0:00

This is the BBC. This

0:03

podcast is supported by advertising outside

0:06

the UK. Last

0:08

episode of the current run.

0:10

On the script, I've written the word in

0:12

tour because every

0:14

time I try to write the word intro, it

0:16

comes out as in talks. That's it. Yeah. Do

0:18

you have words that you just always

0:21

type out wrong.

0:22

Oh, yeah.

0:23

What's your because you're in soldier. Well,

0:26

how do you type it? Oh, god. Every time

0:28

I'm

0:28

like, I'm just s. And then

0:30

I basically don't know what's next. I

0:33

did. So my

0:35

thesis back in the day was on development

0:37

of the retina. Mhmm. So

0:39

that required writing the word retina a

0:41

few times. Mhmm. Not a single time

0:44

in the last twenty five years, have I written the

0:46

word retina without first writing Retian?

0:48

Really? I've I've put a I've

0:51

put an auto

0:51

correction. Oh, you know what actually? So I did

0:53

have one in in my thesis, but it was

0:56

viscous. My thesis was all about viscous

0:58

fluids. A lot of them were viscous.

1:01

Like, it have really serious implications. Is

1:03

is it considered a vicious fluid? Mm-mm. Okay.

1:05

Yeah. Alright then. Anyway, That's

1:07

nothing to do with the program that you're about

1:10

to hear. The final episode in this

1:12

this run, which is one of those

1:14

ones, Johannes, extraordinarily

1:18

overexcited. What I will say

1:20

is that I I just asked a producer to

1:23

cut out some of my squeals. She

1:26

did. I hope he doesn't. I hope he doesn't. You're

1:28

about to find me and sift feeders.

1:35

After our major masks nerd out

1:37

in the pie episode from last series,

1:40

loads of

1:41

you. Rotin to ask for some

1:43

more maths. Did they loan? Yes.

1:45

They did, Adam.

1:46

Yes. They did. This is what the listeners

1:48

want. And you know what? I know you're gonna enjoy this

1:50

by the end. Will I you enjoy the pie

1:52

episode by the end? No. I did. You did. And

1:55

today, we're venturing into

1:57

extra weirdness because, you know,

1:59

pie ultimately, it's just a number that's

2:01

slightly bigger than three. So see. It's

2:04

it's it's interesting, but it's not that weird

2:06

one 4. Quite easy to grasp. Yes.

2:08

Today, I don't know. We are gonna go for

2:10

stuff that are much stranger much

2:12

more difficult to conceptualize and

2:14

yet numbers that are indispensable

2:17

in maths, engineering and physics. Today,

2:20

Adam, we're talking about

2:22

imaginary numbers. No fancy. Really?

2:26

Yes, we are. Thirty minutes,

2:29

buckle up, strap in. Okay.

2:31

Wait wait. Let me you know the

2:33

number

2:33

nine. Right? What's the square root of nine? Three.

2:36

Correct. What's the square root of minus nine?

2:40

Minus three. Right. Well, it can't

2:42

be. The square root of of

2:44

minus nine can't be minus three

2:46

because, yeah, minus three times minus three

2:48

is nine. I have an answer

2:49

for you. Mhmm. The square root of minus

2:51

nine is there isn't one.

2:53

You've well, you that is what people

2:55

thought for a long time. Sensible people. Sensible people.

2:58

But people also thought that you couldn't divide

3:01

one number over the other. People thought that there wasn't a

3:03

zero for a really long time. And so

3:05

mathematicians don't like the idea of having

3:07

a question without an answer. And

3:09

it turns out that there is an answer

3:11

to what is the square root of minus nine, but

3:13

you can only do it if you know

3:15

what the square root of minus

3:18

one

3:18

is, and that my friend is the topic of staying.

3:20

Right. Okay. Well, listen to Peter

3:23

Scott did ask in an email should

3:25

you have a program on the square root of minus one?

3:27

Well, Peter, I'm going with

3:29

no, but it may be that I've already lost this

3:31

back. In indeed, you have because the answer

3:33

is yes. And we've got two fellow math

3:35

nerds to help us navigate through this weird

3:37

landscape. We've got doctor Michael Brooks,

3:39

author of the mass that made us. And

3:41

doctor Ellen Noxa 4 loss for a physics from

3:43

Kings College London. Currently, a visiting

3:46

fellow at the University of Pittsburgh, and you

3:48

last heard her on the Pie

3:50

program. Mhmm. Okay. Well, listen.

3:53

Michael will start with you. We've

3:55

established that the square root of minus one

3:57

is a

3:59

it's an impossible concept. Who

4:02

first came up with this not so idea

4:04

and what were they up to? So

4:06

it's a guy called Jerome Cardano who was

4:09

working in Italy in the sixteenth century.

4:11

And he was solving some cubic equations like

4:13

he do. And he discovered that

4:15

basically, in the middle of his calculations, he

4:17

had a root square root of minus fifteen to

4:20

deal with. And it sort of stopped him

4:22

in his tracks, and he said, that's not right.

4:24

You know, you can't do that. He knew that you can't

4:26

have the square root of a negative number. And

4:28

he sort of said this seems like it's an

4:30

impossible case was what he wrote down. So he

4:32

said, you know, this is arithmetic subtlety

4:34

or there's something odd about all of this. But

4:37

he went on to kind of, you know, work out

4:39

a sort of side step. He basically carried on with

4:41

the calculation and got to an answer that worked in the

4:43

end, so it was fine. But he did at least

4:45

acknowledge that this was actually

4:47

a real thing that was really there in the mess,

4:49

which previous mathematicians like

4:51

Herron of Alexandria, had said,

4:54

oh, I've just got something wrong and scrubbed out the

4:56

negative. Hey, which is I I would've gone

4:58

with Karen of I am wondering if

5:01

I hear anything about him him

5:03

a

5:03

bird? Is that a hero?

5:04

He was he was a man. Okay. That was man.

5:06

Not an actual hero. Thank you. That's useful information.

