Number system with an irrational base

  • Thread starter BenVitale
  • Start date
  • #1
70
1
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.
 
Last edited by a moderator:

Answers and Replies

  • #2
CRGreathouse
Science Advisor
Homework Helper
2,820
0
There's not much to say -- they work just like number systems with rational bases.
 
  • #3
70
1
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 [Broken]
 
Last edited by a moderator:
  • #4
CRGreathouse
Science Advisor
Homework Helper
2,820
0
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.
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
41,833
961
"Base 1" is easy: 1, 11, 111, 1111, 11111 are the numbers that, in base 10, would be called 1, 2, 3, 4, 5.
 
  • #6
arivero
Gold Member
3,320
69
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.
 
  • #7
70
1
Has anyone explored base 3/2 ?
Write in base 3/2 the numbers 1, 2, 3,...,10, ... 20,...
 

Related Threads on Number system with an irrational base

  • Last Post
Replies
9
Views
3K
  • Last Post
Replies
4
Views
3K
  • Last Post
3
Replies
73
Views
17K
Replies
2
Views
2K
  • Last Post
Replies
3
Views
3K
  • Last Post
3
Replies
59
Views
15K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
8
Views
4K
  • Last Post
5
Replies
124
Views
23K
Replies
6
Views
7K
Top