Wikipedia:Reference desk/Archives/Mathematics/2019 January 7

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January 7

What's the most ambiguous decimal year? (UTC)

That is, the moment(s) of each 400 year cycle which give the most disparate results depending on whether the decimal of times of the form 2019.06249 AD is interpreted as "percent of that calendar year (365 or 366 days) since the last January 1.0" or "percent of 365.2425 days since the last average January 1.0".

What's the most ambiguous where the other interpretation happens first? Sagittarian Milky Way (talk) 00:11, 7 January 2019 (UTC)

The simple way to approach this is to consider the time of the vernal equinox. In year 303 of the cycle it occurs latest compared to the mean calendar year of 365.2425 days (e.g. at 19:15 (GMT) on 21 March 1903). By year 96 it has regressed to e.g. 14:02 (GMT) on 19 March 2096. During the remainder of the cycle it oscillates between these extremes. With five decimal places (as in your question) the difference between GMT and UTC becomes significant - the question becomes much more difficult to answer in terms of UTC because the placement of leap seconds in UTC is, to all intents and purposes, random. 2A00:23C0:7903:B200:BDFD:FD93:24C1:D5C6 (talk) 14:13, 7 January 2019 (UTC)

So if I set Jan 1.0 1901 as meaningless bookkeeping 0, Jan 1.0 1902 is 0.2425 days early, Jan 1.0 1903 is 0.485 days early and Jan 1.0 1904 is 0.7275 days early, for an average of 0.36375 days early. Jan 1.0 1905 is 0.03 days late so the next set of 4 will end 1908 and average 0.33375 days early. There are 200/4=50 sets of 4 between Jan 1.0 1901 and Jan 1.0 2100 inclusive so the average of those 200 year starts should be the average of 0.36375-0.03*24 and 0.36375-0.03*25 days early or 0.37125 days late. Jan 1.0 2101 should be 0-0.03*50-1=0.5 days late so the next 200 years should be 0.37125+0.5=0.87125 days late except 2201-2300 have an extra day cut off so the 400 year average is (not surprisingly) still 0.37125 days after Jan 1.0 1901. Jan 1.0 1904 would then be 1904-0.0030082753239 the complicated way. Days after that would be divided by 366 = >365.2425 the simple way since 1904 is a leap year so more days wouldn't help. Jan 1.0 2097 is 49 normal Olympiads or 71589 days after Jan 1.0 1901 or 0.0040247233002-0.37125/365.2425=0.003008275323(~8) years after average (2097.003008275324). The simple way makes a 2096 AD day worth less years than average which doesn't help so if treating days as GMT (I forgot about leap seconds) the answer is 1697.0, 1904.0, 2097.0, 2304.0 and so on. If it's 365.2425*86,400 SI seconds of unchanging time vs. GMT calendar days then 2497.0 is most ambiguous for ~4 centuries, then 2897.0 and so on. That was easier than I thought.. Sagittarian Milky Way (talk) 03:13, 9 January 2019 (UTC)