5:09

So Cadauma then say he doesn't

5:12

ignore this. He he recognizes that

5:14

it's a real thing, but also acknowledges that

5:16

it's a it's a slightly quirky

5:18

concept. What happens next? He basically says too

5:20

hard for him to deal with. So he he

5:22

kind of talks about it, writes a bit about

5:24

it, and and then sort of moves on,

5:26

and he's he's working with his student

5:29

Ludovico Ferrari to kind

5:30

of, you know, advance the frontiers of maths.

5:33

And and and so he just doesn't wanna

5:35

get distracted, so he just leaves You

5:36

know how you said that? Just doing some

5:38

cubic equations. So so it's like a standard thing

5:40

that you do on a Wednesday afternoon. Right? I mean, I know

5:42

that not everyone has an exciting and life as

5:45

I do.

5:46

But the around this time, the Renaissance,

5:48

there's a lot of this going on, isn't it? Yeah. Because

5:51

these things are really important. So if you can

5:53

solve in your first of all quadratic equation,

5:55

so x squared, and then the cubic

5:57

with x cubed, x to the power of

5:59

three. And then the quartic x to

6:01

the 4. If you can solve these things, you work

6:03

out ways to solve them, you can make a lot of money

6:05

from it. First of all, because all the financiers

6:07

want to employ you because it helps them to calculate

6:10

good rates of interest. So as today,

6:12

you know, the mathematicians do got hoovered up

6:14

by all the bankers and finance

6:16

institutions. But

6:17

also can of course. No colorators.

6:19

Not even algebraic notation. So everything's

6:22

written out in words, literally like

6:24

words describing, you know, what we would see as

6:26

an equation. And also, you know,

6:28

you can get university teaching posts by

6:30

solving equations that other mathematicians

6:32

can't solve. So if you know the solution,

6:35

to a pubic equation, you can challenge somebody

6:37

who's got job you want, challenge them to a mathematical

6:40

duel in the street, where you set each

6:42

other thirty problems. And

6:44

people watch you try and solve them.

6:46

And and you lose or win a job

6:48

on on the basis of

6:49

this. Whoa. Whoa. Whoa. Did this happened. It's

6:51

just really a real phenomenon. Yeah.

6:53

So Cardano nearly got tangled

6:55

up in one, except that he so there was

6:57

a guy called Nicholas Cartaglia who wanted to

6:59

challenge him. Because he said that Caldano had

7:02

published a solution that Cartaglia owned.

7:04

So and it was wild. So so he

7:07

he got very upset and challenged Caldano to

7:09

a duel after lots of nasty letters. Mastool.

7:11

Mastool. And and and Carvana

7:13

said, I'm not doing that. But but

7:16

the his student Ferrari said, I will I'll

7:18

do it and took him on. And because Ferrari

7:20

had worked out the solutions to the quadric equations,

7:23

and Totalia hadn't, Ferrari

7:25

just saved thirty with thirty

7:27

questions that were

7:28

just, like, quartet, but he knew how to solve

7:30

and the other guy didn't. And and totally

7:32

didn't even turn up. I'm deeply in favor

7:34

of bringing this back. I think next Prime Minister.

7:37

Right. What we should do is get them in Downing

7:38

Street, give them Blackboards each, and just set them

7:40

off and go.

7:41

Wow. It'd be amazing. It

7:44

seems to be wonderful. This was

7:45

like a public spectacle.

7:46

Right. They also used to hang

7:48

people as public spectacle. I don't know how

7:51

well And that wasn't

7:51

maybe much entertainment as well. I don't know whether that's

7:53

in the plus column or the negative column. But

7:55

but then this this idea, when

7:58

does it become imaginary rather than

8:00

impossible. So Cardano

8:02

doesn't do anything with it. And then couple

8:04

of decades later, Rafael Bombardier

8:06

sort of works out the maths of doing in your

8:08

complex numbers with imaginary parts and

8:10

real parts. But then nobody really does much with

8:12

it for really a couple of centuries. So

8:14

then dayparts turns up, calls them imaginary

8:17

in kind of very derogatory

8:19

kind of

8:19

way. Like, you know, what's the point of this? So okay.

8:22

Descartes essentially just throwing some

8:24

eighteenth century shades -- Yeah. -- on

8:26

this, which is fine. But the thing is

8:28

is that you shouldn't get sidetracked and

8:30

or should you by by thinking of them

8:32

as imaginary, even though they

8:35

are technically different from real numbers.

8:37

Should we is it do you think that we can we

8:39

can load Adam's brain with the

8:41

difference between the two?

8:43

So real numbers are the ones that

8:45

we're usually pretty familiar start off with the integers

8:48

and you start off counting and then you learn about

8:50

fractions and eventually you have a long line.

8:52

Right? All of the numbers that you're familiar with

8:55

and eleven. In from school. Eleven

8:57

is a real

8:58

number. Four

8:58

two point

8:59

three four is a real number.

9:00

It's gonna take us really long time

9:02

if we go through of them. Twenty

9:05

three, that's not a real number, is it?

9:07

Look, I mean, zero is also a real number

9:10

and minus ten is a real number

9:11

two. So they're not quite just the numbers that you've

9:13

used to count, you know, blocks in your kids'

9:15

toy box. Oh, alright. How should

9:17

we think of this? I don't want

9:19

to say impossible. I say, perfectly

9:22

perfectly fine number.

9:24

Well, I mean, you could think of, you know,

9:26

eye is just something that we've made up that's terribly

9:28

terribly useful

9:29

Right? You need some solutions to some equations.

9:31

You wanted to know what the square root of a negative

9:33

number was, and we just kind of made up

9:35

this number and popped it

9:36

in. And all of a sudden, a lot of things become a lot

9:39

easier, and that's sometimes how maths works.

9:42

You know, just making you can't make up numbers.

9:45

Can't just make up numbers to make your equations work?

9:47

Can't. There are numbers you can't make

9:49

them up. Zero. Yeah. Where is

9:51

it?

