Non-Negative Integer Binary Concatenation: Is This an Irrational Number?

[Jimmy Snyder]

I am interested in the following number which is obtained by concatenting the binary representations of the non-negative integers:

.011011100101110111...

i.e. dot 0 1 10 11 100 101 110 111 ...

This is a little bigger than .43 and I assume it irrational since no pattern of bits repeats forever. I assume that I am not the first to become interested in it, so I wonder if it has already been given a name.

 

[arildno]

It is actually called Jimmy's snide Remark.

 

[Jimmy Snyder]

It is actually called Jimmy's snide Remark.

Is this a pun on my name, or did I inadvertently say something snide?

 

[eljose]

when watching that i have a doubt...

once i read in a number theory book that existed a series (i think it was over primes) that gives you a sequence of primes in the form:

$$0.p100p2000p30000p4....$$ or something similar i think it

was related to the calculation of the series $$f(x)=\sum_{p}10^{-p}$$

 

[Jimmy Snyder]

The number that I posted has this obvious characteristic: Every finite sequence of 1's and 0's is found in it. Therefore, every idea that can be written down in finitely many bits is there. For instance, all of the works of Shakespeare are there, complete with typesetting commands. And yet it's just a single irrational number like $\pi$ or e.

 

[shmoe]

It looks to be one tenth the Champernowne constant, the base 2 version that is. What's usually called the Champernowne constant is the base 10 version, just writing all the numbers in order, 0.123456789101112131415..., and of course you can do this for any base. They're built to be normal numbers (and are of course irrational).

 

[Nimz]

You mean one half of the base 2 Campernowne constant?

 

[arildno]

Is this a pun on my name, or did I inadvertently say something snide?

Do you think Newton declared the units of force to be Newtons?
Do you think Gauss called Gauss' theorem Gauss' theorem?

 

[Jimmy Snyder]

It looks to be one tenth the Champernowne constant.

Thanks shmoe, you and this forum are an invaluable resource.

 

[Jimmy Snyder]

Do you think Newton declared the units of force to be Newtons?
Do you think Gauss called Gauss' theorem Gauss' theorem?

It was intended as a joke. Sorry you didn't get it.

I assume that I am not the first to become interested in it, so I wonder if it has already been given a name.

 

[arildno]

Anyhow, it was an interesting number, I'll grant you that.

 

[shmoe]

You mean one half of the base 2 Campernowne constant?

I mean the Champernowne constant divided by 10.:tongue:

Thanks for the correction.

 

[Jimmy Snyder]

I mean the Champernowne constant divided by 10.

According to this article, there is more than one Champernowne constant.
http://en.wikipedia.org/wiki/Champernowne_constant
In particular, the number I spoke of in the original post is in the notation of that site $C_2 / 2$. I am not concerned about the factor of 1/2, and so I transfer my interest to $C_2$. Thanks again to everyone for your interest and help.

 

[shmoe]

According to this article, there is more than one Champernowne constant.
http://en.wikipedia.org/wiki/Champernowne_constant
In particular, the number I spoke of in the original post is in the notation of that site $C_2 / 2$. I am not concerned about the factor of 1/2, and so I transfer my interest to $C_2$. Thanks again to everyone for your interest and help.

That's what I meant by the 'base 2 version'. Divided by 10 was a weak attempt at correcting my one tenth gaff with some binary humour.

 

[Jimmy Snyder]

That's what I meant by the 'base 2 version'. Divided by 10 was a weak attempt at correcting my one tenth gaff with some binary humour.

There are 10 kinds of people. Those who know binary when they see it and those who don't. I didn't, but now I do.

 

[eljose]

-I don,t know if Gauss called his law "Gauss Theorem" (perhaps by humility scientist don,t give his name or baptize it with other people name) but Gauss and Newton were very arrogant, in fact you will now the "Egregium theorem by Gauss" in latin egregium meaning supreme or best..so i don,t think he was very "humble" ...:) :) :) :)