental modern physics.

A big thank you to all my Patreon supporters, and to Members of the Wild Egg maths channel.

0:01

hello everybody I'm Norman walburger

0:03

thanks for joining me on this adventure

0:05

we're looking at special relativity from

0:06

a new point of view today complex

0:08

numbers for physicists essentially

0:10

giving us a algebraic framework in which

0:13

we can think about the physics of

0:15

special relativity in a novel fashion as

0:18

we'll see so I am going to be presenting

0:20

a very different point of view as to

0:22

what actually special relativity is

0:24

really all about okay very different

0:27

point of view for them the standard

0:28

story and today is something of an

0:31

introduction to that a gentle

0:33

introduction familiar territory of

0:35

course with complex numbers but I'm

0:36

going to present it in a very novel way

0:38

so there'll be a lot of interesting

0:40

novel things even for physicists who are

0:42

very familiar with this thing so this is

0:45

you should think about it as being part

0:46

of the story of Chromo geometry which is

0:49

a three-fold symmetry in planar geometry

0:51

which I've developed

0:53

there's sort of three colors blue red

0:55

and green blue is euclidean red green

0:58

are relativistic and they're going to be

1:00

playing a very major role but the story

1:02

starts with the blue version which I'm

1:05

going to be talking about today and it's

1:06

just ordinary

1:08

complex number Theory but presented in a

1:11

very rational finite computationally

1:13

explicit way

1:15

okay so what is a complex number for us

1:17

it's a pair of rational numbers T comma

1:21

X

1:22

and we'll have basis vectors called one

1:25

and I so this can be thought of as T

1:27

times one plus x times I let's just

1:30

forget that one and just write it as t

1:31

plus x i and what is this thing well you

1:35

can think about this as just being a

1:37

short form for this two by two Matrix t

1:39

x minus x t Okay and then the

1:44

um the space of these things is a

1:46

two-dimensional Subspace of the space of

1:48

two by two matrices actually it's a sub

1:51

algebra okay and then the the relations

1:54

are just immediately uh taken from the

1:57

relations of two by two matrices so we

2:01

have addition of complex numbers in the

2:02

usual way scalar multiplication and

2:04

here's the multiplication uh in these

2:07

coordinates so it's the usual

2:08

multiplication of complex numbers t x

2:11

times r y is T times R minus x times y

2:15

comma T times y plus x times R okay so

2:20

now let's introduce complex conjugation

2:22

and so if Z is TX the complex conjugate

2:26

which we will write as Z Star okay

2:29

more in line with what's called star

2:31

algebra's Z star is T minus X and then

2:35

the crucial property that the complex

2:37

conjugate of product Z times W is the

2:40

product of the complex conjugates

2:43

and when we multiply Z times Z star

2:46

well then it's the difference of squares

2:48

we get t squared plus x squared times

2:50

one and zero I component we'll just

2:52

write it like this t squared plus x

2:54

squared and that by definition is the

2:55

quadrants of Zed okay so in this

2:59

rational version we don't want to be

3:01

taking square roots

3:02

except in an approximate fashion but

3:04

theoretically it's much better to stick

3:07

with the quadrants that's the

3:08

fundamental metrical quantity being

3:11

asserted here and here's the basic

3:13

relation this is probably the most

3:14

single important

3:16

property of complex numbers that the

3:18

quadrants of the product Z times W is

3:21

the quadrants of Z times the quadrants

3:23

of w and it's a nice exercise just to

3:25

see that that's a consequence of this

3:26

and the definition and another place

3:29

where the quadrants is important is in

3:30

terms of defining the inverse of a

3:33

complex number Z inverse is one over Q Z

3:36

times Z star of course provided that the

3:39

quadrants is not zero which over the

3:42

rationals occurs precisely when we're

3:44

just having the zero complex number zero

3:48

comma zero

3:49

but I just want to alert you to the fact

3:52

that this is very algebraic and in

3:54

principle we could do this over other

3:55

fields okay we could do this over finite

3:58

fields in particular where it gets a

4:00

little bit more interesting because then

4:02

this condition that this is you know

4:04

only zero when we're talking about the

4:06

zero zero Vector that's not necessarily

4:09

the case and so things get a little bit

4:10

more interesting there but we'll stick

4:12

with rational numbers because we're

4:14

talking to physicists okay so now let's

4:16

talk about how we visualize complex

4:18

numbers we're going to introduce time