9:51

Didn't It's didn't that no things. Can

9:53

you see it? Didn't that no. Can

9:56

you hold it? Is this a song? I

10:00

It should be. I'm gonna release it as a b side.

10:03

But it just you know, these mathematical

10:06

convenience is because your equations don't work.

10:08

Yeah. Yeah. That's why they're

10:09

imaginary. It's not imaginary.

10:11

No. But it gets weird because, you know, they turn

10:13

out to be useful. So you have this thing that you

10:15

make up solve an equation that it turns out to lots of

10:17

other things too.

10:18

Alright. Okay. I feel like

10:20

I I need a recap just for my own purposes here.

10:23

Stu I, which is

10:26

the imaginary number that

10:27

we're taking minus one. The square root of minus

10:29

one, which is invented, doesn't exist,

10:31

is impossible to conceptualize, but it does

10:33

follow certain rules. Mhmm. And

10:37

we know that it came about from people like

10:39

Cudano and the Heron guy who

10:41

just ignored it to solve

10:43

some pointless equations. And

10:47

it and it and at

10:50

some point in history, it switched from being an number

10:52

two, an imaginary number, which

10:54

is my home assistant

10:55

voice. Thank thank thank

10:56

you very much. I don't know. With that said,

10:58

I'm on top of this. What a

10:59

wonderful summary. We appreciate it. But

11:01

you know what okay, all of this stuff

11:03

about being pointers, this number actually turns

11:05

out to be wildly wildly wildly useful

11:08

because here is professor Jeff O'Connell

11:10

from Aloni College in California to

11:12

tell us why. Being a teacher of mathematics,

11:16

I've always found it interesting that

11:18

we teach imaginary numbers

11:21

to algebra students, but

11:23

there's never a there's never

11:25

an application. And it isn't until

11:27

you get in to talking about things

11:29

like differential equations and physics

11:32

where imaginary numbers become

11:34

this fantastic tool that

11:36

we use in order to solve problems.

11:38

And it isn't that the

11:40

problem starts off with imaginary numbers

11:43

or even ends with imaginary numbers

11:46

but all of the tools that we use in

11:48

the middle, imaginary numbers are

11:50

very much a part of that. For

11:52

example, when you are modeling

11:55

maybe the suspension of a car.

11:57

That is what we call a spring mass system.

12:00

When you're driving, you go over a bump and

12:02

the car oscillates a little bit

12:05

to kind of absorb the shock of that

12:07

bump and give the people in the car a

12:09

smooth ride. So when we

12:11

create the equations to model

12:13

that behavior, the equation

12:16

doesn't have any imaginary numbers in it,

12:19

and then the answer doesn't

12:21

have any imaginary numbers in it. But

12:23

in the middle getting from the equation to

12:26

the answer many times

12:28

we have to use imaginary numbers in

12:30

order to get to that answer.

12:32

Adam's pulling a face. And

12:35

then I helped me unpack that a little bit though because

12:37

Jeff was talking about using I, using

12:39

in margarine numbers, in modeling

12:41

things that that oscillates. So, you know, bouncing

12:43

springs or

12:44

pendulums. Tell us why it's useful

12:46

in oscillations. So it

12:48

turns out that you can use these numbers

12:51

to think about anything that's got to do with angles

12:53

and circles and periodic and to give

12:55

you a little bit of an idea about how that might work.

12:58

I need you to go back and imagine our real

13:00

number line again. So all the

13:02

way going down to the negative numbers, up into positive

13:04

numbers, lay your normal numbers out on

13:06

the line. Now, stick

13:08

another axis on through the zero

13:11

going vertically this time. So we're

13:13

gonna turn this into a pair of axes. And

13:15

the vertical axis is going to go

13:17

all the way up into positive multiples y,

13:20

and all the way down is negative multiples y.

13:23

So once we've got our two axes,

13:25

our real axis and our imaginary axis,

13:28

I want you to think of every single

13:30

point on that plane on those axes

13:33

as representing a number. And now those

13:35

numbers, we're gonna call them complex numbers.

13:37

Adam may like this title better than imaginary or

13:40

impossible. Complex numbers are basically

13:42

a mix of a real number and

13:44

an imaginary

13:45

number. Yeah. So okay. can

13:47

deal with that. So on the anything

13:49

which is an interaction between the real numbers and the

13:51

imaginary numbers is called

13:52

complex number. Exactly. Right. So three plus

13:54

2i or eleven minus exight.

13:56

It's

13:57

sort of the coordinates on that plane where

13:59

you find yourself.

14:01

Now I want you to imagine putting four big red dots

14:03

on plus one minus one plus I and minus

14:05

I and drawing big circle all the way

14:07

around your axis. It's

14:09

going to turn out that

14:11

that circle which has radius

14:13

one, has a very special relationship

14:16

to the angle. And the the points

14:18

on that circle have a special relationship to the angle,

14:20

and that's closely related to

14:22

the nature of these complex numbers. And

14:25

once you start to understand that, you get

14:27

the opportunity to use these

14:29

complex numbers, these things with a real bit and an

14:31

imaginary bit, to model

14:34

anything that has to do with angles and

14:36

circles, and it turns out that springs

14:39

and waves, and all

14:41

of the beautiful things in partial differential

14:43

equations in physics are a bit like

14:45

that. So you get this tool that is just step

14:47

tackley's for many, many hundreds of years

14:49

after people try and solve these funny equations

14:52

and invent this number

14:53

I. Okay. Here's the thing, Adam. I know

14:55

I know I can read your expression of work

14:58

with you long enough. know exactly what's going

15:00

on in your head. I know you're not happy.

