## Mathematics Throughout History## Math and Calendars in History |

## What was the problem with lunar-based calendars? |

Nothing is perfect, especially a lunar month. The biggest drawback with using a lunar calendar was the fractional number of days, which makes a calendar quickly go out of synch with the actual phases of the Moon. The first month would be off by about a half a day; the next month, a day; the next month, a day and a half; and so on. One way to help solve the problem was to alternate 30 and 29 day months, but this, too, eventually made the calendars go out of synch.

To compensate, certain cultures added (intercalations) or subtracted (extracalations) days from their calendar. For example, for more than a thousand years, the Muslims’ lunar calendar has had an intercalation of 11 extra days over a period of 30 years, with each year being 12 lunar months. This calendar is only out of sync about one day every 2,500 years: To see this, mathematically speaking, the average length of a month over a 30-year period is figured out in the following equation: (29.5 × 360) + 11 / 360 = 29.530556 days, in which 11 is the number of intercalated days, 360 is the number of months in a 30-year cycle (12 months × 30 years), and 29.5 is the average number of days in the calendar month, or (29 + 30) / 2.