Number Line - Numberphile Numberphile English English (auto-generated)

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Science & Technology 252.6K views 5.3K likes 317 dislikes 30 Jul 2013

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I want to take a little bit of a

SIMON PAMPENA: I want to take a little bit of a departure

departure from looking at specific

from looking at specific numbers and ask the question,

numbers and ask the question how do you

how do you measure numbers?

measure numbers this was one of the most

This was one of the most challenging things I came

challenging things I came across

across studying maths at uni, was this notion of how to

studying maths at uni was this notion of

measure, mathematically measure.

how to measure mathematically measure so

So this is what I'm going to do.

this is what I'm going to do I'm going

I'm going to show you the real number line.

to show you the real number line yeah

Yeah, you're familiar with that right?

you're familiar with that right so think

So think about this.

about this right this is the real number

This is the real number line. It's there in your head.

line it's there in your head so here's

So here's my question.

my question what's the distance between

What's the distance between 3 and 4?

3 & 4 what's the measure how long is

What's the measure? How long is that section? BRADY HARAN: 1.

that section well when how did you get

SIMON PAMPENA: 1. How did you get that? BRADY HARAN: Because I know the number that I need to add

that because I know the number that I

to 3 to get to 4 is 1.

need to add to 3 to get to 4 yeah is 1

SIMON PAMPENA: OK, that's good.

okay that's good

So you know the number.

so you so you you know have a number so

So you know the answer, basically, you're saying?

you know the answer basically you're

BRADY HARAN: Yeah.

saying yeah yeah ok ok imagine you

SIMON PAMPENA: OK, well, imagine you didn't know the answer.

didn't know the answer so imagine this

So imagine this is the first time you've ever tried to

is the first time you've ever tried to

measure the distance between 3 and 4.

measure the distance between 3 & 4 what

What would you do?

would you do like I'd make a piece of

BRADY HARAN: I'd make a piece of wood the distance

wood the distance between 0 & 1

between 0 and 1. And then I would take it to 3 and 4 and see how many of them

all right and I would take it to 3 & 4 there you go how many of them do I have

do I have to put on there.

to put on that okay that's good that's

SIMON PAMPENA: OK, that's good. That's good. I like that.

good I like that okay that's good it's

It's actually good that you brought in reality.

actually good that you brought in

Because that's going to really bite you.

reality because that's going to really bite you it's going to really bite you

It's going to really bite you. But that's cool.

but that's cool that's cool I like the

That's cool. I like the fact you said that.

fact you said that so basically you said

So basically you said, OK, well, I'm going to look at a

okay well I'm going to look at a bit of

bit of a number line which I know what the length is

a number line which I know what the

because it starts at 0.

length is because it starts at 0 okay

And you're going to go and drag it over

and you're going to go and drag it over

until you measure it.

until you measure it but another way you

But another way you could do it is you could say, this

could do it okay is you could say this

number, take that number.

number take that number so you're

So you're comfortable with that.

comfortable with that you know that that

You know that that distance there is 4 take 3.

distance there okay is 43 now this is

Now this is what's called naive.

what's called naive and it's not a

And that's not a putdown. It's naive.

put-down okay it's naive it's a naive

It's a naive way of measuring numbers.

way of measuring numbers but okay I'm

I'm hopefully going to take you down the rabbit hole now. 3.5 is a point right there.

hopefully going to take you down the rabbit hole now three point five is a point yeah it's a point right there I'm going to

I'm going to put a little kind of, little bit of

put a little kind of little bit of chalk

choke wire around it.

3.5 is a point.

okay three point five is a point do you

Do you agree that it exists?

agree that it exists three and a half

you're happy with that

BRADY HARAN: I'm happy with that. SIMON PAMPENA: You're happy with that, OK. What's the length of 3 and 1/2?

okay what's the length of three and a half what's the length of it yeah from

BRADY HARAN: What's the length of it? Well, to give you a length, I need a start

all I need us to give you a length I

point and an end point.

need a start point and an end point

SIMON PAMPENA: No, I'm just talking about the point, just

known and then I'm just talking about

the point, one point.

the point just the point one point so

So what's the length of a point?

what's the length of a point point start

BRADY HARAN: Points don't have length.

SIMON PAMPENA: What's another way of saying that?

what's another way of saying that three

3.5 take 3.5 is 0.

point five take three point five is zero

We use exactly the same logic that we used here for the

we use exactly the same logic that we

segment of line.

used here for the segment of line we're

We're going to use the same logic for the point.

going to use the same logic for the point okay now this sounds like I'm

Now, this sounds like I'm being trivial, maybe even annoying.

being trivial maybe even annoying but I

But I am getting somewhere.

am getting somewhere I'm making a point

I'm making a point about points. Because what now is the length of 3 to 4 if we

about points okay because what now is the length of three to four if we remove

remove the point 3.5?

the point three point five what is the

What is the length of that line segment?

length of that line segment the length

BRADY HARAN: The length is still 1. SIMON PAMPENA: Why?

is still one why because the thing you

BRADY HARAN: Because the thing you removed took up no space. SIMON PAMPENA: That's right.

removed took up no space that's right so

So this idea is that we've got a line segment, 3 to 4, which

this idea right is that we've got a line segment three to four okay which is made

is made up of a continuous set of points.

up of a continuous set of points I mean

There's only points between 3 and 4.

there's only points between three and

But if we remove 3.5, we haven't changed it.

four right but if we remove three point five we haven't changed it we haven't

We haven't changed it at all.

changed it at all and part of the reason

And part of the reason behind that is because the real

behind that is because the real number

number line has this property that there is no next number.

line has this property that there is no next number what's the what's the number

What's the number after 3.5?

after three point five three point five

BRADY HARAN: 3.50000000000000--

zero zero zero zero zero zero zero zero zero zero zero zero zero yeah exactly

SIMON PAMPENA: Yeah, exactly. But you're not even saying that correctly.

but even you want you're not even saying

Because the way that we describe numbers doesn't contain the awesomeness of what's going on here.

that correctly because the way that we describe numbers doesn't contain what's the awesomeness of what's going on here

Why is this a problem? Because we are using stuff that we haven't proved.

why is this a problem because we are we are using stuff that we haven't proved

We're using naive notions of measure.

we're using naive notions of measure

That's why that's a problem.

that's why that's a problem we're making

We're making assumptions. This almost broke maths where I'm going with this.

assumptions this almost broke maths where I'm going with this

It almost broke maths.

it almost broke Matt

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