Snyder's number

1. Mar 21, 2006

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.

Last edited: Mar 21, 2006
2. Mar 21, 2006

arildno

It is actually called Jimmy's snide Remark.

3. Mar 21, 2006

Jimmy Snyder

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

4. Mar 21, 2006

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 $$f(x)=\sum_{p}10^{-p}$$

Last edited: Mar 21, 2006
5. Mar 21, 2006

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 writen 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.

6. Mar 21, 2006

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).

7. Mar 22, 2006

Nimz

You mean one half of the base 2 Campernowne constant?

8. Mar 22, 2006

arildno

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

9. Mar 22, 2006

Jimmy Snyder

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

10. Mar 22, 2006

Jimmy Snyder

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

11. Mar 22, 2006

arildno

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

12. Mar 22, 2006

shmoe

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

Thanks for the correction.

13. Mar 22, 2006

Jimmy Snyder

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.

14. Mar 22, 2006

shmoe

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.

15. Mar 22, 2006

Jimmy Snyder

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.

16. Mar 24, 2006

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" ....:) :) :) :)