section_path: Calculations/Leap year calculations
section_name: Leap year calculations
paragraph_text: Leap year calculations
  
  To determine whether a Jewish year is a leap year, one must find its position in the 19-year Metonic cycle. This position is calculated by dividing the Jewish year number by 19 and finding the remainder. (Since there is no year 0, a remainder of 0 indicates that the year is year 19 of the cycle.) For example, the Jewish year 5785 divided by 19 results in a remainder of 9, indicating that it is year 9 of the Metonic cycle. The Jewish year used is the anno mundi year, in which the year of creation according to the Rabbinical Chronology (3761 BCE) is taken as year 1. Years 3, 6, 8, 11, 14, 17, and 19 of the Metonic cycle are leap years. The Hebrew mnemonic GUCHADZaT גוחאדז״ט refers to these years, while another memory aid refers to musical notation.
  
  Whether a year is a leap year can also be determined by a simple calculation (which also gives the fraction of a month by which the calendar is behind the seasons, useful for agricultural purposes). To determine whether year n of the calendar is a leap year, find the remainder on dividing by 19. If the remainder is 6 or less it is a leap year; if it is 7 or more it is not. For example, the remainder on dividing by 19 is 7, so the year 5785 is not a leap year. The remainder on dividing by 19 is 14, so the year 5786 is not a leap year. This works because as there are seven leap years in nineteen years the difference between the solar and lunar years increases by 7 ⁄ 19 month per year. When the difference goes above 18 ⁄ 19 month this signifies a leap year, and the difference is reduced by one month.
  
  The Hebrew calendar assumes that a month is uniformly of the length of an average synodic month, taken as exactly 29 + 13753 ⁄ 25920 days (about 29.530594 days, which is less than half a second from the modern scientific estimate); it also assumes that a tropical year is exactly 12 + 7 ⁄ 19 times that, i.e., about 365.2468 days. Thus it overestimates the length of the tropical year (365.2422 days) by 0.0046 days (about 7 minutes) per year, or about one day in 216 years. This error is less than the Julian years (365.2500 days) make (0.0078 days/year, or one day in 128 years), but much more than what the Gregorian years (365.2425 days/year) make (0.0003 days/year, or one day in 3333 years).