4:21

space

4:22

not space time time space because time

4:25

comes first space comes second

4:30

and it's a different picture from what

4:33

you're used to so you're going to have

4:34

to just change gears a little bit but I

4:37

really encourage you to think about this

4:39

as an alternative because it makes

4:40

understanding what happens in special

4:42

relativity so much easier okay this

4:45

naturally syncs with the usual story for

4:48

special relativity so we have two x's

4:51

there's the t-axis that's our first axis

4:53

and then here's the x-axis that's our

4:55

second axis okay the point one is up

4:58

there the point I is over there

5:02

okay a general complex number say Z

5:05

equals T come X has a t component over

5:08

there an X component over there and this

5:10

component really is a multiple of I so

5:13

properly we should put x times I so

5:16

that's also t plus x i what else do we

5:20

see here well here's the complex

5:21

conjugate it's the reflection in here uh

5:24

shouldn't it should be over here

5:26

somewhere shouldn't it yes sorry

5:28

um

5:30

I don't know what it's doing there so

5:32

let me just move that

5:34

okay

5:36

pretend that's not there

5:39

okay

5:45

okay there is

5:47

the complex conjugate

5:49

Z Star what else do we see the quadrants

5:53

Q is t squared plus x squared you can

5:57

think about that as the quadrants

5:58

between the origin

6:00

denoted by o and this point here and in

6:03

rational trigonometry we like to denote

6:05

that with a little rectangle just

6:07

alerting us to the fact that we're not

6:09

talking about distance which is

6:10

generally an irrational quantity so we

6:12

prefer not to think about that the

6:14

quadrants is the metrically natural

6:16

quantity as you physicists already all

6:19

know so um that's how we indicate that

6:23

um there's a one more very interesting

6:25

thing to uh introduce here and that's

6:27

the slope or the velocity of this

6:31

complex numbers Z equals TX I'm using

6:34

velocity here because we're thinking

6:37

about it in terms of physics also so

6:39

where is that on this diagram well

6:42

here this line here is the line when T

6:45

equals

6:47

one so we might put that there that's T

6:49

equals one and where this line here

6:53

through Z meets it okay that's going to

6:57

be a point whose coordinates are

7:01

well the first coordinate is one and

7:03

let's call the second coordinate V

7:06

so what is this V well

7:09

1 to V is going to be the same as T to X

7:13

so the slope V of Z is X over t

7:19

and this corresponds to the usual

7:22

definition of slope in terms of Delta y

7:24

over Delta X okay Y is the second

7:26

coordinate X is the first coordinate

7:28

usually in this context X is our second

7:31

coordinate T is our first coordinate so

7:33

it's the second coordinate over the

7:34

first coordinate and naturally for T not

7:37

equal to zero

7:40

okay so yeah it's a little bit different

7:44

but um

7:46

it's really good to think about this

7:49

version of complex numbers especially if

7:52

you're interested in physics and special

7:54

relativity so here's a little

7:56

proposition that the velocity of a

7:59

product Z times W is

8:02

velocity of Z Plus velocity of w over 1

8:05

minus velocity of Z times velocity of w

8:09

where

8:11

we introduce also a new notation for

8:15

this particular combination we'll call

8:17

this the circle sum of V of z and v of w

8:22

and we'll denote that Circle sum with

8:24

this little sum with a circle around it

8:28

okay so let's prove this so suppose Z is

8:31

the complex number T comma x w is

8:35

R comma y then V of Z Plus V of w over 1

8:40

minus V of z v of w is what the velocity

8:44

of Z is X over T the velocity of w is y

8:47

over R so we have that sum on the top

8:49

and one minus the product on the bottom

8:51

we multiply through by T times R we get

8:56

x times R plus T times y up in the

8:59

numerator when we multiply through here

9:01

we get TR minus x y and that is the

9:05

velocity of Z times W because here is z

9:07

times w

9:09

okay T times R minus x times y t times y

9:12

Plus x times R so the velocity is this

9:16

second component

9:17

which is the same as that divided by the

9:19

first component which is the same as

9:21

that

9:23

so this circle sum plays a an important

9:26

role okay let's talk about the very

9:28

important rational parameterization of

9:30

this unit circle the unit circle is

9:33

those complex numbers whose quadrants is

9:36

one because of that basic relation Q of

9:39

z w equals Q of Z times Q of w this is

9:42

closed under multiplication and it's

9:45

sort of obviously also closed under

9:47

inverses and so this is actually a group

9:49

okay and

9:53

it's really important in this context to

