Well you are watching this video so you
like numbers, and I like numbers since I
was very little. And one particular
sequence I really loved it was was
powers of two. So let's start with 1,
right, 1 is 2 to power 0 so it has to be
here, and then we have 2, 4, 8, 16, 32 and
then we keep on going: 128, 256, 512, 1024, 2048-
oops we ran out of space. Well let me
make things easier and only concentrate
on the first digit of these guys. We keep
1, 2, 4, 8 and then we only keep 1, 3,
6, 1, 2 and 5. 1, 2 - and so on.
So so let's collect all this data and
put this all together and that's what we
get, here is the sequence of numbers I'm
going to talk about, let's look at it.
Let's take 1, 2, 4, 8, 1, 3, 6, 1, 2, 5
which we had here - let me
mark it like this. And then we see the
same stretch, and then we see the same
stretch - very curious. And one more time.
And we try to do it one more time but
then there is little 7. And we have a
box here, and here we had two things
which are different. And if we keep on
going it happens to be absolutely the
same as this one. But if we- if we're here
something else happens, we have an
extra 6 which wasn't there before. So
the question is what what is going to
happen in the long run? We have a huge
huge huge number of these guys and maybe
maybe we can run a little experiment and
count how many 1s, how many 2s, how
many 3s and so on. So let's have a
little table 1, 2, 3, 4, 5, 6, 7, 8, 9. So in the
first stage we have three dots here,
because there are three 1s. Two dots
because we have two 2s, and then we have
a 3 and the 4 and a 5 and a 6, 8. So 7
is missing, 9 is missing. But then
something else starts going on, this this
is going to change. The question I want
to ask is suppose I
have a billion digits, then do we have
more 7s than 8s? Or maybe more
8s than 7s?
What about 6 or 5s? So how to
actually make sense of it? - (Brady: So will, will)
(the 7s and 9s make a come back?)
(Will someone break away and have a lead or?)
Exactly, we have quite a few 8s here, and
7s and 9s are trailing behind;
but maybe will spring ahead. And
another question is, we have a sequence
which seem to be periodic. We have this
pattern going on. What if, in the long
long long run, we have another pattern?
And it will start going forever.
Well first of all let me state the
question, and mathematicians when they
state a question they sometimes use
some notation, right? You probably got used
to because you watched 100 videos of this
channel already. So so let me, let me talk- we're
about talking about, let me call it a sub
n. n is a natural number and a sub n is
the first digit of 2 to the n. Here we
have a sub 0, and then a sub 1, and a
sub 4- sorry a sub 2, a sub 3 and so on.
Ok? And then we look, we take very big
natural number N and look at the number
of of little ns say between 0 and N such
that this a sub n is equal to k, where
this k could be 1 or 2 or 3 or 4 or 9.
Okay so what is it if n is very big? And
specifically let me take this number
and divide by n, and this is going to be
the probability you hit 2 if k is 2, or 4
if k is 4 okay? So out of these numbers
what is the proportion of the ones that
have a value k? So you have this huge
seq- huge sequence that will saw
already and we can count the number of
1s, and that's what I did over there, or
2s or 3s. And what is going to
happen if we increase the length of the
sequence? Mathematicians usually say
where does it go to if n goes to
infinity? Which means, take a very- and
here k is fixed. So for example k could
be 1 or 2 or 3. So that's a question
you can ask if you look at the sequence
and some of you have seen a video about
the Benford law, and that's the answer,
amazingly. So you already know the answer.
So what happens is that you have more
1s than 2s, more 2s than 3s, more
3s than 4s- - (Brady: Even here with our 7s and 9s?)
That's right, eventually 7 catches up
and starts growing and it beats 8.
And this- the answer that we have those
numbers, which is logarithm base 10 of
1 plus 1 over k, which is exactly the
ones appearing in
Benford's law. This log 10 of 1 over k,
what does it mean? It means that the
number of all- the proportion of 1s
eventually goes to 30.1%, well
approximately. And here we have 17.6% and
here we have 12.5, and 4s get
9.7, and here we have approx- well
it's it's not exact figure it's
approximate, and here we have 6.7,
5.8, 5.1,
and then 4.6. So it gets
smaller and smaller. - (Brady: Obviously you're not)
(surprised by this, because you're a)
(mathematician, but-) - Well I guess I would
say I was surprised by this event
initially when it turned out to be the
case, but then there's a proof for some
surprises. - Like so many of our videos
this was recorded at the Mathematical
sciences Research Institute in Berkeley,
California. To see more from this
interview, including some more proof,
there's a link on the screen and also in
the video description. And you can also
watch our previous video on Benford's
Law. - ...the probability goes up and up and up
through the thousands, and then down
through two thousand three thousand...