"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

## Tuesday, May 25, 2010

### Tangibility of the transcendental

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 lambda:

$\lambda(y\in\mathbb{N})= \text{sup}\left\{x\in\mathbb{N}:\left\lceil\frac{y}{{10}^{x}} \right\rceil =1\right\}$

Now we can express the numbers as infinite sums:

${\mu }_{\mathbb{N}}=\sum_{k=1}^{\infty}k\cdot {10}^{-\left(\sum_{i=1}^{k}\lambda \left(i \right)\right)}$

${\mu }_{\mathbb{P}}=\sum_{k=1}^{\infty}{p}_{k}\cdot {10}^{-\left(\sum_{i=1}^{k}\lambda \left({p}_{i} \right)\right)}$

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:

$CE=\sum_{k=1}^{\infty}k\cdot {10}^{-\left(k+\sum_{i=1}^{k} \lfloor{\text{log}}_{10}{p}_{i}\rfloor\right)}$

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 lambda function acts a lot like the floor of base 10 logarithm of y, plus 1. 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, we need to add k back to the sum.

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 feel would be a herculean task.

Iteration said...

Your notation is a little confusing. It looks to me that you define the value of the function lambda evaluated at some natural number y to be a set, in particular the set of all x such that the ceiling of y/(10^x) is equal to 1.

So for Lambda(1), we obtain all values of x such that ceil[ 1/(10^x)]=1. This is then clearly all non-negative x (if we restrict allowable values of x to the integers, which you should specify).

Mariusz Popieluch said...

You're right that a specification of the range of Lambda could be given. I actually initially included that specification. But notice that restricting the range of Lambda to non-negative integers forces the domain to be a subset of N. Hence specifying x to be a positive integer is redundant. Notice that for each y there's a unique x satisfying the definition. Having said that I will take your advice and add the redundant restriction for clarity.

What I have to fix though is the restriction on the range to exclude zero (Z+) since there is no x such that ceiling(0/10^x)=1.

Thanks for the comment Nick. Hope the thesis is coming along well(?).

Mariusz Popieluch said...

I had a second look at it, and you're right! If I don't restrict x to Z+ then we could have another x satisfying the equality eg:

ceiling(447/10^3)=1

and

ceiling(447/10^3.2)=1

So indeed x wouldn't be unique.