repeating digits ( transcendental numbers )

Brian Mc Laren writes:

> It has been speculated that the digits in the
> decimal expansion of transcendental numbers
> never repeat. One subscriber has even stated as
> much on this forum. However, this proposition
> has never been proven mathematically.
>
I'm quite sure this has been proven
but as I do not know any proof i will prove it myself:

any number with only a finite number of non-zeros is rational
proof:
if there are only zeros after the decimal point
it's integer and we're done
else
just write down all the digits up to ( and including ) the last non-zero
leave out the decimal point
and divide by a 1 with as many zeros as there were digits after the
decimal point
the result is the same number written as a fraction of two integers
we're done

any number the digits of which repeat is rational
proof:
any number the digits of which repeat every n digits
is an integer multiple of 0.0...010...010... ...
( a 1 at every nth position )
this 0.0...010...010...010... ...
can also be written as 1/9...9
( 1 divided by a number with n 9s )
an integer multiple of a rational number is rational
so we're done

any number the digits of which repeat starting from some position
is just the sum of a number with only finitely many digits and
another one the digits of which always repeat

so we know for sure that that if the digits in the decimal expansion
of a number repeat this number must be rational

as a transcendental number can never be rational
its digits never repeat

---------------------------------------------------
by repeating I mean:
starting from some position
a fixed sequence of digits repeats forever
forgive me if I took you ( Brian Mc Laren ) wrong

if you only meant that some sequences of digits repeat sometimes
it is easy to prove that this will always happen
as there are only finitely many different sequences of a given length
there must be a repeating sequence of this length

for example: there are 100 sequences of two digits
so after 202 digits ( 101 sequences )
there must have been a repetition

---------------------------------------------------
BTW:
I really appreciate Brian's postings

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