Uploaded by numberphile on 01.04.2012

Transcript:

DR JAMES GRIME: So today, I want to tell you that almost

all integers contain the digit 3.

So the integers are the whole numbers.

So I'm saying that almost all whole numbers

contain the digit 3.

It sounds strange, doesn't it?

Yeah, because I can think of infinitely many numbers that

contain a 3.

I've got 3, 33, 333, 3,333, and so on.

I can think of infinitely many numbers that

don't contain a 3.

7, 77, 777, and 7,777, and so on.

So how can almost all integers contain the digit 3?

So 23 contains the digit 3.

310 contains the digit 3.

9,002,023.

So what about the numbers less than 10?

How many numbers less than 10 contain the digit 3?

It's just 3, isn't it?

Yeah, it's just one of them.

Right.

So one of those, just the number 3 itself.

So what's the proportion of numbers?

Well, it's one out of ten.

So it's 1/10.

It's 0.1.

It's 10%.

What's the next size up?

Let's look at numbers less than 100.

We've got 3.

We've got 13, 23, 43, 53, 63, 73, 83, 93.

You may have noticed I skipped off the 30s there.

So we've got 30, 31, 32, 33, 34, 35--

so I reckon we've got 19.

19 out of 100, that's 0.19.

It's 19% is what I'm saying.

So the percentage has gone up.

Let's try numbers less than 1,000.

Well, we've got the 19 we said already, right?

But we've got all those numbers again,

but with a 1 in front.

Like it's 103 and 113 and then we get 200 and something.

We get 400 and something, 500, 600, 700, 800, 900.

And then we can include all the 300s,

which I skipped there.

So just the 300, 301, 302, 303, right?

So it's 271.

What's the percentage?

It's about 0.271--

27%.

It's going up.

Now, did you notice the trick that we used

for that last one?

I said, think of all the numbers with a 1 in the front.

It was the same copy but with a 1 in the front, with a 2 in

the front, and then it was all the 300s.

You can do this in general.

The numbers containing a digit 3-- let's call that T for 3.

Let's call these 10.

Less than 10 to the power n.

So these are the numbers containing a digit 3, less

than 10 to the power n.

If you want to find the next one-- and

that's going to be 10,000.

You take the last of your previous number, times it by

9, and add on 10 to the power n.

And you can find the next number.

You can do that.

So that would be the next one we're looking for.

So there's a little formula there.

So let's do it.

All right then, I'm going to use my calculator.

So 9 times 271, plus 1,000.

And I get that number, 3,439.

What's the proportion?

It's about 0.3439.

It's going up all the time-- about 34% now.

It's going up all the time.

In fact, there's a nice way to work out how many numbers

there're going to be with a 3.

Imagine if I was allowed to write my numbers like this,

with zeroes.

So that would be a zero, but with all these zeroes there.

This is a four-digit number.

So here's another one, and here's another one.

How many four-digit numbers are there?

It's actually 10 to the power 4.

10 choices there, 10 choices there, 10 choices, 10 choices.

How many of them don't have a 3?

Well, if I'm not allowed to use 3, then I only have nine

choices for that place, nine choices for that place, nine

choices for that place, nine choices for that place.

So that would be 9 to the power 4.

If you want to find out how many integers that contain the

digit 3, it is 10 to the power n, minus 9 to the power n.

See, these are all my n-digit numbers minus the ones that

don't contain a 3.

The proportion, then, is going to be this thing divided by

this thing--

just 10 to the n.

So it's this.

And what is that proportion?

10 to the n divided by 10 to the n is 1, minus 9 to the n

divided by 10 to the n.

Here's another way to write that--

9/10 to the n.

That's the proportion that contain a digit 3.

And just what happens?

Well, as this gets really, really big, this fraction gets

smaller and smaller and smaller.

And it will go to 1.

That will disappear.

It will barely be worth mentioning.

And that is 1.

So it goes to 1--

we say, as n gets big.

So this is getting closer and closer to 1 minus 0.

As that gets closer to 0, as it gets big, this gets closer

and closer to 1 minus 0.

So the whole thing is going towards 100%.

100% of numbers contain the digit 3.

And this is the proper technical definition of when I

said almost all numbers contain the digit 3.

"Almost all" sounds like a wishy-washy vague expression.

But in mathematics, it has a proper meaning.

It means that the proportion is going toward 100% as you

get more and more numbers.

BRADY HALAN: Would it be equally true to say almost all

numbers contain the number 5?

DR JAMES GRIME: That would be equally true.

And you can do the exact same thing to say almost all

numbers contain a 5.