All numbers can be represented accurately with an infinite string of digits, whether they are rational or irrational. This seems trivial for rational numbers and especially for "round" numbers, but it's easy to be confused by writing conventions and the coincidence of numeric base that we use. In base-10, we omit zeroes, so that 1.5 is really 1.50 with repeating zeroes to infinity. There's no information in the infinite string of 0's so we can omit them and still accurately represent the number (we compress it by writing the repeating parts in shorthand.) As for "not round" rational numbers, the vast majority of their patterns will not be immediately obvious to humans, given our limited pattern recognition limitations and the numeric base that we use. In base-10, the ratio that produces 0.142857-repeating is not obviously 1/7, but in base-7 it is - because in base-7, it's 0.1.
Some numbers have been proven to be irrational. The earliest of these for which we still have a record was Euclid's reductio ad absurdum for the square root of 2. The same has been done for pi and e, among other constants; but as yet, there is no generalized method for proving a number's irrationality.
A proof that there can be no general method for proving a number's irrationality, or at least whether the irrationality of some numbers may never be proven, would be worth having. Using a very unorthodox practical proof: there are finite particles in the universe with which to compute these numbers, and they will exist for a finite time; not nearly long enough to run through all the operations to produce ratios even for all the (infinite) irrational numbers between 0 and 1, regardless what those operations are.
What is interesting about this problem is that ultimately, proving rationality means recognizing some periodicity (or for irrationality its absence) and therefore that more generally proving ir/rationality is a pattern recognition problem. Other problems which are essentially pattern recognition problems are Kolmogorov complexity, i.e. compressibility, which although archiving applications do it all the time is in an absolute sense non-computable; and tiling problems, infinite (plane-covering) solutions of which are famously shown to be undecidable in principle by any algorithms. Are there other properties that these non-computable pattern-recognition problems have in common?
Modality is a bitch (updated)
10 hours ago