"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Saturday, August 24, 2013

Tangibility of the transcendental 2 - updated.

I'm re-posting this blog entry with a crucial correction to the λ function, which previously wasn't properly defined. 

"I was doing a bit of reading about transcendental numbers today, and had an idea of constructing my own well defined transcendental. I was inspired by the Liouville constant. Anyway, my idea was to construct a number which is a base 10 concatonation of the digits of consecutive prime numbers:
mp =  0.2357911131719232931...
or the natural numbers:
mN = 0.12345678910111213...
The construction proceeded as follows. First we define the function λ:\mathbb{N}\mathbb{N}. Intuituively λ(n) gives the number of digits in n, i.e. the "length" of n in base 10.

Now we can express the numbers as infinite sums:

Where pn is the n'th prime.

Initially I thought this to be an exercise in creative procrastination, since firstly the idea is dead simple, and secondly who would be silly enough in going to the trouble of formulating such numbers. Surely only a fresh graduate with too much time on his hands.

To my surprise the idea behind those numbers has been already formulated by not one, but a few mathematicians. My mN is in fact a constant previously formulated by D. G. Champernowne. And it turns out that mp is already known as the Copeland-Erdos constant:

Notice the striking similarity in the formulation - I have formulated mp completly oblivious to the Copeland-Edros formulation. The similarity lies in the fact that my λ function acts a lot like the floor of base 10 logarithm of y. In fact the relation is the following:

Which makes the sums in the exponents of the definitions of mp and the Copeland-Edros constant equal, since for each i we subtract 1 i.e. we subtract 1, k times, so we need to add k back to the sum. And consequently,

Well, I guess I feel better knowing that I'm in good company deeming quite obscure and prima facie pointless ideas valuable. I knew that both mN and mp are irrational, but now I also know that mN is in fact transcendental (mission accomplished!), and hence don't have to provide a proof, which I suspect would be a herculean task."

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