15:02

But here is the thing. You know how

15:04

mathematicians describe kind

15:07

of moving around doing mathematics like your,

15:09

you know, very thick thick thick it. And

15:11

you can't see where you are and it's all like

15:13

in unusual. And then there's a moment

15:15

where you turn a corner, and in

15:18

front of you, you see this beautifully

15:20

landscaped garden. And everything

15:22

is absolutely perfect. And

15:25

you can see completely where you've been

15:27

and this new perspective shows you how

15:29

all of these things are completely connected. What

15:32

Eleanor has just described there is

15:34

circles, it's imaginary numbers,

15:37

it's real numbers, it's triangles, it's

15:39

angles, It's exponential, so it's just

15:41

hiding in there. You can't quite see it. It's pie is

15:43

in there. All of them together form

15:46

this unbelievably the

15:48

eutiform equation, which is

15:50

e to the I pie plus

15:52

one equals

15:54

zero. And

15:56

You're just gonna have to believe me. A

15:59

few weeks ago, we did a program about a

16:01

phantasia and the inability for some

16:03

people to be able to imagine things in

16:06

their head. All I

16:08

can think of now is the garden you

16:10

just described -- Yeah. -- everything else

16:12

you just said sounded like this.

16:17

Okay. So you're in the garden. Right? You're in the

16:19

garden. Mhmm. Oiler is standing in this

16:21

garden. And she's looking

16:23

around. Is the heron there? The heron. It's

16:26

I don't think the heron was

16:27

invited to the heron's in my car. Heron

16:29

disinvited the herons back in the thicket.

16:32

What disinvites himself by ignoring the important

16:34

clues. Mhmm. But, you know, he's standing

16:36

in this garden and he looks around and he's

16:38

got pie in one corner.

16:40

Right? Love that number. Great number. He's got

16:43

imaginary numbers I in another corner. He's

16:45

got zero. He's got one, and he's got exponential,

16:48

EDX merchandising number. And he looks around and

16:50

he's like, holy moly, these are

16:52

the five most beautiful

16:55

broadenly numbers in mathematics. And there

16:57

is one equation, which links them all

16:59

together.

17:00

It's ridiculous, isn't it? It

17:02

is ridiculous. It's so about

17:04

this game. I'm I'm feeling less confused because

17:07

this garden is quite a peaceful place. It's a beautiful

17:09

garden. There's only five things in it. You've

17:11

all said something interesting which is and

17:13

and and the the clip from Jeff has also said

17:15

that that having

17:17

to use imaginary numbers in order to solve

17:20

equations which have real world applications

17:23

you that the input are

17:25

real numbers and the output

17:27

are real numbers, obviously, because the imaginary

17:30

numbers aren't real. But in

17:32

the middle, you're using imaginary numbers --

17:34

Yeah. -- in order to get from see, it's

17:36

like going from London to birmingham

17:38

via a wormhole. Kind of.

17:42

Yeah. And that it takes you into another dimension.

17:44

Mhmm. You know, which is what Elena was talking about.

17:46

That that axis that goes away from

17:48

the the normal number line is Imagine

17:51

you're just going into another

17:52

dimension, but you come back again in a very

17:54

useful place.

17:55

You know what, honestly, your wormhole idea is actually

17:57

is pretty good. It sort of is, isn't

17:59

it in there?

18:00

Well, you need imaginary numbers to model wormhole.

18:02

That's for sure.

18:06

Come on. I know he's doing so well there.

18:09

Okay. But here's the thing. We've talked

18:11

about this sort of theoretical connection,

18:13

this this wormhole that you get to go into.

18:16

But the thing is is that there

18:18

might That makes it sound like it's a math

18:20

trick, but those weird

18:22

properties that imaginary numbers

18:24

have don't just turn out to be

18:27

useful for nice fancy mass

18:29

gardens. Because you can actually

18:31

also you can also use imaginary

18:34

numbers

18:35

in order to make quite a lot of money.

18:38

Right. Buying imaginary

18:41

money

18:41

is that's this is not an unproblematic

18:44

concept. Mhmm. How does this work? It's

18:46

real money. You can make money, but you just

18:48

use imaginary numbers. And this is the entire

18:50

basis of the twenty twentieth century

18:52

electronics industry. Comes

18:55

from actually a a guy called Charles

18:58

Proteus Steinmetz. Proteus was his nickname

19:00

because when he was growing up, he was so clever.

19:03

That all his mates at school thought that

19:05

if they yeah. I just touched him, like, yeah, he used

19:07

to touch the Greek God Proteus. He

19:09

would impart wisdom to them. So they they said he was

19:11

just so amazing. So he grew up in

19:13

Prussia. He came over to America in

19:17

eighteen eighty nine, somewhere around

19:19

then. And he came

19:21

right at the time when everyone was trying to work out how

19:23

to electrify America, so how to build

19:26

all the, you know, the electrical infrastructure like

19:28

generators, you know, the the electrifying

19:30

houses and cities and how you how you do all

19:32

of this. And there was a big debate going

19:34

on between Edison and Tesla

19:37

at the time. About whether it should

19:39

be alternating current a c

19:41

or direct current d c. Edison

19:43

was d c. Tesla was a

19:45

c. And there were sort of

19:47

problems with both in some ways. But

19:50

what what they couldn't do with

19:52

a c, which was really difficult, was

19:54

modeling how circuits would behave.

19:57

So the mathematical model of a circuit,

19:59

you know, going from the generator all the way through

20:01

to your light switch to your light bulb was actually

20:03

really difficult with AC because

20:05

alternating current varies all the time,

20:08

which means you have to build in a sort of time

20:10

variation into your equations.

20:12

And then you add in components like capacitors

20:15

and inductors, and they add a phase shift

20:17

to all those waves. So it gets really messy. And

20:19

stymets came in and

20:21

said, oh, it's easy. You just use complex numbers.

20:24

So what has got excited about that? You you

20:26

you did get really excited there for who Tesla

20:29

was AC. Tesla was AC. Edison

20:31

was DC. Yeah. And what was Angus Young's

20:33

position? Sorry.

20:35

Can't you be much on this conversation to be better than

20:37

ours? The thing is though, I mean,

20:40

alternating current, positive,

20:42

negative, positive, negative,

20:43

positive, negative

20:44

-- Yeah. -- it's that oscillation and

20:46

it's you think about something spinning around in

20:48

a circle, if above the

20:49

line, you are positive and below the line, you're

20:51

negative, it's it's the same thing. Yeah.

