Various entries in the books write the date with a series of digits followed by M41.

Example: "853779.M41" on page 8 or 9 of the Achilus Assault. What date is that?

I'm guessing it's the 779th year of the 41st millennium? So year 41,779? Or do they mean 40,779? And what does the 853 mean?

Your guess is correct!

The numerals before that are "check numbers" that indicate the reliability of the message, depending both on the distance to Terra (where the event will eventually be logged) as well as the degree of "directness", meaning by how many astropathic relays it has passed.

The 6E rulebook had a lengthy explanation of this in its fluff appendix.

Hmmm, why is it not included in Core? Weird. Anyway, the first digit is the check number, which is often omitted. The next 3 digits are the year fraction, and the last three digits are indicating the year itself. Your example, 853779.M41 is around the end of year 779 of the 41st millennium, that is 40,779 (millennia work the same way as centuries, the first goes from 0001-1000, presuming they kept our Common Era where there is no year 0 - if there is, the turning point is a year earlier, the millennium spanning between 0000-0999).

Right!

And here's the picture from the TT book:

In more detail.

The check digit, as noted, is a measure of distance from Terra and/or accuracy (all dates are Terran standard, so distance and accuracy are the same thing). 0 or 1 for the check digit means the recorded event takes place on Terra itself, and is thus as accurate as possible. 7+ tends to cover approximate and estimated dates rather than reported events.

The year fraction - noted as the Chronosegment in the 6e 40k rulebook - is a thousandth of a Terran standard year. Consequently, each segment is roughly 8 hours and 45 minutes long. It's fair to assume that nobody except the Administratum actually uses that value for practical purposes - it's a record-keeping tool. That said, I can imagine some worlds using it for work cycles - each segment is a work shift, each person gets a work shift, a sleep shift, and a shift for leisure and prayer.

I'd imagine segmenting the year in such a way would be a godsend (sorry Emperor send) if you had a spread sheet with literally billions of entries from all over the galaxy and you were told to put them in chronological order. That said travel and communication is so slow relatively speaking in the Imperium that I doubt such precise dates matter much to generals and other tactical commanders.

When Gming I use a convention of using the official time at the beginning of a mission but then also a 'local' time amd date stamp. Well I use this for relevant handouts anyway.

thanks all, makes more sense now.

I'd imagine segmenting the year in such a way would be a godsend (sorry Emperor send) if you had a spread sheet with literally billions of entries from all over the galaxy and you were told to put them in chronological order. That said travel and communication is so slow relatively speaking in the Imperium that I doubt such precise dates matter much to generals and other tactical commanders.

When Gming I use a convention of using the official time at the beginning of a mission but then also a 'local' time amd date stamp. Well I use this for relevant handouts anyway.

Exactly so. When you're talking about tracking events on an interstellar scale minutes become largely irrelevant anyway.

one year fraction 'tick' is about nine hours in practice, which makes it a nice "later that day" value.