9:55

parametrize this unit circle

9:58

now most physicists will automatically

10:01

think in terms of cosine Theta sine

10:04

Theta in terms of an angle

10:07

but if you learn some rational

10:08

trigonometry you realize this is not

10:10

generally the optimal thing to do there

10:13

are some places where it's quite

10:15

important especially if you talk about

10:17

harmonic motion or uniform uniform

10:20

motion around a circle in that case

10:23

Theta is playing the role of time and

10:25

that's quite natural but if you're just

10:27

interested in the geometry of the circle

10:29

and in this case sort of the geometry

10:31

and algebra of complex numbers that's a

10:34

red herring and it's better to avoid

10:36

angles okay so I'm going to give you a

10:39

little bit of something which I call

10:41

Vector trig which I'll have to talk

10:42

about at length somewhere else so it's a

10:45

it's a an alternative to rational

10:47

trigonometry which is a little bit less

10:49

rational okay but it's a little bit more

10:51

focused to engineering and physics

10:53

applications

10:55

so uh this rational parameterization of

10:57

the circle which goes back you know in

10:59

some sense basically to to Greek times

11:01

is crucially important and I've talked

11:04

about it in many places on my channel I

11:07

talk about it a lot because it's so

11:09

important that people understand this so

11:13

um let's go over it again there's

11:15

actually a rather Rich story here which

11:18

is not really properly appreciated by by

11:21

most students of mathematics okay so

11:24

here's our unit circle there's one

11:26

there's I

11:28

I want to parametrize this thing here

11:30

okay so one way of doing this is to take

11:33

this tangent line that we've already

11:35

seen the the line where T equals one and

11:39

choose a point on it let's call it one

11:42

comma h

11:43

okay and let's take this complex number

11:46

W and consider it's Square so the square

11:50

in this case will be something like this

11:52

W Squared it'll be the form 1 minus H

11:55

squared 2H you just square that thing

11:57

okay and then what we look at is this

12:01

thing here which is this W squared

12:03

divided by the quadrants of w

12:06

okay so um

12:09

we're not able to divide by a length

12:12

because our length is not a rational

12:14

quantity so taking a square root of a

12:17

quadrant is something we prefer not to

12:19

do and so in this theoretical

12:21

development we avoid that so we get

12:23

around that by working out with W so

12:25

it's not as if we take W and we we

12:27

normalize it by dividing by its length

12:29

no what we do is we take W Squared and

12:32

we divide by its quadrants because the

12:34

quadrants is rational

12:36

okay so then you get something that's

12:38

actually on the unit circle so W squared

12:40

over quadrants of w the quadrants of w

12:42

is obviously one plus h squared so you

12:44

get this combination 1 minus H squared

12:46

over 1 plus h squared 2H over 1 plus h

12:48

squared let's call that e of H

12:51

so

12:52

I've written this down probably dozens

12:54

of times okay it is so important to

12:57

understand this parameterization

13:00

now there's a different way of thinking

13:04

about this another way of thinking about

13:05

this is to take this point here one

13:07

comma H and go directly down onto the

13:09

x-axis then you're going to get the

13:11

point H times I and another way of

13:14

getting this point is to start at the

13:16

point -1 and take a line through it and

13:20

this point h i and see where it meets

13:22

the circle

13:24

it's a line that passes through the

13:26

circle at one point so the other

13:28

intersection is guaranteed there's no

13:30

square roots that need to be invoked in

13:33

order to get that second point of

13:34

intersection

13:35

and that line there turns out to be

13:38

parallel to this one and another way of

13:40

thinking about what we're doing is we're

13:42

we're taking this line here and we're

13:44

basically reflecting one in it okay

13:47

that's another way of getting at this

13:49

point so it's quite actually a rich

13:51

story this this point here is connected

13:54

in in different ways to things okay it's

13:56

connected by being a reflection in this

13:58

thing here it's also um

14:01

on this line here which has the same

14:04

same direction as this thing here

14:08

once we have Zedge that's now a point on

14:10

the unit circle then well we know what

14:13

its quadrants is it's quadrants is one

14:15

but its velocity will be X over T so

14:19

it'll be 2H over 1 minus H squared

14:22

okay so we're getting the velocity there

14:24

of that thing in a rational fashion and

14:28

what we get this way is that every Point

