98 and Grafting Numbers - Numberphile

Uploaded by numberphile on 20.01.2012


MATT PARKER: I'd like to talk to you about the number 98.
98 is a bit large for people to think it's
quite a cool number.
However, if you take the square root of
98, it equals 9.89949.
What I like is that the first two digits are the same as the
digits in the number you were taking the square root.
This works for other numbers.
If you take the square root of 99, it equals 9.94999.
Starts with 99.
And when I saw this, I saw 98 first.
I thought, wait a minute, I wonder if there are other
bigger ones.
And it turns out there are.
If you take the square root of 9998, that equals 99.98999.
And there's your 9998.
And all these numbers-- and you can keep going.
9999, if you square root 9999, it starts 99.99.
And there's other stuff going out there.
But it starts with those.
And this continues.
The next one, it's always an even number of nines.
So 9999, and then ending with an 8.
And 9999, ending with a nine.
You've got these numbers where, if you take their
square root, they begin with exactly the same digits.
Admittedly, only in base 10.
But I was just doing this in Excel.
I made a little spreadsheet, and I was putting
in all these numbers.
And I found these.
And I went, I wonder if there are more?
And I couldn't find any more where the square root starts
with exactly the same digits as the number itself.
But then I went, you know what, I wonder if there any
numbers where just after the decimal point, maybe just the
digits after there are the same as the number itself.
And there are.
I'll just do a couple.
The first one is 77.
Because if you do the square root of 77, it starts with 8,
which is a bit disappointing.
But then 77 appears right there.
And, in fact, any number--

