another number for you Brady I'm loving
this number this number is is great is
so cool 1 4 2 8 5 7 I want you to um
just give me a
number 13 okay fine 13 I'm going to
multiply this by 13 okay okay I can't do
that me Ed so let's now take the last
six there's six numbers here right I'm
going to take the last six of these and
I'm going to add it to the first one
that's
1+
8571 41 now I can do that that is
8571
42 which if you look
carefully is a permutation of this guy
it's 1 4 2 8 5 7 give me another number
ready give me another
number 112 right 112 very good so 1 4 2
8 5 7 * 112 equals right let's do it
okay I noticed something about 112 you P
for choosing 112 but we see something
else okay 112 I think must be a multiple
of seven in fact it is right you ruined
it with this what the multiples of seven
bring this multiples of seven are the
only ones they do something else they do
something let's let's let's run with it
let's run with it okay multiples of
seven we'll do it we take the last six
okay
15 plus 999
984 equals 9 9 9 9
99 okay give it we'll do one more one
more okay 18 18 okay not a multiple
seven thank you
2,571 426 do the same thing take the
last six and I get 5 7 1 4 2 8 which is
1 4 2 8 5 7 if I take any anything
that's not a multiple of seven anything
that's not a multiple of seven and I
multiply this number by
it and then I do this procedure of uh of
sort of
taking the last six and adding the
previous one the other ones I always get
a permutation of
142857 in particular if I take the
numbers one to six there's six nut
digits here and I multiply uh this by
them I will always get a permutation of
157 so for example obviously If I
multiply it by one I get 142857 that's
trivial If I multiply it by 2 I get 285
714 which again is it pmut
mtip by three I get 4 28 571 and so on
and so forth and I can keep doing this
and if you go above seven when if you go
above seven then you you need to do this
business of sort of taking the last six
and then adding it but you'll always get
this uh this this cyclic structure this
is an example of a cyclic number in fact
they're common right not particularly
okay so this is the only one that
doesn't begin in zero the next one
actually after this one is
0588
235
29411 7
64 7 this is another cyclic number okay
this has got uh 16 digits you play the
same game you can take in this now you
would multiply it between H what's
important is the fact it's got 16 digits
so if you do multiples of 17 this goes
wrong but if uh if you do 1 to 16 you'll
get a permutation of it and then you can
play this game as well H but with 16
rather than six digits so so here's the
other weird thing about it if you
multiply it by any multiple of seven you
always get these all the nines If I
multiply this guy this has got SE this
has got 16 digits If I multiply this by
17 I'll get all the nines why does it
work okay let's see why it works I'm
going to need more
paper okay well the important thing
about that number that one 142857 is it
appears in the decimalization of a sth
okay if I take a sth right then a 7th is
equal to
not
142
857
142857 to and so on keeps repeating and
this is critical why is this critical
well now think what happens when I
multiply this by 10 I multiply it by 10
107 I multiply it by 10 all these
numbers just shift along one right and
now but what do you know about this this
is just
107 is
1 plus 37
right okay so from this you can infer
that
37th is equal to not.
42857 1
4 and the point is as you can see the
multiplying by 10 just shifted
everything along one okay and then we
pull off this original one but it got
replaced by another one and getting
shifted along and then you're left with
a permutation it works perfectly and it
keeps working now we can do the next one
so um multiply it by 10 again so do
multiply by 10 do 37th and I get again I
just shift everything just get shifted
Along by one but this I know is 4 +
27 so I get rid of the fours and I
conclude that 27 is just .2 2 8 5 7 1 4
and again it was just everything got
shifted by one but because it was the
stuff that you
removed that replaces it you just end up
with permutations and that's why it
works that's essentially the essence of
why it works and actually from this
structure you can see why um what the
structure of all these cyclic numbers
has to be so you take a prime number you
take its reciprocal one over a
prime okay
it doesn't work for all prime numbers
but for some prime numbers it does
actually we'll come back to that in a
minute but you'll get something not
times some your cyclic number and your
cyclic number gets repeated okay this is
a p minus one a number of length pus one
so here it was a seventh and these these
this blocks had a length of of six okay
so if I multiply this by 10 to the^ P
minus 1 in this case by 10 the
6 I get 10 P -1 / p is equal to well my
cyclic
number point my cyclic number so you got
all you get the same thing now you take
these two away from one another and you
conclude that your cyclic number is
equal to 10^ the P -1 -1 over p and this
is where those nines appear okay CU If I
multiply this by P the prime number
which in this case was seven or any
multiple S I get something that's
related to the nines right because it's
10 to the sun power minus one which is
going to be all the nines right so
that's why it works all the cyclic
numbers have this form not all prime
numbers will give a cyclic number when
you construct this guy so the prime
numbers that work that give you if you
construct this that will give you a
cyclic number are P = 7 7 17 19 23 29
there're the first few there's more okay
are there an infinite number
there are an infinite number yes um in
fact and this is this is one of these
things that I just think number theorist
how do they they dream up these things
we even know how many what percentage of
prime numbers uh give cyclic numbers
when you construct this it's 37.3 n5%
roughly of uh prime numbers will give
cyclic numbers in this way I
mean why we saw how if we multiply it by
multiples of seven and then we do our
little construction that we discussed
that we get back all the nines okay all
the nines appears everywhere in this
what else can you do let's look at it
142
857 okay let me take the first two
numbers let me take the next two 28 let
take the next one 57 I'll add them all
together look what I
get have some of that all the NES right
let me do this let me take
three 142 + 857
add them together what do I get
999 right let we even do this let we do
four right I get 1 14
two8 +
5714 and I have to use every number the
same number of times 28 57 what's that
nine nine
nine
nine that's amazing it's amazing just
love it what is going on it's all
related of course to this very special
property that it's basically these
numbers can be written as 10 to some
power minus one divided by the um by the
prime number but it's clearly just uh
you know it's all built in there and
they all have this property they'll all
do this um you can also work in
different bases you know you can have
CYCC numbers in different bases what
happens then is the cyclic numbers
instead of being we're we're working in
base 10 as we're familiar with but you
just change this number from 10 and then
you get the CYCC numbers again you can
play the same games and they and but of
course they're cyclic in that particular
base
and instead of getting all the nines you
might get all the threes or all the
fours depending on whatever base you're
in it's amazing number it doesn't even
stop there there's one more quirkiness
that I've got to show you 1 4 2 8 5 7
okay now we're going to take the first
three numbers
142 going to square it to 857 squared
what do we get 714 285 what's that it's
another permutation of our 1 4 2 8 57
it's amazing
and this number actually it's we're not
the only people to have noticed that
this this number has sort of quirky
qualities there's something called an
anagram now this is some crazy Armenian
spiritual leader from like wartime uh
called G Jeff and uh he basically was
able to explain everything in life and
the cosmos through this number it's
total nonsense the less said the better
but he did draw a nice pretty picture
which I think we can draw which sort of
just gives us our last little thing to
do with this this nice so here we go I
draw that is a rubbish Circle but it'll
do we just draw nine up there one two
three four five six 7 eight okay so
we're going to multiples of three we're
not interested in boring get rid of them
not that any numbers are boring
obviously but uh and we just draw a
triangle okay now the remaining numbers
we just sort of draw this nice pretty
picture through them one to four to
two to
eight to
five to seven and then back to one again
and this is called an anagram and
apparently the properties of this can be
related to cells and all sorts but less
said about the better but it's a nice
picture isn't it so 142857 Cy amazing I
love it