14:30

Z on the unit circle except for Z equals

14:32

minus one is of the form Z equals e of H

14:35

so as this point here sort of wanders up

14:37

and down

14:39

or across this line T equals one or

14:42

conversely if this point h i marches

14:44

back and forth on X we're going to be

14:46

covering the unit circle and the only

14:49

point that we don't cover that way is

14:52

minus one itself which you can sort of

14:53

think of as being associated with this

14:55

point going to Infinity if you like okay

14:58

so there's a correspondence here between

15:00

the

15:01

point on the unit circle called Z and

15:04

this quantity H and I find it convenient

15:07

to use the term half slope for this so

15:10

we'll call H the half slope of Zed

15:14

that's uh sort of in keeping with what's

15:18

going on there

15:20

okay so the rational parameterization of

15:23

our blue unit circle okay this rational

15:27

parameterization of the circle that

15:29

we're talking about is really a

15:31

classical construction called

15:34

stereographic projection that's another

15:35

way of thinking about it what we're

15:37

thinking here is that we have this line

15:39

here and we have this point on the

15:41

circle and we're projecting this line

15:44

onto the circle just by Rays emanating

15:48

from that point there okay so this gives

15:51

us a nice way of thinking about this the

15:54

circle we've already said that the

15:55

circle is a group so the multiplication

15:58

of complex numbers is closed under this

16:01

circle so it's natural to ask well how

16:04

does that look geometrically

16:06

now if you're a classical physicist you

16:09

automatically want to think about angles

16:10

and you know you're adding angles but

16:13

please try to move away from that

16:16

because that's not a rational

16:19

construction the angle associated with

16:22

typical lines is always almost always

16:25

transcendental it's not something that

16:27

you can get at exactly okay we want to

16:30

stick with the mathematics where we can

16:31

write everything down exactly and for

16:34

that purpose this rational

16:35

parameterization is far superior so we

16:38

get a formula which is a little bit more

16:40

complicated than just being a sum of

16:42

angles but the advantages is that

16:44

everything is rational so we can write

16:45

everything down completely and precisely

16:47

without approximation

16:49

okay it's really important because also

16:51

when we move to special relativity

16:53

exactly the same things apply okay

16:56

all right so let's say we have two

16:58

points on this circle say U1 and U2 okay

17:02

so what is the product U1 U2 well it

17:05

turns out to be down here and

17:09

we can uh think about that in in at

17:12

least two different ways okay they're

17:14

both very nice one is in terms of the

17:17

rational parameterization so here are

17:19

the half slopes corresponding to U1 and

17:21

U2 there's H1 there and H2 there right

17:23

so we take these lines here's a minus

17:26

one we join that up there and that's H1

17:28

times I and here's H2 times I

17:32

so here's a nice exercise that if you

17:34

take the the points U1 and U2 which are

17:37

the E of the H1 and the E of the H2 and

17:40

you multiply them as complex numbers

17:42

you're going to get e of H1 Circle some

17:45

H2 again

17:46

that same formula that appeared as the

17:50

the way of combining velocities or

17:53

finding the velocity of a product so

17:55

velocity at half slope are a little bit

17:58

very similar except one sort of

18:01

not exactly twice but the half slope is

18:04

sort of half of this of the velocity

18:07

in some sense okay so H1 Circle sum H2

18:11

that is I remind you H1 plus H2 over 1

18:14

minus H1 H2 so it's a very pleasant

18:16

exercise to verify that this is the way

18:19

you multiply on the unit circle in these

18:24

rational coordinates okay

18:26

Now France lememeyer uh

18:30

in modern times has realized a very

18:33

profound and and beautiful visual way of

18:37

understanding this

18:39

I've had correspondence with him so he's

18:41

attributed this understanding actually

18:43

to 19th century people I forget exactly

18:45

who but I'm going to associate it with

18:48

him because he's the the person that

18:50

taught me about this and also sort of

18:52

has brought this into the four in modern

18:54

times

18:55

so the idea is that we join up U1 and U2

18:59

and here's the one which is the identity

19:01

in this group okay so what we're going

19:02

to do is simply take the line through

19:05

one which is parallel to the line

19:07

through U1 and U2

19:09

that's another line through the

19:12

um through this point on the unit circle

19:15

so the other point of intersection is

