Number system with an irrational base

[BenVitale]

Could you provide a link to a 'number system with an irrational base'?

I only found this link http://www.jstor.org/pss/3029218

The link shows a small part of this number system ... I would to know more about it.

 

[CRGreathouse]

There's not much to say -- they work just like number systems with rational bases.

 

[BenVitale]

Actually, I'm interested in the unusual bases, such as,

- Base 1
- Fibonacci base system
- Irrational bases: pi base, e base, Phi base

The Fibonacci base system is easy.

Base 1 : I haven't looked into it, yet.

Irrational bases
---------------
Bergman investigated irrational bases in 1957 [Bergman, G. "A Number System with an Irrational Base"] ... I don't have access to Bergman's article. Have you read it?

base pi and base e not so common - they are impractical.

phi is irrational and is solution to x^2 - x - 1 = 0
pi and e cannot be roots of a polynomial with integral cefficients.

This statement caught my attention:

It was shown e to be theoretically the most efficient base out of every possible base.

On Page 7/32

Source: http://www.artofproblemsolving.com/Resources/Papers/FracBase.pdf

 

[CRGreathouse]

For base 1, search for "unary"; you'll find a lot of things using it, though probably not too much discussing it directly (again, there's not much to say).

"Base efficiency" in that sense relates to expected length of representation times number of symbols (the per-symbol entropy, really, when we look at non-integer bases). It's not hard to do the calculation on your own here.

 

[HallsofIvy]

"Base 1" is easy: 1, 11, 111, 1111, 11111 are the numbers that, in base 10, would be called 1, 2, 3, 4, 5.

 

[arivero]

Harmonic basis is funny:

0+ a/2! + c/3! + d/4! + e/5! +

or something son. For each n, the coefficient must be an integer less than n.

 

[BenVitale]

Has anyone explored base 3/2 ?
Write in base 3/2 the numbers 1, 2, 3,...,10, ... 20,...