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Imagine a civilization which has surmounted almost all of the limitations of biology and the universe. They have become truly immortal, put behind them all their petty differences, and dug down into the core of the laws of physics and somehow figured out how to stop entropy and product free energy. Now they must face their biggest challenge: time notation.

For the first 100,000 or so years, it's fairly easy to just keep up with the current form of time notation. Just update the code of apps to support 5- and 6-digit years at the 10,000- and 100,000-year mark.

But as the millennia turn to eons and longer, eventually writing down the full year becomes somewhat infeasible, and our civilization realizes that it must come up with a way to cope with year numbers increasingly becoming longer.

Eventually, the year passes 101000, 10100000 and more, eventually reaching into the millions and billions of digits, becoming incredibly difficult to log.

Our civilization wants to keep a well-detailed history and doesn't want to resort to things like only storing the last four or five digits of the year and just relying on people to know what era something occurs in by context.

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There isn't really a problem with using very long numbers. Even the feeble computers of nowadays can manage that. The problem is that human minds find it difficult to handle very large numbers themselves, which is why we use scientific notation and similar systems. Now, if you're assuming that your aliens have the same kind of mental limitations as humans -- which is not unreasonable at the start of their civilisation -- and retain that limitation for at least 10^6 years, which seems rather doubtful to me, then it's worth trying to solve the problem.

The usual solution to this problem is to use some kind of era system, because humans find it easier to deal with lists of names, or of short numbers, than with one very big number. For a somewhat weirder system, look up the Mayan Long Count Calendar, which is real, and has nothing to do with apocalypses. You might also care to look at Tumbler Arithmetic, which was formulated for an index-all-the-worlds's-information project.

So, how do you build an era system methodically, rather than have it evolve out of a reigns-of-kings system? Start with the basic unit, of maybe 1000 years. If you specify a date with just a year number, it's in the current era.

Then you need a naming system for blocks of 1000 years, where you need names that have an easily memorable sequence. Those tend to be culturally specific: as English-speakers, we might use A-Z, and maybe drop a letter to give 25, rather than 26, but humans who use other writing systems would find this difficult, and translating it into their own alphabet drags in the problem of collating sequence translation between different alphabets, and you really don't want to go there.

Then, given you have groups of 25,000 years, or maybe up to 100,000 years depending on what you use to name them, you can use numbers for the groups, say up to 1000 of them in an era. You have year numbers within the era of the form 207 D 778, which may be easier than 5129778, but the difference doesn't seem huge to me.

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Rollover. Simply after hitting 100000, start over from 1. Call them cycles and number each cycles . Normally, in an cycles people will talk about that cycle and for a short time previous one. No one will mistake year 99992, they will all understand that it is the previous cycle. Now for the record keeping. Make your records structured so that you need to set the cycle once in a system and you keep using the year only. If necessary, you could save it as 99992c-1 to refer to previous cycle.

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Like many questions which involve the breaking of laws of physics (in this case, the march of entropy), there are many solutions. Don't restrict yourself to any one!

One conclusion may be that, with free energy, they have all the time in the world to read off Stardate 15984843515543438181324576751519186.5213846857468461558 and realize that that's just 2 days before a great festival on that planet. Time is going to mean something different to people in a post-entropy world.

A well detailed history may be a term that needs further expounding. It would be horrible to have a document with 1000000 digit dates, followed by an entry "I had soup today." It's a wasteful approach.

Despite your goal of not resorting to context, it really is the answer you are looking for. If you divide time into slices (like years), the number of bits you need to describe a year exactly is... well... the number of bits you need to describe a year exactly. You can't cut corners. Mathematics prevents it. You have to permit some level of imprecision, use context, or deal with writing out enough bits of data.

One solution might be to weave that context into the document itself. Perhaps any given event only has a partial date, but the cataloguers which made the history took great care to ensure that anyone who wants a more precise date can determine it by cross referencing other related events. This would permit clever compression of the date data by spreading it across multiple entries in the history.

Once you have some form of context, implied or explicitly woven, to let you determine the exact date, the recorded dates near the entries could have less precision. Given your use of exponential notation like 10^1000 and 10^1000 already, such notation might be very effective. It might be recorded as years before some major event which serves as a meaningful epoch for that particular volume of history.

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Really, this is impossible to answer. Either you write the full number, or the last digits so it's clear in context, or you round, but you can't have the full number and at the same time not the full number.

Using large-base systems helps, say a base 62 system using alphanumeric symbols. But then you don't get around writing the whole number down. There is no shorter way. I'm pretty sure you can proof that for a set of symbols the shortest (in terms of symbols used) way to represent numbers is just using the symbols as digits.

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