19:17

well defined it's rational and that's U1

19:20

U2

19:22

so in this picture here uh here will be

19:25

H1 Circle sum H2 and you can see in this

19:28

case here if you take H1 maybe it's like

19:31

0.3 or something at H2 maybe there's one

19:34

point three or something and you you add

19:37

them okay you can get something out here

19:39

and then you have to divide by one minus

19:41

H1 H2 well the project is going to be

19:43

positive

19:44

um and when you take one minus that

19:46

thing you get something less than one so

19:48

you're dividing by something less than

19:49

one you're getting something bigger so

19:50

that's why instead of going here you're

19:52

getting out out there okay

19:54

and while we're doing exercises

19:57

um it's really interesting to think

19:59

about generalizing this important

20:00

formula so let's say you have three half

20:03

slopes H1 h2h3 and you take the complex

20:06

numbers the unit complex numbers

20:08

Associated to each of them so e of H1 e

20:10

of h two e of H3 so the exercises to

20:13

show that that's e of H1 Circle sum H2

20:15

Circle some H3 which has the following

20:18

form it's a quotient on the numerator we

20:21

have H1 plus H2 plus H3 like a linear

20:23

term minus this cubic term H1 h2h3 in

20:28

the denominator we have one minus an

20:31

expanded quadratic sort of symmetric

20:33

quadratic expression H1 H2 minus H1 H3

20:36

minus h2h3

20:39

okay that's very nice and actually it

20:41

leads to a very nice generalization

20:43

which you might like to explore and see

20:46

if you can prove it

20:48

again this is also this is all great

20:50

stuff because when we go to special

20:53

relativity when we go to the red and

20:55

green versions of all this then

20:57

everything kind of just translates over

21:00

in a very pleasant way so our

21:02

understanding that we're developing here

21:04

naturally just slots into our

21:07

understanding of the algebra required to

21:10

understand special relativity

21:12

now I want to just share with you some

21:15

tantalizing looking aheads at some

21:18

relativistic kinematics kind of stuff

21:20

but in this curious sort of euclidean

21:23

setting just still in the realm of

21:26

ordinary complex numbers

21:28

so

21:30

um the starting point here is to think

21:32

about this two-dimensional space the way

21:34

a physicist would in terms of time space

21:37

so we really think about this as say

21:40

being a Time axis and this is being a

21:42

space axis so we're thinking of a

21:43

particle moving along this

21:45

one-dimensional space

21:47

and this line here okay which is you

21:50

know at this slope here it's

21:52

recording where this particle is at

21:54

various times so you know here the maybe

21:57

the particle is here at some point okay

21:59

and uh you know where is it there well

22:02

maybe it's at this time here so in that

22:04

case this point here corresponds to this

22:06

particle being here at this particular

22:08

time and this world line here is telling

22:10

us that this particle is moving

22:12

uniformly in this positive direction

22:14

okay along the x-axis

22:17

so associated with such a particle maybe

22:21

it has some kind of size or or mass or

22:25

some kind of thing that tells us how big

22:28

it is in some sense

22:30

in that case we can try to do some

22:33

kinematics think about energy or

22:35

momentum Etc and so let's just introduce

22:38

rather informally and Loosely some

22:42

Concepts

22:43

that you know figuring special

22:45

relativity in this sort of simple

22:47

context so let's imagine that the

22:49

momentum of this thing is given by a

22:52

particular Vector okay and the vector is

22:55

you know along this direction of the

22:58

world line so we choose some point on on

23:00

this thing let's say that one there okay

23:03

that's p

23:04

it's a point on this world and so it

23:07

determines a vector you could think of

23:09

the vector emanating from the origin to

23:11

P okay and the bigger that Vector is

23:14

it's always going to be in this

23:15

direction but the bigger the vector is

23:16

somehow the more massive or the bigger

23:19

this particle is meant to be in some

23:21

sense in some sense which maybe we are

23:23

not very clear about now

23:26

let's denote this point by a capital P

23:30

okay and suppose that it has t and x

23:34

coordinates capital E and little p

23:37

so that this is the point e and this is

23:40

the point Little P

23:43

times I

23:45

so

23:46

um

23:48

then if we take the the quadrants of P

23:52

okay which is like P dot p in terms of

23:55

geometry okay it's the square of the

23:58

first component plus the square of the

24:01

second component so e squared plus P

24:03

Squared that's going to be the quadrants

