Can we expand numbers on the left to produce irrational numbers?

Question

I have seen examples of irrational numbers that are expanded on the right, after the decimal point:

e.g. $\pi = 3.14159265...$

But can we expand numbers on the left side as well?

e.g. Is $82757395734548...5473987349857348.0$ with digits bijective to $\mathbb{N}$ a number?

Syntactically, does an element of a free monoid with the set of generators being the decimal basis represent an irrational number or any number at all in decimal basis? Where can I find more about these numbers?

Answer 1

I suggest you look into p-adic numbers, which is the most well-known way to make sense of "numbers notated with an infinity of digits to the left". But it's not exactly what you're proposing here.

Your idea about monoids is a red herring, I'm afraid. The free monoid contains only finite words, which in this context encode the ordinary finite integers.

And notice that "3.14159265..." is not really syntax. The real decimal expansion of $\pi$ is an infinitary object that can't be written down in full, so "3.14159265..." is either just an evocative hint of the real thing or a practical approximation.

Answer 2

What you find is called a $p$-adic number.

http://en.wikipedia.org/wiki/P-adic_number