It takes the Earth an average of 365.242189 days to orbit the Sun. The orbit changes slightly from year to year, so the length of time to complete the orbit is not exactly the same each year. Astronomers call one orbit a “tropical year” or “astronomical year.” If you want to get technical about a year, you have decide on the orbital reference point and the exact angle of measurement.https://www.public.asu.edu/~mjwhite/Sidereal%20and%20tropical%20years.pdf

**The Most Common Calendar**

Solar calendars use the Earth and Sun (not the moon, seasons, or a fixed number of days) to determine the length of a year. Most countries, including the United States, use the Gregorian calendar to measure the unit of time known as a year. https://www.timeanddate.com/calendar/gregorian-calendar.html The Gregorian calendar is a solar calendar.

In 1582, Pope Gregory XII introduced the Gregorian calendar as an updated version of the Julian calendar. Eastern Orthodox Christian religions still use the Julian calendar to mark religious holidays. The Julian calendar had eleven minutes too many in it. The extra eleven minutes caused some problems. By the 1570s, the Julian calendar was out of sync with the astronomical year by ten days.

The Gregorian calendar does not perfectly align with an astronomical year, either. It is off by approximately 26 seconds! By the year 4909, the Gregorian calendar will be off by one day. https://www.history.com/news/6-things-you-may-not-know-about-the-gregorian-calendar

**Use Math to Decide if the Gregorian Calendar is the best Calendar**

The months in solar calendars aren’t tied to an astronomical cycle. People could group the 365.242189 days into many different variations. The Gregorian calendar groups the days into 12 months. Is this the best grouping? Let’s investigate.

Seven months have 31 days, 7 x 31= 217 days.

Four months have 30 days, 4 x 30= 120 days.

One month has 28 days most years and 29 days on leap years, 1 x 28 =28 and 1 x 29 = 29

217+120+28= 365 days and 217+120+29= 366 days in a leap year

You may wonder why not just make every month the same length. 365 days is not divisible by twelve. If the twelve months had exactly equal number of days every year, each month would be 30.43 days. Partial days in a month would get a little confusing.

There are several solutions to the problem of evenly dividing 365-day year into months.

1) Have five months with seventy-three days each. Three hundred sixty-five is divisible by five and seventy-three.

2) Have months of varying lengths such as in the current solution.

3) Have thirteen months of twenty-eight days and one extra day not in a month. 365-1=364, 364/13 = 28 To compensate for the extra day (or two days in leap years), make summer solstice a special day.

This last solution is called the “International Fixed Calendar” and was used by Kodak from 1928-1989.

**Using the Leap Year to Correct for the Fraction of the Day per Year**

To offset the fraction of the day per year, the Julian calendar introduced a leap year concept. Every four years, the calendar had one extra day calendar. (4 years x .25 day = 1 whole day).

Remember the extra eleven minutes in the Julian calendar? They had calculated the extra fraction of the day as .25 instead of .242189, which accounts for the extra eleven minutes. The Gregorian calendar corrected this eleven minutes by using fewer leap years. To be a leap year, the year number must meet three requirements.

1) Divisible by four

2) Divisible by four but **not divisible** by one hundred, like 2020

3) Divisible by four **and** four hundred (and by default one hundred)

In other words, the Gregorian calendar says it is a leap year *if*the is a year divisible by four… *unless* it can be divided evenly by 100 and then it is **not** a leap year…*unless* it is divisible by 100 **AND** 400, then it **is** a leap year.

Remember, divisible means it can be evenly divided and the quotient is a whole number. 10/2=5, so ten is divisible by five. 10/4=2.5 (not a whole number), so ten is not divisible by four.

Let’s look at some example years to understand leap years better.

Year 2018 / 4 = 504.5 It is **not** divisible by four so it is **not** a leap year.

Year 2016 / 4 = 504, but 2016/100 = 20.16 Since 2016 was **not** divisible by 100 it **was** a leap year.

Year 2000 / 4 = 500, 2000/100 **is** divisible by 100, so only using the first two criteria it would not be a leap year. However 2000 / 400= 50, and since it was divisible by 400 it **was** a leap year.

Year 2100 / 4 = 525, 2100/100 = 21, but 2100 / 400 = 5.25. 2100 is divisible by four, it is divisible by 100, but it is not divisible by 400. Therefor year 2100 will **not** be a leap year.

Thus, February 29^{th} 2,000 did exist but February 29th 2100 will not exist.

Babies born on February 29^{th} 2096 won’t get to celebrate their first birthday until 2104!

Wow!

At Mathnasium of Parker, we love to show students how math is a relevant part of life. If you thought this was interesting, you might also enjoy reading the following articles.

Inspire Your Child in Geometry Using the Art of Origami

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