24:04

between here and here okay and if the

24:08

circle here that I've drawn which is

24:09

just a dilate of the unit circle if that

24:13

is passing through this point M on the T

24:16

axis let's say then this quadrants is

24:20

the same as that quadrants which is

24:21

going to be M squared

24:23

okay so there's some going to be a

24:25

relation here between this e coordinate

24:28

this P or Pi coordinate and the M that

24:32

defines the size of the circle on which

24:35

P lives

24:38

okay now there's a few other

24:41

points of interest or vectors of

24:43

Interest so if here is that familiar T

24:46

equals one line then where this world

24:50

line meets that line okay that's going

24:53

to be a point of the form 1 comma V and

24:56

this V we recognize as the velocity of

24:59

this this world line essentially okay so

25:02

that's good notation and we could call

25:04

that point say capital V

25:07

just that's a little bit at odds with

25:09

what physicists do but it's I think more

25:13

natural in this setting um okay the

25:16

other point that's of interest is when

25:18

we think about intersecting this a line

25:21

with the unit circle

25:23

now

25:25

from a rational point of view this line

25:27

does not necessarily have to exactly

25:30

meet the unit circle okay so there's an

25:33

assumption here when I draw that point

25:35

here there's a bit of an assumption

25:37

um that we were you know that there's

25:40

actually a point on here so if there is

25:42

that point let's call it U

25:44

okay so that's going to be a point on

25:46

the unit circle which is in the

25:47

direction of p in that case the

25:50

relationship between U and P is the same

25:52

as the relationship between 1 and M so a

25:54

dilation of a factor of M will take us

25:56

from U to P so p is equal to M times U

25:59

and another way of um saying that

26:03

probably I should put quotes here

26:05

because I'm invoking a square root here

26:07

but I'm doing the same kind of thing

26:08

that physicists like to do okay so

26:10

another way of saying that is this

26:12

Vector U is a normalization of the

26:14

vector v okay if we take the vector v

26:16

and we divide by its length length in

26:19

quotes then we get the vector U so

26:21

there's the vector v and the length of

26:23

this thing here will be the quadrants

26:25

which is 1 plus v squared square rooted

26:27

square rooted in quotes that's why

26:29

there's a square root of 1 plus v

26:31

squared here

26:32

okay so that's just another way of

26:34

thinking about what this momentum Vector

26:36

that's what a physicist would call it a

26:38

momentum Vector would be so in

26:41

coordinates it's e p but now in terms of

26:44

uh its connection with with U or V it's

26:47

M over square root one plus v squared 1

26:50

comma V

26:51

so from this we get two equations

26:53

because we can equate the E coordinates

26:55

so e is equal to M over 1 square root 1

26:59

plus b squared and if we use a sort of

27:02

simple calculus binomial expansion of

27:05

this this is like you know 1 plus v

27:07

squared to the minus a half then we

27:09

could get um M times uh like there's

27:12

always be one minus uh one half V

27:15

squared so we get M minus one half MV

27:18

squared plus some other stuff

27:21

and then maybe you could recognize this

27:24

and say well okay that's uh this is like

27:26

a

27:27

some kind of you know this is the energy

27:29

maybe maybe this is like a mass this is

27:31

maybe like a kinetic energy rest Mass

27:33

kinetic energy

27:35

and the other equation P equals m v over

27:38

square root one plus v squared this also

27:41

is like a familiar to physicist

27:44

um it's like momentum equals mass times

27:47

velocity this is the velocity okay of P

27:51

um but it's not really exactly the mass

27:53

because there's sort of this factor of

27:55

square root one plus v squared on the

27:57

bottom there so

27:59

um sometimes physicists call this

28:01

something like this relativistic Mass

28:03

but of course physicists don't think

28:05

about

28:06

special relativity in in the context of

28:09

complex numbers like this they think

28:11

about a corresponding story in in the

28:14

the red version of this but what I want

28:17

to emphasize to you is that what's

28:19

happening there in that special

28:20

relativity story has a very natural

28:22

analog over here that we can study just

28:24

in the context of you know high school

28:28

algebra with ordinary complex numbers

28:32

okay so

28:34

um I think that's pretty interesting and

28:36

there's lots of interesting directions

28:38

here as we look now towards the

28:40

relativistic versions of this algebraic

28:43

story

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