# Normal numbers

1. Mar 25, 2005

### Zurtex

I was just wondering if it was possible to prove anything about the normality of the number:

$$\sum_{x=0}^{\infty} \left((P(x) \mod b)\left(b^{-x}\right)\right)$$

Where P(x) is a Polynomial with integer coefficients and b is the base of decimal representation. Is anything even known for simple polynomials such as P(x) = x^2?

2. Mar 25, 2005

### mathwonk

that looks like a pretty normal number to me.

3. Mar 25, 2005

### Zurtex

Well thinking about it is fairly obvious that for P(x) = x^a it is normal to base b and I would imagine not too difficult to prove rigourously. For that matter P(x) = x^a + c would seem to always be normal to base b as well.

Hmm, just a matter of curiosity I suppose, I've always been interested in the normality of numbers since I first heard about it.

4. Mar 25, 2005

### Zurtex

Am I being stupid here?

Have I simply defined a rational number , anyone?

5. Mar 25, 2005

### shmoe

Looks that way. P(x) = P(x+k*b) mod b for all integers k, so you have a repeating decimal.

6. Mar 26, 2005

### Zurtex

Hmm, o.k fair enough, but lets suppose I start here in thinking about normal numbers. The way I've defined rational numbers here doesn't stray too far from being able to define all real numbers. So does anyone know if this defines all rational numbers or at least how I would start about proving if it does?

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