The Hebrew calendar (Hebrew: הַלּוּחַ הָעִבְרִי, *HaLu'aḥ Ha'Ivri*), also known as the Jewish calendar, is a unique and mathematically sophisticated system. Unlike the purely solar Gregorian calendar or the purely lunar Hijri calendar, the Hebrew calendar is luni-solar: it harmonizes the cycles of both the Moon (for the months) and the Sun (for the years and seasons).

This calendar is used globally by the Jewish people to determine the dates of all religious observances—from the weekly Sabbath to the annual festivals of Passover, Yom Kippur, and Rosh Hashanah.

Part I: Historical and Religious Epoch

The Hebrew calendar's foundation is rooted in the biblical narrative and an ancient scholarly tradition.

1. The Epoch: Anno Mundi (AM)

The numbering of the Hebrew year is based on the traditional rabbinic calculation of the time elapsed since the Creation of the World.

• Starting Point: The calendar begins at the year of creation, known as Anno Mundi (AM), or "Year of the World." This epoch is traditionally set at 3761 BCE.
• Significance: This reckoning connects the passage of time directly to the foundational text of Judaism, serving as a constant reminder of the spiritual age of the world.
• The Day's Start: A fundamental difference from the secular day is that a Hebrew day begins at sunset (the start of the evening), following the biblical description: "There was evening and there was morning, one day."

2. From Observation to Fixed Calculation

For centuries, the calendar was set by physical observation in ancient Israel:

• Months: The beginning of each new month was announced by the Sanhedrin (the Jewish High Court) only after witnesses physically sighted the thin crescent of the New Moon.
• Years (Intercalation): The decision to add a "leap month" was based on observing agricultural conditions (e.g., the ripening of the barley for the Passover sacrifice) to ensure the spring holiday of Passover always fell in the correct season.

This observational system ended in the 4th century CE. To ensure the continuity of Jewish practice following persecutions and the dispersion of the Jewish people, the sage Hillel II established a fixed, calculated calendar based on precise mathematical rules, which remains in use today.

Part II: The Mathematical Core of the Luni-Solar System

The genius of the Hebrew calendar lies in its mechanism for solving the major conflict between the lunar and solar years.

1. The Month Cycle: *Ḥalaqim* and the *Molad*

The foundation of the calendar is the precise length of the mean lunation (the time from one astronomical New Moon to the next).

• Unit of Time: The system uses the ancient unit of time called the ḥelek (plural: *ḥalaqim*), which is equal to 3 1/3 seconds. There are 1,080 *ḥalaqim* in one hour.
• Mean Lunation: The average time between new moons (*Molad*) is precisely defined as: 29 days, 12 hours, and 793 *ḥalaqim*.
• Month Lengths: Hebrew months alternate between 29 days and 30 days to approximate the 29.5-day lunar cycle.

2. The 19-Year Intercalation Cycle (Metonic Cycle)

To prevent the calendar from drifting away from the seasons, a leap month is added periodically. This is based on the ancient Metonic Cycle.

• The Problem: A 12-month lunar year (approx. 354 days) is about 11 days shorter than the solar year (approx. 365 days). Over time, this difference would cause spring festivals like Passover to eventually fall in the winter.
• The Solution: The calendar uses a 19-year cycle in which seven leap years are inserted. This means that 12 years have 12 months, and 7 years have 13 months.
• The Leap Years: The extra (30-day) month, Adar II (*V'Adar*), is added in the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of the cycle. This precisely balances the difference, ensuring the holidays always remain in the correct season.

3. The *Dechiyot* (Postponement Rules)

The most complex mathematical layer involves the four Postponement Rules (*Dechiyot*). These are rules designed to prevent the Jewish New Year (Rosh Hashanah, which falls on the first day of the month of Tishrei) from falling on certain days of the week, which would cause certain holidays (like Yom Kippur or Hoshana Rabbah) to conflict with the Sabbath.

The *Dechiyot* require the New Year to be postponed by one or two days if the *Molad* falls on a Sunday, Wednesday, or Friday. These rules ensure social harmony and religious observance are maintained throughout the annual cycle.

Part III: Year Types and Conversion

The Hebrew calendar has an exceptional number of variable year lengths due to the *Dechiyot* and the leap month system.

Conversion between the Hebrew calendar and the Gregorian calendar is complex because of these annual variations and the difference in the start of the day.

While the 19-year Metonic cycle ensures that the two calendars align closely every 19 years (meaning a Jewish birthday will often fall on the same secular date as the 19th birthday), the *Dechiyot* and the daily time difference mean the coincidence is not guaranteed.