So the latest mathematical fact taking the internet by
storm, well, you may have seen it.
Let's check this out.
I don't know if you have seen it.
It's been all over the internet.
I'm going to take 1 divided by 998,001.
If you take that fraction, what you get, if you do the
decimal for it, you get something rather remarkable.
You get every single three-digit number.
It starts with 0 point, and it goes 0, 0, 0.
So that's 0.
And it goes 0, 0, 1.
There's 1.
0, 0, 2--
0, 0, 3--
0, 0, 4--
0, 0, 5.
And it will give you every single there-digit number
except for one.
It doesn't give you 998.
Every single three-digit number except 998.
OK, so let's go on.
So it goes 6, 7, skip a few.
Then it'll go 9, 9, 6--
9, 9, 7.
There is no 998.
It goes 9, 9, 9, and then it starts again.
It'll go back to 0, 0, 0, and then 0, 0, 1, and goes to
start again.
It repeats.
So, we had a lot of people email us about this, and one
of our favorite YouTubers, Vsauce, gives us a little
shoutout and asked us about it as well.
And so, we had to make a video about this.
We have seen this before.
It's not as remarkable as you may think.
So this is a repeating decimal.
For start, I want to show you that this is part of a family
of similar facts.
So, one thing to notice is 998001.
This is actually 999 squared.
Now, this is why we've got a little family of facts here,
because if I took 1 over 9,801, this is actually 1 over
99 squared.
Same sort of idea.
And what you would get are all the numbers, all the two-digit
numbers for that.
So you would get 0.0001020304.
So it's the same idea.
You get all the two-digit numbers except
for 98 in that case.
Very similar idea.
9998001.
It's the same sort of idea.
Again, this is 1 over 9999 squared.
And you get all the four-digit numbers.
What about 1 over 9 squared?
Now, that's the one I really want to show you, because we
wanted to make this a bit simple, OK?
Let's not go straight into it.
Let's take the simplest example.
1 over 9 squared.
So, 1/81.
That's what we're going to look at, 1 over 9 squared.
And that will give you all the one-digit numbers.
0.01234567.
It missed 8.
9, and then back to 0 again.
It repeats that pattern.
So, it's the same idea.
So, let's see if we can work out why this is true, and then
this idea will work for the other examples that
we've seen so far.
Right.
Do you want another piece of paper?
I do need another piece of paper.
Let's get you more paper.
So, the easiest explanation for this is that we could make
any repeating decimal that we wanted to make.
So, if you wanted to make, say, this.
Let's say it's going to start with naught point, and let's
take some digits.
Let's call it a1, that's the first digit.
a2, the second digit.
a3.
And let's say it's, I don't know, n digits long.
And then it repeats, so you get the same thing.
It goes dot, dot, dot, and you get that repeated over and
over again.
You can make that quite easily.
All you have to do to make that is to take this little
sequence, a1, a2, a3, up to an, and divide it by the same
number of 9s.
You've got n nines here.
9, 9, 9, whatever number that is.
So those two things would be equal.
For our example, then, for 1/81, we wanted 0.012345679,
dot, dot, dot.
012345679 over 999999999.
And that would give you that decimal representation, and
that does simplify.
That makes a nice 1/81.
Now, you could make any repeating decimal using that
method, but there is something special about that, because it
simplifies so nicely.
It makes such a nice number.
So, not all fractions like this would make a nice number,
so why is it so nice?
Let me try and show you that.
So here's a little formula.
If you study mathematics, here's a little formula that
you will learn, perhaps at some point, anyway.
Let me show you what it is.
A little expression.
It starts with 1 plus 2x plus 3x squared
plus 4x cubed plus--
and that can go on forever, OK?
So imagine this little equation.
Now, here's a little cool thing.
If your number x is less than 1--
in fact, this works for negative numbers as well if
they're greater than minus 1.
What I'm saying is if the size--
and that symbol means the size of x-- if it's less than 1,
this actually really simplifies to
something pretty cool.
It becomes 1 over 1 minus x squared.
And isn't that cool?
That complicated thing that goes on forever, actually if
your x is less than 1, it's the same as that, will give
you the same answer.
That's pretty cool.
That's pretty cool, isn't it?
Right.
So, we're going to use this to work out our fraction.
OK, so if I let x equal 1/10, what do I get?
On this side--
I think I'll write it as a column--
I get 1, 2x.
So two times 1/10, that's 0.2, and you add them together.
This one is 3x squared, which is 0.03.
Plus 4x cubed equals 0.004.
And you keep going, dot, dot, dot.
Keep going forever.
All this on the left is equal to--
on this side, if x is 1/10, well, let's see.
1 minus 1/10 is 9/10 squared.
9/10 squared is 81/100.
And because it's 1 over that fraction, you flip it.
This is equal to 100/81.
So all this is 1.23456 equals 100/81.
The one we're interested in was actually just the same
thing, but just 100 times smaller.
You just had to shift it across a couple of places.
So that's why you've got such a neat, cool
fraction out of that.
Really simple.
But why is it missing the 8?
Now, that's interesting.
Why is it missing the 8?
Check this out.
Let's go a few down.
Let's do the 8.
Naught point-- let me see, I'm going to need six
of those, then 8.
Let's do the next one.
Naught point seven more.
Then there's the 9.
And if I want to add 10, it's going to 0--
try again--
point.
Now 0000000--
now, I want to write 010 here, but there is no symbol for 10.
I can't write 010.
I have to write 10, like that.
Now, if you add this all together, you just add up the
columns, just a column at a time, can you
see it carries on?
Like you learned at school, this carries on.
So this 1 plus 9, the 9 turns into a 10, so that carries on.
So you get 1 plus 8, and that becomes a 9.
Your 8 becomes a 9.
Your missing 8 turns into a 9 because of all
this carrying on.
So all this is like a row of dominoes .
You get a carry on, carry on, carry on.
Your 8 turns into a 9, it disappears, and you get that
number we discovered.
And the same sort of idea will work for all those other
fractions that we looked at.
So the 1/998,001 is the same idea.
You can prove the same thing with the same idea.
It's a cool fact, isn't it?
It's a cool, little fact.
Yeah, I like the fact.
And it's not completely trivial, but you can make any
repeating decimal that you want to make, so that's not
particularly remarkable.
Why has it suddenly gone on the internet?
Who knows?
You get these things ever now and again.
These things go viral.
Mathematicians can roll their eyes and go, oh, no, not this.
Oh, you're boring.
But every now and again, for no reason as far as I can see,
you get this happening.