Today’s Abstruse Goose comic got me thinking:
Does an “infinite palindromic” number (other than the obvious $x\times1.\overline1$) make sense?
In any conventional number system, the answer (as far as I know) is “no”. But on the other hand, I can trivially construct a context free grammar to generate such a sequence:
\begin{align} \color{gray}{P} \rightarrow& 0\ \color{gray}{P}\ 0 \\\\ |\ & 1\ \color{gray}{P}\ 1 \end{align}
This generates an infinite palindromic sequence of $0$s and $1$s. My question: is there a number system which allows me to do calculations with such a sequence?
That is, have we got a number system which tells me the result of $k + k$, $k = 101{…}101$ (sounds simple: $202{…}202$ … but what about $10 \times k$?) or that can solve equations such as $x^2 = k$?