20:53

It's the same story. Same thing. It's it's sort of

20:55

you look at it like that, you think, why didn't anybody

20:57

see this before? But Steinmetz came in and

21:00

just said, oh, you know, I can solve all of these problems.

21:02

Gave them the complex maths that

21:04

that would do it. All of a sudden, all the electrical

21:06

engineers were like, oh, we can do this. And

21:09

AC just won, like, won the day immediately

21:11

because all of a sudden, you could use really easy

21:13

equations to model AC. And AC

21:16

has the advantage that you can transmit

21:18

it from the generator to where you're using it

21:20

without much loss or with much less loss than

21:22

with

21:23

DC. So Edison was sort of out

21:25

at that point.

21:25

And and imagining numbers are absolutely

21:27

crucial to working out the the the modeling

21:29

of a c for this.

21:30

Absolutely crucial if you want something you can

21:32

actually manage.

21:34

Well, I'm I'm slightly persuaded by that

21:36

that argument.

21:37

The daypack are the

21:38

ones who are working. Amazing thing

21:40

is that so you go from there, and it's like, oh, we

21:43

just electrified America really easily basically

21:45

using, you know, Tesla stuff all his

21:47

hardware and Steinmetz is

21:50

brilliant math. So you get the

21:52

birth of radio, you get everything sort of taking

21:54

off at that point. And then people are doing electrical

21:56

engineering degrees. And you've got this guy

21:58

called Bill Hewitt who does a master's

22:00

degree in electrical engineering.

22:03

And he uses imaginary numbers.

22:05

And Bill Hewlett takes this to

22:07

his friend, Dave Packard, who And

22:10

hold on a minute. Exactly. David

22:12

Packard has a garage that they

22:14

can build this thing that that Hewlett

22:16

has kind of designed, which is an audio oscillator,

22:19

basically sound generator. And

22:21

and they so they start building it in David

22:23

Pecosgarage. They 4 this company called Hewlett

22:25

Pecard. They bring out their

22:27

first sort of electronics box, which they

22:29

called the HP two hundred a, so that people

22:31

didn't think that they just like, that was their first

22:34

invention. So they wanted to make it sound like they

22:36

were, you know, they've been back on the production line for ages

22:38

and ages. And then so

22:40

then when they got the HP two hundred b

22:42

going, The Walt Disney Company

22:45

bought eight of them, used them in

22:47

the first broadcasting in cinemas of

22:49

Fantasia

22:50

to recreate that amazing sort of symphony

22:53

sound.

22:53

They built on

22:54

the back of imaginary numbers. So

22:55

what what what are they actually what is the a what

22:57

was it? The HP two hundred b. HP hundred

22:59

b is basically a way of generating sound. So

23:02

Walt Disney were looking for something that would

23:04

faithfully recreate the sound of a symphony

23:06

orchestra in a cinema. So

23:08

they had to it was basically the first proper

23:10

decent sound system. What Hewlett

23:12

Packard

23:13

did? No.

23:13

What sound is Adam? No. No. No.

23:16

No. No. No. You're talking

23:18

about films. I can get on

23:20

board. I'm beginning to be persuaded by this.

23:22

Well, that's it because anything anything

23:24

that rotates or oscillates. So helicopter

23:26

blades, all of the sunset. I mean, this is

23:28

everywhere that you look, where

23:30

there's anything rotating or oscillating, and

23:33

that includes Adam. I think we're

23:35

gonna get you on this last one. That includes

23:38

things that are fundamental to our

23:40

universe. Doesn't it Illinois? Yes,

23:42

it does. So we're gonna

23:44

get to quantum mechanics -- Mhmm. -- who terrifies

23:47

everyone. But but I'm gonna give

23:49

Adam a little bit of a little bit of support

23:51

here. So I I'm gonna push back to you how

23:53

to work right here on this wave stuff. It

23:55

is mathematically extraordinary and

23:57

beautiful and edible, but all this stuff ends

23:59

up connected. And then we can describe currents

24:03

and sound waves and water

24:05

waves. Using complex numbers.

24:07

But I'm gonna make you feel better because you don't

24:09

absolutely have to use complex numbers

24:11

for any of those

24:12

things. We can do without. It's

24:15

not as nice. Thank goodness for that.

24:18

It's not as nice. We'd have struggled to electrify

24:20

America. But, you know, if

24:23

you want to do clunkier mathematics, less

24:25

beautiful mathematics, there are

24:27

ways to describe oscillations without

24:29

having to use complex

24:30

number. We can just use our old fashioned angles

24:32

and our signs and cosigns and things that we knew.

24:35

If

24:35

you don't know about beauty,

24:37

Well, that's really

24:37

interesting. A really hard time. Mhmm.

24:39

So so is the imagining numbers

24:42

are they they simplify the complexities

24:45

of of actually making calculations which pretty hard

24:47

to

24:47

do. Yeah. So there it's a mathematical convenience.

24:50

Yes.

24:51

Yes. Until we get to quantum mechanics.

24:53

I've got bad news.

24:55

So I was gonna throw you a bone, but now I'm gonna take

24:57

us back in the other direction. Wow. So

24:59

early in the twentieth century, people are puzzling

25:01

about how to model really small stuff. Atoms,

25:04

electrons, atomic

25:06

structure, etcetera. Right? This is the birth of quantum

25:08

mechanics. And what we do know by that point

25:10

is that those things behave exceedingly oddly.

25:13

And that the physics we're gonna have to use to describe

25:15

those things is going to look nothing like all

25:17

of our nice previous physics, which is

25:19

often gonna be wavy physics, for example.

25:23

And a whole bunch of people are working on this in

25:25

the mid-20s. There's

25:27

one version of it come up with Bijesenberg

25:30

called Matrix Mechanics. That

25:32

looks pretty alien to everything

25:34

4 the sis that used used to. But at

25:36

the same time, Owen Schrodinger, is

25:39

working on shoehorn and quantum mechanics

25:41

into a really comforting familiar form.

