You can provide definitions for irrational numbers, but they'll be expressed in terms of infinite serieses (what the hell is the plural form of series?).
So, if ever you try to expand them, you'll need to truncate at some point.
There will however, be situations where you can perform operations directly on the serieses, and in those cases (but not the ones requiring expansion and truncation) you'll be able to perform the operations losslessly.
(And anyway, the distance from any irrational number to the 'nearest' rational is always infinitessimal, so it's generally a moot point unless you're doing calculus.)
Series. It's from the Latin 5th declension (res, dies, species, etc.) for which the nominative singular and plural are the same, which carries over into English.
In Latin that was seldom a problem like in English, because these nouns were seldom used in the nominative case, i.e. as agents of verbs.
They were used much more frequently in other cases, like accusative (singular seriem, plural seriees).
In general the fact that English and many other languages borrow the Latin nouns in their nominative form, because of the rather useless tradition that this is their dictionary primary form, can be considered as a serious mistake, which frequently makes awkward the use of such words.
The termination "-s" does not have anything to do with the meaning of the word "series". It is just one of the most frequently used markers for the agent of a verb. Even if it is written without delimiting spaces after "serie", it does not belong to this word any more than for example the word "at" in "at home" would be considered to belong to "home". Like "home" can be encountered in various contexts, e.g. "at home", "in the home", "to the home", "for the home" and so on, "serie-" can be found in various contexts, e.g. "serie-es", "serie-m", "serie-ebus", "serie-ii" and so on.
What corresponds in Latin to an English word is the Latin word stem, without any of the terminations that may be attached to it. The Latin terminations correspond in English to either explicit prepositions or to implicit role markers provided by the fixed order of the English words.
When a Latin word is borrowed into English, it should have been borrowed as its bare stem, not together with its nominative termination. Moreover, for the nouns that denoted inanimate things, the nominative was rarely used and the accusative case was the main form. Even for animate but non-human beings the most important case was still the accusative. This is the reason why in the Romance languages most of their nouns correspond to the accusative forms of the Latin nouns and not with their nominative forms.
If the Latin nouns were borrowed as bare stems, there would never be conflicts between their singular nominative marker "-s" and the English plural marker "-s".
So, if ever you try to expand them, you'll need to truncate at some point.
There will however, be situations where you can perform operations directly on the serieses, and in those cases (but not the ones requiring expansion and truncation) you'll be able to perform the operations losslessly.
Of course, no one would ever bother with that - even NASA truncates pi to 3.141592653589793 (https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal...).
(And anyway, the distance from any irrational number to the 'nearest' rational is always infinitessimal, so it's generally a moot point unless you're doing calculus.)