so 9797, and then 997997.
And this pattern continues all the way down.
Each of those, this one is 98.
You ignore that.
It then goes 9797 and then I think
it's like 95 or something.
But this here, so there's 77.
There's 9797.
This one here starts 998.
We ignore that.
And then you've got 997997, and then a bunch of stuff
after that.
But there it is.
I find these weird properties of numbers fascinating.
And again, I'd never heard of this.
No one ever talked to me about numbers where the decimals in
the number appear in its square root.
I looked online, I couldn't find anyone else talking about
these sorts of numbers.
I asked some friends.
No one had heard of these.
And I was like, well, I've got to call them something because
I was making notes on them.
I was looking into different types.
And so I called them grafting numbers, because you start
with a number and then its root just grows off from it.
However, up until now, they're being quite boring.
Because there's these two families.
There's the 98 family and the 77 family, which is a quite
predictable series of numbers.
And I wanted to see if were more exciting ones.
So I thought, how am I going to do this?
Up until now, I've been putting this into Excel and
doing a spreadsheet.
And Excel is great, because you can see the numbers that
you're doing calculations.
And you can search for numbers.
But you hit the limits quite quickly.
And so I thought, you know what, I've been meaning to
learn how to program in Python, which is
a programming language.
If you've done any programming, I highly
recommend Python.
Don't go through life without being able to program.
It's a whole area of a way of thinking and a way of doing
things that underpins all our modern technology.
At least, for a little while, out of interest.
It's like everyone should skydive once, because we're
the first generation of humans that can jump off
that kind of a height.
You know, just a--
you know, this weekend I'm going to fall
from a plane and survive.
No other human's been able to do that.
So I programmed a little script.
And I found all the numbers up to 10 million which appear
somewhere in their root before you go past the decimal point.
Because once you fall off that decimal point, it
is a long way down.
It looks like it stopped, but it's just checking bigger and
bigger numbers.
See, it's found a few more.
And it's checked the first million at this point.
I'll show you the ones I've found.
There's another-- and I'm not going to bother with the
square root sign anymore, just to make life easier.
764, its root is 27.64054 and so on.
And there it is there.
764 appears a little way in.
And in fact, 765 does the same thing, 27.6586 and so on.
And there it is there, 765.
5711 gives you 75.57115, and there it is
there, and that continues.
The next one down was 5736.
That gives you 75.73638 and so on.
And there it is there, 5736.
So in addition to the ones I showed you before, these are
all the other numbers that appear somewhere at the
beginning of the square root.
Now, mathematicians can be very patronizing towards some
of the sciences.
In fact, physicists themselves refer to biologists as people
doing stamp collecting, because all they're doing is
looking at things and just labeling them.
They're not doing any of the underlying patterns or the
reasoning behind it.
And I, as much as I make fun of biologists, I also do like
stamp collecting.
I like biology, for the record.
And in maths, there's some great fun doing stamp
So these numbers are meaningless, to be honest.
This only works in base 10.
There's no rhyme or reason.
It could be a complete accident that 5711, its square
root starts 5711.
There could be absolutely no meaning.
But I enjoyed writing a computer program, as such, to
search through and to find all these numbers.
But then the mathematician in me thought, what
if there is a pattern?
What if there is something underneath these?
And so I suddenly looked at these.
I went, wait a minute.
This one here, 76394, looks a lot like
this one here, 7639321.
In fact, it's the first five digits of this, and then
rounded up, as such.
And then I went, wait a minute, what if I take the
first three digits of this and round that up.
And that's over here, 764.
And so it looked like there was this pattern of numbers.
And so I wrote a different program to just look for the
numbers of this form.
And I'll show you what I found.
If you continue on, this pattern does keep going.
The next one down-- where we up to-- gets to 31.
The next one is 763932023.
That appears in its square root.
The one after that was 76393202251.
The next one is 7639320--
0, I'll make sure I get this right--
0225003, from memory.
And then the next one is a bit longer, and a bit longer, and
a bit longer.
And it keeps going.
And there is this sequence of numbers that have
the grafting property.
And I thought, well, what is it?
And if you just take this string of numbers, it turns
out this string of numbers is 3 minus the square root of 5.
If you work out 3 minus the square root of 5 as a number,
it equals 0.763932.
And then it's exactly the same sequence as far in
as you care to go.
So this family of grafting numbers are
based on this constant.
And so I decided this--
I call it a grafting constant.
And if you take that number and multiply it up.
So if you multiply it, well, first of all, by 1,000 and
then round it up, you get the first one.
If you multiply that by 100 and round it up, you
get the next one.
Multiply it by 100, get the next one, and then so on.
You get-- there's a whole family.
Actually, if you want to write that mathematically, we'll put
that in brackets.
You're then going to have to multiply this by 10 to the
power of 2n minus 1.
That will generate your next term up.
And in maths, to say round up, we do what's
called ceiling brackets.
So I'm going--
I'll do--
it's like, I draw them like this.
There are variations.
It's calculate that, round it up to the next number.
And you get a whole family of grafting numbers.
So there you are.
I was very pleased because I went from just being the
mathematical equivalent of a trainspotter, pulling out
these random numbers, to finding some that had an
underlying pattern, and the constant that generates them.
And that made me disproportionately happy.
And this may be number four.
I've not come across it, but most things in maths have the
number four.
I was at the Maths Jam weekend, which is the Maths
Jam conference.
So math jams are monthly gatherings in pubs to do maths
and talk about fun things.
Once a year we get together, so 130, not mathematicians,
just people who like maths, were in a room.
And everyone gets five minutes to talk about something that
they assume everyone else will find interesting.
And I talked about these numbers.
And some people just found the programming aspect of finding
these, and just categorizing them, was interesting.
But the mathematicians in the room were a bit more pleased
when I could show that there's a family which has an
underlying pattern to them.
And I don't know what happens.
I've thought, you could look at them.
You could do cube roots.
You could do this in different bases.
There's always another way to generalize this
and see what happens.

BRADY HARAN: What do you think about this?
What do you think I think about this?
MATT PARKER: I-- goodness knows.
My theory was that, as a mathematician, my theory is
not that unusual.
Because I figure if you find something interesting, someone
else will find something interesting.
And I found these numbers interesting.
And well, I'm sure I'm unusual in a lot of ways, but there
must be some other people who find this stuff fascinating.
And so, I thought I'd share it with them.
And some people did.
They thought it was great.

I sat on the train and learned computer programming.