# Irrationality of a real number $b$ that is related to a real number $a$ by the umbral Taylor series.

Given the sequence $\{a_0,a_1,...a_x\}$ then we may represent $a_n=\sum_{i=0}^{\infty} \Delta^i (0) {n \choose i}$. Where $\Delta^i(0)$ represents the operation mapping $a_n$ to $a_{n+1}-a_n$ iterated $i$ times then evaluated at $0$.

Now suppose $a_{n}$ represents the $n+1$ th digit of the decimal representation of a number $a$ in base $10$. I am interested in the coefficients $\Delta^i(0)$ as they also form a sequence $\{a_0=b_0,\Delta^1(0)=b_1,\Delta^2(0)=b_3,...\}$. And from this sequence (disregarding signs) we may form a new number:

$$b=|b_0|.|b_1||b_2||b_3|.....$$

For example $a=\pi$ gives the sequence $\{3,-2,5,-11,24,-44,60,...\}$ and $b=3.2511244460...$.

I'm interested in the irrationality of $b$. Is $b$ irrational or not? I suppose that $b$ is irrational if and only if $a$ is irrational. I suppose if $a$ is irrational then the $b_n$ is an infinite sequence, but I'm not not sure how to prove this infinite sequence does not have repeating digits. Furthermore is that even enough.

• Irritated operators... will they come and kill me ? Iterated is better – Gabriel Romon Jul 7 '16 at 13:07
• Thanks, I can't spell @LeGrandDODOM – Ahmed S. Attaalla Jul 7 '16 at 13:08