25:44

So he wants to make it look like

25:46

a wave equation. And he manages to.

25:49

He manages to write down Schrodinger's

25:51

equation, which looks like a wave equation. And

25:54

when you teach this stuff to undergraduates, right, you

25:56

kind of pull the wool over their eyes and you show them this equation,

25:58

you go, it's fine. You know how to solve wave

26:00

equations using complex numbers. This

26:03

is just the same thing. And

26:06

to some extent, it is. But schrodinger's

26:09

wave equation doesn't just to use

26:11

complex numbers to solve for

26:13

a wave. It gives you a wave

26:15

with a complex value, with an imaginary

26:18

value. 4 it's amplitude.

26:20

So my water wave, right, its amplitude

26:23

is like how far off from the middle it is, how high

26:25

my wave is. And if that's five

26:27

feet, right, five is a nice real number.

26:30

The waves that get involved in quantum mechanics,

26:33

their amplitudes are given by

26:35

multiples of I, complex numbers.

26:39

And that looks

26:41

like an application

26:42

of imaginary numbers that we can't just get

26:45

rid of. I'm

26:47

doing the face again. I mean Yeah.

26:49

I I was I got on board with the ACDC

26:52

in America. got on board with the

26:54

general rotating things, but

26:57

the amplitude of a of

26:59

a quantum state is

27:01

not whole number like the amplitude

27:04

of a or like a water wave

27:05

is, it's an imaginary number.

27:08

Yes.

27:08

I mean, Schrodinger is unhappy too. So

27:10

Well, that makes me feel much better. And

27:13

as a result, he put the cat in the box

27:15

and now the cat's dead.

27:17

Exactly. I mean, the piece in the cat is alive

27:19

and dead. Are pretty intimately related

27:21

to these these pesky complex numbers. But

27:24

it's now pretty widely accepted that in quantum

27:26

mechanics, you just have to have states

27:28

of the system that are directly described

27:30

by complex numbers. Now, of course,

27:32

that means that we don't have nice,

27:35

neat, easy ways of interpreting that

27:37

state. That's part of why we get so

27:39

puzzled by quantum mechanics. But

27:41

it doesn't look like we can change it,

27:43

essentially, Adam. These

27:45

things cannot be imaginary

27:48

only. They are in there

27:50

embedded in the fundamental state

27:53

of the universe. They are not

27:55

a mathematical convenience. They are

27:58

not just a made up answer to

28:00

an equation that no one else can solve.

28:03

They are totally and completely

28:05

integral. And they're

28:07

there in the garden forever.

28:09

Yes. 4, we're back in the garden.

28:11

We've just discovered them effectively. So

28:13

they're not impossible. They're not imaginary. They

28:15

are real things. Absolutely. You're

28:17

all nodding enthusiastically like I've

28:19

had an epiphany.

28:22

Well, you're on that note. think it's

28:24

it's time to thank our guests. Thank

28:26

you to Dr. Ellen Ninnox and

28:28

Dr. Michael for joining us and have

28:30

me persuade Adam. So,

28:34

Dr. Ruddy, when it comes to imaginary numbers,

28:36

can we say case closed?

28:38

No, professor to fry. I'm still not very

28:40

happy. That wasn't the question asked if we could

28:42

say case closed. Well, okay.

28:44

You know what? I'm gonna do it because we

28:46

can. Yes. Because imaginary numbers were it

28:48

to solve a problem and called imaginary

28:50

by Descartes as an insult, but they

28:52

are wildly useful in anything that oscillates

28:54

or rotates Is it say

28:57

it? And found in fundamental equations that describe

28:59

the universe, and therefore, are not imaginary,

29:01

but very much real. Thank you.

29:04

Imagine Mary. I

29:07

mean, yes. I am persuaded

29:10

Obviously, it's important. I did do my say levels,

29:13

and I sort of know what an imaginary number is. But the

29:15

history is really interesting. Sorry about the garden

29:17

thing. Sorry about the Heron thing. I'm really

29:19

sorry about the ACDC

29:20

joke, which I think only four percent

29:22

of the audience will get. Hang on, Adam.

29:24

Are you saying are you saying this podcast

29:26

is like a performance. Are you

29:28

saying this are you saying you're not always absolutely

29:31

completely and totally true to your real character?

29:33

I'd say that that I mean, I do you

29:36

know, we've talked about as many times before.

29:38

I do find this stuff conceptually. Glorious,

29:42

No. Fantastic. No. Beautiful.

29:44

And I can see you doing that. And

29:46

that makes me more anxious

29:50

and weirded out. And but at the

29:52

same time, I don't wanna

29:54

be you know, I value

29:56

scholarship and accent academia and and

29:58

expertise. And I don't wanna be guy going, oh,

30:01

your people are not in the real world and your stuff is

30:03

completely made

30:03

up. And at the same time, I'm looking at it going, I'm

30:05

taking a lot of what you're saying on trust.

30:08

Well, I mean, you can

30:10

if you want to, but you can

30:12

just go into the garden yourself. It

30:15

does sound like a nice garden.

30:17

Such a lovely garden.

30:18

Is that a real metaphor that has already existed?

30:20

I I think I'm not

30:22

the first to use it, but I wouldn't say it was

30:24

exactly official.

30:27

The cut this could be

30:28

You know, the thicket the overgrown thickest.

30:30

It's quite a thickest.

30:31

You did say a thick thickest moment. Yeah. Let's

30:33

go with that. It's

30:34

just been until twenty eighteen. Anyway,

30:37

thank you. enjoyed that. Yeah. I did. I really enjoyed

30:39

that. My ideas too. And I'll it's one

30:42

of those things where I'll attempt to explain

30:44

it to someone in

30:46

four days time -- Mhmm. -- and go,

30:48

well, it's because it's like, you know, there there's

30:50

a tape. There's two axes. And

30:53

if you draw it on the axes, there's a heron,

30:55

and on one, and there's a Ferrari on

30:57

the

30:57

other, and and money is

30:59

based on it and so printer. And

31:01

it's it's And there's a crowd

31:03

watching people solve equators. Exactly.

