Hebrew Calendar

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For the Civil calendar a new day begins at midnight and extends for twenty-four hours.  In the Jewish calendar, the day begins and ends at sunset, from Genisis 1:5
  ‘And there was evening and there was morning, one day.’
According to the Jewish calendar a person born after sunset on Monday night, January 1, 2001 is considered to have been born on Tuesday, January 2. His Jewish birthday is Tuesday, 7 Tevet 5761, which corresponds to January 2, 2001.
By the same token, according to the Jewish calendar one who died on Monday night, January 1, 2001 is considered to have died on Tuesday, 7 Tevet 5761. In future years the Yahrzeit (anniversary of death) of that person must always be observed on the Jewish date: 7 Tevet.

The following months in the Jewish calendar always
have twenty-nine days:
Iyar, Tamuz, Elul, Tevet, Adar,

The following months always have thirty days: Nisan,
Sivan, Av, Tishri, Shevat.

Heshvan and Kislev are "swing" months. They may have twenty-nine or thirty days, the number depending on whatever adjustments are required.  Heshvan is sometimes called Mar Heshvan (mar means "bitter, sorrowful") because no Jewish holidays fall in this month.

In a leap year (which occurs seven times every nineteen years), a second Adar is added to the calendar, making a total of thirteen months in the year. In a common (non-leap) year, Adar has twenty-nine days; in a leap year, Adar I has thirty days and
Adar II has twenty nine days.

Following are the holidays of the Jewish year, broken down month by month. An asterisk * after the name of the holiday indicates that it is of biblical origin. The date it commences and the number of days it is to be observed are stated in the Bible.

It is important to note that whereas Israeli Jews observe all biblical holidays for the number of days indicated in the Bible, in the Diaspora, only Reform and some Conservative congregations follow this practice. Most Conservative and all
Orthodox congregations in the Diaspora observe Passover, Shavuot, and Sukkot for an extra day because of the uncertainty of the calendar in early times.

Tishri     (month 7, year number increased)

Rosh Hashana,* on the first day of the month, for one day  (but the holiday is observed for two days both in the Diaspora and in Israel.)
Fast of Gedaliah, on the third day, for one day.
Yom Kippur,* on the tenth day, for one day.
Sukkot,* on the fifteenth day, for seven days.
Hoshana Rabba,* on the twenty-second day, for one day. (Reform)
Shemini Atzeret,* on the twenty-second day, for one day.
Simchat Tora, on the twenty-second day, for one day. (Reform)

Heshvan    (month 8)

No holidays

Kislev     (month 9)

Hanukah, on the twenty-fifth day, for eight days.

Tevet     (month 10)

Asarah B'Tevet, on the tenth day, for one day.

Shevat     (month 11)

Tu B'Shevat, on the fifteenth day, for one
day.

Adar         (month 12, Adar II month13)

Fast of Esther  on the thirteenth day, for one day.
Purim,* on the fourteenth day, for one day.
Shushan Purim,* on the fifteenth day, for one day.

Nisan     (month 1)

 Passover,* on the fifteenth day of the month, for seven
 days.

Iyar        (month 2)

Lag B’Omer, on the eighteenth day, for one day.

Sivan     (month 3)

Shavuot,* on the sixth day. for one day.

Tamuz     (month 4)

Shiva Ascar B'Tammuz. on the seventeenth day. for one
day.

Av            (month 5)

Tisha B'Av, on the ninth day, for one day.

Elul           (month 6)

No holidays.
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