Is it possible to convert irrational p-adic numbers to a standard number? Rationals and negative rationals are relatively straightforward, but is there a way to know that for instance $\ldots 2100121201\_{p=3} = 10.0111010220\ldots\_{3}$?



Converting p-adic to decimal
If by "standard" you mean "real", it is not possible : $p$-adic numbers and real ones are really different. By the way, what do you really mean by the "decimal" $10.0111010220...\_{3}$ ?


$\mathbf{R}$ i s a completion of $\mathbf{Q}$ for a precise metric, the usual one. $\mathbf{Q}\_p$ is the completion for an other metric, the $p$-adic one. Those metrics give deeply different topologies, hence different behavior. Example : reals can have two decimal developments, $p$-adic numbers not ; $p$-adic have ultrametricity property, reals not ; etc. It is for that it is interesting to consider both real numbers and $p$-adic numbers. They are all the different ways to "do analysis" on $\mathbf{Q}$.


[Miller's notes](http://www.osti.gov/eprints/topicpages/documents/record/173/1305544.html) give a swift introduction and good references to $p$-adics. A very lovely introductory and intuitive text is the one of [Alain Robert](http://www.sens-neuchatel.ch/bulletin/no18/art18-2-robert.pdf), if you read french -- otherwise, explore his bibliography.


