Is it possible to have a number that extends to the left of the decimal point in mirror image of an irrational number? Such as <...95141.30000...>, to write pi as a mirror image.
No. Why? Let us assume that we don't claim that the result is a 'number,' as it's infinite (problematic). We could make this more precise, saying that we want some sort of way of placing a 'mirror' in the middle of $\pi$ s.t. we have something like $ ... \alpha _3 \alpha _2 \alpha _ 1 | \beta _1 \beta _2 \beta _3 ... $ and such that $\alpha _i = \beta _i$.
But even this is not very meaningful, and you included the key problem in your question. Eventually, you will have an infinite string of zeroes on one side - but not possible on the other side (as the expansion of pi does not terminate).
I wrote that knowing that it's hardly sensible to describe, not because I enjoy fancying things that are awkward (which may or may not be true), but because there is an interesting related fact. For any finite number $k$, there exists a place to put the 'mirror' in the expansion of $\pi$ such that $\alpha _i = \beta _i \quad \forall \; i \in [0, k]$. And that's pretty cool, and even related to the question.
protected by Qiaochu Yuan Sep 23 '11 at 16:36
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