31:05

When he started when Michael started talking about

31:07

Hewlett Packard, I was I was listening,

31:10

following him like a Panther. Slightly

31:12

thinking about the job that I had at Hewlett Packard when I

31:15

seventeen, which was just soldering, so it wasn't very technical

31:17

at

31:17

all, but also thinking may

31:20

Does that explain why print has never worked?

31:24

Tell tell you what, though, actually, you're talking about numbers.

31:26

I was actually having a little look through

31:28

the numbers of downloads for our episodes.

31:31

Right? And would you like to know,

31:33

I don't know, third, what the most

31:35

popular episodes that we've ever released is

31:37

the most downloaded. Hundreds

31:40

and hundreds of thousands of people have downloaded

31:42

this and listened

31:43

to. Was it Was

31:45

it infinity? Nope. Was

31:48

it Oh,

31:51

I know. Hairy the hair like,

31:54

really in the first series when we had Atlas on and she

31:56

was really rude. Nope. Chatelier.

31:58

What?

31:58

It was your ASMR track.

32:01

Sure. Yes.

32:03

By quite a long way.

32:05

My one. Your one. What was I doing?

32:07

You you have got way more

32:09

than double. The

32:11

number that I got 4 mine, way

32:14

overdoubled up. Oh, we were making a drink. Yes.

32:16

And with but with the sound, the

32:19

special microphone, that

32:21

that picked up the Where he were in the

32:23

room? Yes.

32:24

Well, I was making an old fashioned. The cott hundreds

32:27

and hundreds of thousands of people have

32:29

downloaded that and

32:30

How weird? Why?

32:31

I don't know. Maybe you're a big star in the ASMR

32:33

world. Oh, man. There's money to be made there. Okay.

32:35

ASMR cocktails by thigh me.

32:37

Mhmm. Yeah. I'm trying to make one cocktail.

32:40

Same time. Well,

32:41

more fashion. Yeah. I think it was a mojito.

32:44

Wasn't it?

32:44

No. You made a mojito. Oh, what did she make?

32:46

Old fashioned. Mhmm. Okay. Oh,

32:48

you shouldn't have told me that because now

32:50

this is like last week when you told

32:52

me that Stephen Soderberg had been reading on book.

32:54

Got really overexcited. That's what this section of

32:56

the podcast is for Adam. It's just a circular video.

32:59

Things that I get very excited

33:01

about. Okay. So for for the hundreds

33:03

of millions of I I think that's what I heard when you said

33:05

it's the hundreds of millions supposed to be joining

33:07

us. you spreadsheet, you've allocated the fund. Gonna

33:10

do that. The people who downloaded that

33:12

noise, can I just say to them? Thank

33:18

you. Right.

33:26

Correspondence. Let's do correspondence. So

33:30

couple weeks ago, we did the episode

33:32

on magnets and

33:34

how do they work? Conclusion don't

33:36

know. Mhmm.

33:36

Mhmm. Mhmm. Well, so

33:39

one of things that I asked was we we

33:41

were talking about what is the

33:44

the thing that carries the magnetic field because

33:46

I was doing that ultra rational thing that I do

33:48

again, which is to say stop making stuff up.

33:50

There has to be a physical property here.

33:52

What, you know, what is the particle that carries a

33:54

magnetic

33:55

field. Okay? Mhmm. Mhmm.

33:57

And the particle is photons, they say, But

34:00

the thing is is that people tend to think that photons

34:02

are equal to light. Right? And a lot of

34:04

people wrote in because they were quite baffled. I

34:07

miss it. Don't Excuse

34:10

me. How do how come magnets work in the dark?

34:12

That's a perfectly reasonable question. See

34:15

where you start laughing and go. Wait a

34:17

minute. Hang on a

34:19

second. So our physicists, Felix

34:22

Fitter, got an answer for us.

34:24

Because now you need to you need to bear in mind.

34:26

Right? This is still quantum magic, but

34:28

here you go because he emailed us with this very cool

34:31

fact. He said, the short answer

34:33

is that magnets generate their

34:35

own light exclamation mark.

34:37

Actually, it'd been in Tara Bank maybe. Remember

34:41

that light is electromagnetic radiation and

34:44

not all light is visible. If

34:46

you've seen if you set a magnet spinning

34:48

so that it completes a frugal turn once

34:51

per second. The changing magnetic

34:53

field will generate a one

34:55

hertz radio wave. To

34:57

generate visible light, you would need to spin

34:59

the magnet about a quadrillion

35:02

times per second. That's ten to the fifteen

35:04

times per second. A one and fifty

35:06

eight million. Is that

35:08

a possible

35:10

thing to do? Let's

35:12

try it. Yeah. That's not

35:14

the correct answer at all, isn't it?

35:16

A reason I was asking that stupid ways because

35:18

when you say numbers like ten to the whatever it

35:20

was, could really have made up number. Then

35:22

quantum physicists go, oh, yeah. Yeah. We do that

35:24

all the time. That's what that's what these bogey

35:28

quantum things do. They

35:30

rotate at that all the time. So

35:32

I it was a genuine

35:33

question. Can we rotate things that tend to the

35:35

world level number? You said time. I don't know why you're asking

35:37

me, mate. I've got

35:39

genuinely no wonder. And this is probably

35:42

not. This is probably not. I was more

35:44

so after the Magnets episodes, and

35:47

we put it up on Twitter. And

35:51

my my friend's a children's author, Anthony

35:54

McGowan, expressed

35:58

dismay that we hadn't asked him

36:00

on the program as a magnet expert. He's

36:02

a children's author. Okay. Excellent award winning.

36:05

Children's author. So he

36:07

put up a list of magnifacts, how to take magnifacts

36:10

on Twitter. just wanna read a few of them out. Please

36:12

do. So some archaeologists believe that Stonehenge

36:14

functions as a giant electromagnet. Okay.

