Re: Hello There?

Nam Nguyen wrote:
[...]
|I actually don't know (or remember) the definition of a
transcendental's
|being "normal".

It's not just for transcendental numbers. A number is normal
to base b if each sequence d_1,...,d_k of base-b digits appears
with the correct limiting frequence. That is, if a_n is the
number of occurrences of the sequence in the first n base-b
digits of the number, the limit as n goes to infinity of
a_n/n is b^{-k}. A number is normal if it's normal to every
base. Being normal implies the weaker property that each such
sequence (in each base) occurs at least once.

A rational number is obviously not normal to any base, because
the expansion either terminates or repeats. There are
transcendental numbers like the sum of 10^{-k!} that are
obviously not normal. It's not known whether there are any
algebraic irrational numbers that aren't normal, and it seems
somewhat plausible that they are all normal.

We don't know whether pi is normal to any base (although it's
basically expected to be normal to every base), but we do know
that Champerowne's number

.123456789101112131415161718...

is normal to base 10.

|Of course we certainly could create a transcendental
|containing "Hello There" simply by putting it there, say, in front
|of pi's decimal expansion. The motivation for asking the question is
|that "Hello There" is too short; so pi may "accidentally" contain it.

It's not hard to compute the position in Champerowne's number where
an encoding of such a message appears, given the message, so there's
not much magic to that.

Given a specific such message, encoded as n decimal digits, and some
arbitrarily chosen irrational number such as pi or log 2, it would be
a little surprising if the decimal expansion didn't contain the message
somewhere. You expect it to appear first at somewhere on the order of
the 10^n-th digit, after all. For it not to appear, there would have
to be some (somewhat surprising) property of pi or log 2 preventing it.

The fanciful idea of the novel has it, of course, that certain kinds
of coherent messages occur much *sooner* than one would expect from
this kind of information theory alone. Otherwise there's no point
in it. I didn't read the novel, so I don't know whether he had it
that pi had been altered so as to include them (which is logically
impossible) or that it just had a pre-existing harmony with some
eternal intelligence.

Keith Ramsay

.

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