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You probably saw several different answers because there are several different ways to define the number of days in a year, e.g., a sidereal year or a tropical year. These two at least differ on the order of 10−3 10 − 3 days which is precisely what you reported. You need to pick a type of definition for the year first (and also a definition for a day), then figure out how many days are in ...
The Romans started using the 10-month calendar in 738 B.C. Their months were initially called: To account for the remaining ∼ 60 ∼ 60 days, Mensis Ianuarius (January) was added to the beginning of the year and Mensis Februarius (February) to the end of the year during Numa's reign around 700 B.C. with a leap year every 4 years.
The calendar year, however, by its very definition is a whole number of days; if it's the same time of day, then there must have been an integer number of days in between.
It is 365.2564 days. The Tropical Year is the length of time it takes for the Sun to complete a cycle around the Ecliptic and return to the position in the cycle of seasons; e.g. from Vernal Equinox to Vernal Equinox. It's about 365.24217 days, about 20 minutes shorter than the Sidereal year because of the precession of the equinoxes.
We then add the number of the current day of the month, and subtract 30. So what does N1 do? Well, lets look at the values returned compared to days in a non-leap year: A bit messy, right? Now, remember that you're subtracting 30 from the total at the end to get N, and we're adding in the current date.
The Modified Julian Date (MJD) is JD - 2400000.5, or the real number of days since 1858-11-17 0:00 UT. The IAU officially approves either JD or MJD as long it's clear which quantity is given. A number of 366 or less in a bureaucratic or informatics context is a day number or day of the year.
Originally a year would be noticed as the time between two winters (or two harvests, two annual floods etc), although the exact number of days might not be known with any accuracy. As people start to observe the sun more carefully, it would be noticed that the position of sunrise and sunset move regularly.
The average period over that time is about 522 days, so over 4.5 billion years (with those years being our current definition of year ) that's about 3.1 billion orbits. But it's a pretty useless number, I think. It's worth noting that we could not accurately simulate such a large number of orbits with any accuracy.
The 7 days of the week certainly fit somewhat as a quarter of a month. But there was another argument for the ancients to use the number seven, because they knew about seven "planets", seven objects which wander across the sky relative to the thousands of fixed stars. The Sun, the Moon, Mercury, Venus, Mars, Jupiter, Saturn.
@PM 2Ring: I checked this site, but it seems that it is defining various types of one year based on number of days - with some variations. My question is how did we calculated the number of days it takes to come to the same starting position. However, I will read the site text thoroughly, and come back in 24 hours. Please do check.