36:19

Are those the archaeologists who have Netflix series?

36:21

Okay. Go. Technically,

36:24

the crown owns all of the magnets in

36:26

Great Britain but not Northern Ireland. Because

36:28

at this point, I was thinking Are these real

36:30

folks? Let me keep going.

36:32

Magnets can be used in place if fire extinguishes

36:34

by sucking the magnifically charged oxygen from

36:36

the flames. You just might get away with that

36:39

one. It's a myth that magnets

36:41

always point north, magnet fact. Some

36:43

do, but many others don't and can point

36:45

anywhere. These

36:50

are great magnets. I think there's whole book.

36:53

Chimpanzees have been observed attempting to use

36:55

magnets to extract termites from their

36:57

nests. It didn't work. What

36:59

I wanna know is, was he just tweeting these into

37:02

a void? Was he getting any engagement on them?

37:04

I'm having a look now. No retweet. Apart

37:06

from me. Although difficult,

37:09

it is possible to magnetize wood, magnetic

37:12

bars were popular in the Soviet Union. Here's

37:16

one which could definitely be

37:18

because at the time of what you think

37:20

this might work. In eighteen fifty one, a man in the

37:22

US state of Oklahoma Legally married

37:25

his magnet. Amazing.

37:29

I how is his career going? He's

37:33

a multiple award winning. Carnegie

37:35

medal award winning all that. think

37:37

that is enough magnet

37:38

facts. Wait. We've got one more. We've got one

37:40

more. Maybe this one's slightly more

37:42

of a real magnet back. Not not

37:44

casting any his visions on that on

37:46

on that list that you've extensively

37:49

shared. We had this message in from doctor Richard

37:52

Hill. We were talking also about

37:54

levitating magnetically. And

37:56

Richard said, Anatum University, we

37:58

routinely do experiments in which we magnetically

38:01

levitate water. magnetic

38:05

susceptabilities of the various types of soft

38:07

biological tissues are approximately all

38:09

the same around that of

38:11

water bone, however, has

38:14

a different susceptibility. And the upshot is

38:16

that the magnet would levitate the soft

38:18

parts of the person, which would be approximately

38:20

weightless. Whereas bones would

38:23

still feel the force of gravity, and

38:25

the bones would hang down within

38:27

the flesh. In other words, which is the other

38:29

way around from the normal situation of the skeleton

38:32

supporting the rest of the body. This

38:35

I mean, that just sounds like one of Anthony McAllen's,

38:37

ma hashtag magnified. Let

38:40

let me keep going. Until nineteen twenty

38:43

two, the Catholic church banned the use of magnets

38:45

on Fridays.

38:47

Right? Let's take care

38:47

of the week. Past and curious

38:50

occasion unless areas, tote

38:53

de la ghee, rather

38:57

than fries, Curio,

39:00

of the week.

39:07

Strong case for Anthony McGowan to be

39:09

a cheer of the week after the after that

39:11

exceptional entry. I'm sure he'd be delighted.

39:13

But, actually, we do have a

39:15

different magnet themed care of the week this week.

39:18

Adrian Glasser made a machine

39:21

So I'm gonna look at this video. So

39:23

we got the video up

39:24

here. And if

39:26

we press play -- Mhmm. -- and

39:29

it looks like a bit of Raspy

39:32

pie circuitry on top of a computer

39:34

with a screen and with an oscillating wave on

39:36

What are we looking

39:37

at? Let's let's let's hold it. Listen. Let's hold it.

39:38

Listen. Do

39:41

you know what it is? What is a

39:43

rotating magnet? And do you know what

39:45

that does at them? Beautiful and

39:48

succinctly. It ties together

39:50

both the magnet episode and complex

39:52

number episode. Oh god. Oh

39:55

oh, no. I mean, Can I do

39:57

do I have any details

39:58

left? No. We're all gone. It's the end of the Thank

40:00

you very much. Thank you very much for

40:02

doing my job. But he says he says I

40:05

was inspired by your magnets program

40:07

to build this machine that I've been thinking

40:09

about. This is the Arduino nano

40:11

microcontroller driving a DC

40:13

motor.

40:13

Okay. Yep. Yep. Yep. Yep. At relatively

40:16

slow speed. There are eight magnets in

40:18

the three d printed octagonal part that

40:20

rotates around top of the motor shaft. And the magnets

40:22

are spinning around on top of the Mojo.

40:25

Absolutely fantastic. I

40:26

mean, I'm just going along with it. You

40:29

get a pass for me. Anna

40:32

looks absolutely delighted. That

40:35

is the end of the current series.

40:38

We will be back although

40:40

we're not sure in what

40:42

4, when, how,

40:46

or why. I

40:49

don't think we've ever known why of me. Anyway,

40:51

send in your questions as ever to curios

40:53

cases at bbc dot co dot uk, and

40:55

we will see you soon. Are

40:58

you fed up? 4 with

41:00

the news.

41:01

In the last few minutes, I've been talking to

41:03

Michael Gove. It's a snake like mouth

41:05

quivered.

41:06

Slammed, like, wet leather, the

41:08

skewer, skewer, skewer. The news

41:10

dropped and channeled.

41:11

President Biden has used his annual state

41:13

of the union address to tout his administration's

41:16

record.

41:17

Gary, point of silence for you, Justina. little

41:20

from Karlamir. Little golden bee.

41:22

It's

41:22

everything you need to know like you've never had

41:24

it before. Understand. You don't drink. You never

41:27

smoke. You never take a drug, and you're a biter

41:29

than a Later, reminding

41:30

me. So many people a sufferer. Yes. I think

41:32

that's what country deserves. The biggest story

41:34

is with a twist. The

41:35

energy giant BP has indeed announced that their

41:38

underlying profit more than double.

41:40

Seeing it in the heart. I'm

41:42

seeing it in the heart. So may

41:44

just wanna watch it well, but

41:49

by childhood, some crack team.

41:51

Some wizards. Some wizards. Some wizards. These now.

41:54

These sons.