# Base 10 and Base b

1. Sep 19, 2008

### Diffy

Question:
Decimal (10-nary) expansions of real numbers were defined by special reference to the number 10. Show that the real numbers have b-nary expansions with analogous properties, where b is any integer greater than 1.

Attempt at solution:
I think if I show that there is a bijective function between the real numbers base ten, and any other base that will show they have analogous properties.

so let a0.a1a2... be any real number, where a0is any integer and ai i >0 and i /in {0,1,2,...,9}.

Then it has been shown (in the book) that this can be represented as
a0 + a1/10 + a2/102 + ...

I think now I need to show that this number can be changed into base b which I am not quite sure how to do. And even once I have done that, I am not sure that I am any closer to solving the problem.

Any help is appreciated.

Thanks

-Dif

2. Sep 19, 2008

### mathman

The above proof for 10 carries over to base b with the following changes.
1) range of a's is (0,b-1).
2) denominators are powers of b rather than powers of 10.