Transcendental numbers

Brian writes:
> However, perhaps one could say that a transcendental
> number is one which arises *only* as the
> solution of an equation involving logarithms
> or antilogarithms. (Manuel and John Fitch may
> jump on me for this one; it may not be 100%
> true all the time. Can you think of a counterexample?)

That's probably very hard to prove.
It's known that e and pi are transcendental, but not if e + pi is.
I think it's known neither if for instance sin 1 is.

> the most spectacular of which were
> discovered by Srinivasas Ramanujan.

My favourite equality of Ramanujan is so elegant I'll try to write it here:


2
a 2 ____ -a
( -x V pi ` e
| e dx = ---- - --------------------------------------------
) 2 2a + 1 /
0 a + 2 /
2a + 3 /
a + 4 /
2a + 5 / .....


> Transcendental numbers also tend to arise when
> the exponents of an algebraic equation are imaginary.

Even if they are irrational as you write:

> Transcendental irrationals arise when the
> algebraic exponent, for example, is itself
> an algebraic irrational--as in Hilbert's number,
> 2^[sqrt(2)]. )

This is Hilbert's Seventh problem which was proven to be true
partly in 1929 and 1930 and completely by Gelfond and Schneider
in 1934 independently.
b
a is transcendental if a and b are algebraic, a /= 0,1 and
b is irrational.

Transcendental numbers are always irrational as shown by
Franz Kriftner.

> Claude Shannon proved elegantly that the amount
> of information required to generate (or parse) a
> message is proportional to the log to the base 2
> of the number of bits in the message.

That's not exactly true. But this is more the domain of
computational complexity.

> To my knowledge, there is as yet no known algorithm
> by which the entire field of reals may be sieved
> and by which all transcendental numbers will
> always be found.

There is, but you need an infinite amount of time.
The set of all algebraic numbers is namely enumerable.
All rational numbers are enumerable, so you can enumerate
the coefficients of the algebraic equations whose solutions
are the algebraic numbers, and therefore enumerate the algebraic
equations themselves if you do it cleverly.

> Mathematicians widely believe the digits
> in pi to be completely random.

I believe the first million digits of pi pass the chi-square
test for randomness. I'm not sure what this means for the
colour of the spectrum of the digit values; it may be white or not.

> Since information is logarithmically
> proportional to energy (another of Shannon's theorems).

I'm not sure exactly which theorem Brian means. But entropy in
information theory and entropy in physics are different concepts which
-regrettably perhaps- share the same name. I remember my information
theory professor saying he regretted it in any case since it's often
a cause of confusion.

Manuel Op de Coul coul@ezh.nl

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