Subtitles section Play video

• JAMES GRIME: We're going to break a rule.

• We're break one of the rules of Numberphile.

• We're talking about something that isn't a number.

• We're going to talk about infinity.

• So infinity.

• Now like I said, infinity is not a number.

• It's a idea.

• It's a concept.

• It's the idea of being endless, of going on forever.

• I think everyone's familiar with the idea of

• infinity, even kids.

• You start counting 1, 2, 3, 4, 5--

• you might be five years old, but already you're thinking,

• what's the biggest number I can think of.

• And you go, oooh, it's 20.

• You get a bit older, and you go, maybe it's a million.

• It never ends, does it? 'Cause you can keep adding 1.

• So that's the idea of infinity.

• The numbers go on forever.

• But I'm going to tell you one of the more surprising facts

• There are different kinds of infinity.

• Some infinities are bigger than others.

• Let's have a look.

• The first type of infinity is called countable.

• And I don't like the name countable.

• And Brady gave me a little bit of a hmm, just then.

• Because if you're talking about infinity, you can't

• count infinity, can you?

• Because it goes on forever.

• I think it's a terrible name.

• I prefer to call it listable.

• Can we list these numbers?

• All right.

• Let's do these simple numbers, 1, 2, 3--

• BRADY HARAN: You're not gonna do all of them, are you James?

• JAMES GRIME: 4.

• How long have we got?

• BRADY HARAN: (LAUGHING) 10 minutes.

• JAMES GRIME: Right.

• 5, 6--

• so you can list the whole numbers.

• So this is called countable.

• Listable, I prefer.

• All the integers.

• That's all the negative numbers as well.

• So there's 0.

• Let's have that.

• But there's 1 and minus 1, there's 2 and minus 2, there's

• 3, and minus 3.

• Now, that is an infinity as well.

• And in some sense, it's twice as big, because there seems to

• be twice as many numbers.

• But it is infinity as well.

• They're both infinity, and they're both the

• same type of infinity.

• They both can be listed.

• Perhaps more surprisingly, the fractions can

• be listed as well.

• Let's try and list the fractions.

• I'm going to write out a rectangle.

• 1 divided by 1.

• That's a fraction.

• [INAUDIBLE].

• Let's have 1 divided by 2, 1/3, 1/4, 1/7--

• OK, that goes on.

• Let's do the next row and have two at the top.

• 2/1, 2/2, 2/3, 2/4.

• Let's do the next one.

• 3/1, 3/2.

• 4/6, 4/7.

• That goes on and we can keep going.

• So here, I've made some sort of an infinite rectangle array

• of fractions.

• Now if I want to make it a list like this, though, If I

• went row by row, you're going to have a problem.

• If you go row by row, I'll go--

• there's 1, 1/2, 1/3, 1/5, 1/6, 1/7-- and

• I'll keep going forever.

• And I'm never going to reach the second row.

• I can't list them.

• Not that way.

• You can't list them that way.

• You'll never reach the second row.

• This is how you list them.

• Slightly more clever than that.

• You take the diagonal lines.

• Now, I can guarantee that every fraction will appear on

• one of those diagonal lines.

• And you list them diagonal by diagonal.

• So that's the first diagonal.

• Then you list the second diagonal-- there it is.

• Then you list the third diagonal, then you take the

• fourth diagonal, and the fifth.

• So eventually, you are going to do this every fraction.

• Every faction appears on a diagonal, and you're

• going to list them.

• Now, if you take all the numbers, right?

• That's the whole number line.

• Let's try that.

• Look, I'm going to draw it.

• It's a continuous line of numbers.

• These are all your decimals.

• You've got 0 there in the middle, and you'll

• go 1 and 2 and 3.

• But it has a 1/3.

• It will contain pi, and e, and all the

• irrational numbers as well.

• Can you list them?

• How do you list them?

• But hang on.

• We've missed a half.

• So we put in the half.

• Hang on, we've missed the quarter.

• We put in the quarter.

• But we've missed 0.237--

• so how do you list the real numbers?

• It turns out you can't.

• In fact, rather remarkably, I can show you that we can't

• list them, even though were talking about something so

• complicated as infinity.

• BRADY HARAN: Do it, man!

• JAMES GRIME: We need paper.

• BRADY HARAN: We need an infinite amount of

• paper here, I think.

• JAMES GRIME: (LAUGHING) It's a big topic.

• Imagine we could list all the decimals, right?

• We can't, actually.

• But pretend we can.

• What sort of--

• what would it look like?

• Let's pick some decimals.

• 0.121--

• dot dot dot dot dot.

• Let's pick the next one.

• Let's say the next one is 0.221--.

• Next one, let's do 0.31111129--.

• And let's take another one, here.

• 0.00176--.

• Now I'm going to make a number.

• This is the number I'm going to make.

• I'm going to take the diagonals here.

• I'm going to take this number and this number and this

• number and this number and this number.

• And I am going to write that down.

• So what's that number I've made?

• It's 0.12101--

• something, something, something.

• Now this is my rule.

• I'm going to make a whole new number from that one.

• This is the number I'm going to make.

• If it has a 1, I'm going to change it to a 2.

• And if it has a 2 or anything else, I will change it to a 1.

• So let's try that.

• So I'm going to turn this into--

• 0-point.

• So if it has a 1, I'm going to turn it into a 2.

• If it's anything else, I'm going to turn it into a 1.

• So that will be a 1.

• I'm going to change 1 here into a 2.

• I'm going to change that one into a 1.

• I'm going to change that one into a 2-- that was my rule.

• And I'll make something new.

• That does not appear on the list.

• That number is completely different from anything else

• on the list, because it's not the first number, because it's

• different in the first place.

• It's not the second number, because it's different in

• second place.

• It's not the third number, because it's different in the

• third place.

• It's not the fourth number because it's different in the

• fourth place.

• It's not the fifth number, because it's different in the