Stardate (glossary term)
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It was also used by fans to date their zines and letters, in much the same way they used Live Long and Prosper as a salutation
Now for the proper way to write Stardate. In my opinion it's any which way you like. Roddenberry felt we surely wouldn't still be using the current dating methods 200 years from now, and then there would have been the problems of integrating other cultures and planets into the Federation and StarFleet  who undoubtedly would have had dating methods of their own that varied with ours—which would somehow have had to be reconciled. He also didn't want to pinpoint any exact year for his ST adventures of the future, so to solve all his problems he hit upon using a made-up system of dating, which he called Stardate. It was just a quick off-the-top-of-his-head decision to avoid problems when he came up with them-- he didn't "work out" any "real" way of making it work consistently or to have any "real" meaning or "real" way to figure certain dates. Those fans who have researched it find that in the first few shows, the stardates assigned to them (in the Captain's log, for instance) jump around at random and follow no pattern, chronologically or otherwise--he, and then the writers, were just picking numbers and throwing stardates in to make it sound good. But fans have to work out a reason for everything in the show, a way it could work, or the reason why, or how it is possible--for every aspect of the show. So they came up with s quick way to write Stardates that could be used by themselves today and would reflect the current date--which was to put year-month-date thusly 7610.15 ...at least this is the way most fans do it, a few vary it in other ways. Apparently it makes them feel very much a part of Star Trek to use it for dating things-- I myself, like you, get so confused by it that I long ago gave up and went back to old-fashioned dating on my letters. But you do need to know how fans are using it if you wish to know the date they are using. So really both answers you had received are right sofar as I can see!
- "StarFleet" is Helen Young's spelling, which illustrates that the term wasn't entirely normalized in spelling at this time.
- referring to the date on this letter