Note that Adar II is the "real" Adar, the one in which Purim is celebrated, the one in which yahrzeits for Adar are observed, the one in which a 13-year-old born in Adar becomes a Bar Mitzvah. In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations.This calendar, still in use, standardized the length of months and the addition of months over the course of a 19 year cycle, so that the lunar calendar realigns with the solar years.In ancient times, this month was added by observation: the Sanhedrin observed the conditions of the weather, the crops and the livestock, and if these were not sufficiently advanced to be considered "spring," then the Sanhedrin inserted an additional month into the calendar to make sure that Pesach (Passover) would occur in the spring (it is, after all, referred to in the Torah as Chag he-Aviv, the Festival of Spring! A year with 13 months is referred to in Hebrew as Shanah Me'uberet (pronounced shah-NAH meh-oo-BEH-reht), literally: a pregnant year. The additional month is known as Adar I, Adar Rishon (first Adar) or Adar Alef (the Hebrew letter Alef being the numeral "1" in Hebrew).The extra month is inserted before the regular month of Adar (known in such years as Adar II, Adar Sheini or Adar Beit).After many years of blissful ignorance, I finally sat down and worked out the mathematics involved, and I have added a page on The Jewish Calendar: A Closer Look, which may be of interest to those who want a deeper understanding or who want to write a Jewish calendar computer program.
The problem with strictly lunar calendars is that there are approximately 12.4 lunar months in every solar year, so a 12-month lunar calendar is about 11 days shorter than a solar year and a 13-month lunar is about 19 longer than a solar year.
To compensate for this drift, the Jewish calendar uses a 12-month lunar calendar with an extra month occasionally added.
The month of Nissan occurs 11 days earlier each year for two or three years, and then jumps forward 30 days, balancing out the drift.
In addition, Yom Kippur should not fall adjacent to Shabbat, because this would cause difficulties in coordinating the fast with Shabbat, and Hoshanah Rabbah should not fall on Saturday because it would interfere with the holiday's observances. (Before the Common Era), which are commonly used by scholars today.
A day is added to the month of Cheshvan or subtracted from the month of Kislev of the previous year to prevent these things from happening. The "first month" of the Jewish calendar is the month of Nissan, in the spring, when Passover occurs.
The year number on the Jewish calendar represents the number of years since creation, calculated by adding up the ages of people in the Bible back to the time of creation. D." means "the year of our L-rd," and we do not believe Jesus is the L-rd. The American "new year" starts in January, but the new "school year" starts in September, and many businesses have "fiscal years" that start at various times of the year.