# Non-integral bases?

1. Aug 6, 2005

### abhishek

Apologies if this is in the wrong forum.

In a discussion we're having on our compsci forums at uni, about binary numbers, someone brought up the notion of non-integral bases. I take it to mean numbers with bases that aren't integers. Is this right?

I've never encountered such a thing before, so I'm curious about it. I've only seen bases defined as integers - is it valid to think of non-integral bases? Do the same processes for understanding and converting integer bases apply to non-integers?

I have no specific questions, really. How does it work? How is it meaningful? What are some applications of it? Links to further information would be nice.

Searching the web turns up almost no information about this, so I wonder if it goes by any other names too.

Thanks to anyone who can shed some light on the matter.

2. Aug 7, 2005

### AKG

No, I believe it's problematic. Imagine taking base $\pi$. Then $\pi$ would have a finite $\pi$-ary expansion. What's worse is that it doesn't make sense to use decimals in this case. When you have a base n, then the digits in your expansion can take values from {0, 1, .., n-1}. What if you have base $\pi$? There's no reasonable choice.

3. Aug 7, 2005

### lurflurf

This can be done.
call our base b
1234(base b)=1*b^3+2*b^2+3*b+a
as usual digits are chosen as integers such that 0<=digit<=b
any positive real except one can be used as a base.
The two main problems that arise are
1) accuracy can become an issue and rational numbers have not reapeating decimal expansions.
2) non uniqueness. This has to do with algebraic bases.
2=10=1.01010101010101010101... (base sqrt(2))
4=10000=100=11.010000010010... (base sqrt(2))
here are some things in base pi
pi=10 (base pi)
e=2.2021201002111122001 (base pi)
17=120.2200211010202300

4. Aug 7, 2005

### lurflurf

pi=10 (base pi) is not a problem
Even though the base is nonintegral the digits are still integers.

5. Aug 24, 2005

### serge

but the concept raises an interesting question : what can you say about a number wich has a periodic development in base pi ?
such as a = 1.0101010101(base pi) = pi+pi^3+pi^5+...
it is tempting to name those numbers "rationals in base pi"
However, they look like p-adic numbers, because of the infinite right part.
They also look like elements of an hilbert space, or infinite polynomials in pi

what about addition, multiplication, ... of those numbers ? is there a closed operation ?