Calendars

1
Calendars

• Calendars from different cultures

2
Keeping Time

• Today you probably think of a year as 365 days,
  but why do we have 365 days?
• The 365 day year is based on how long it takes
  the earth to revolve around the sun. Venuss year
  is 0.615 earth years, Jupiters is 11.9 earth
  years, and Plutos is 248.5 earth years.

3
Other Calendars

• Not all calendars are 365 days!
• We do not know exactly how long the Inca calendar
  was. The Aztec and Mayan had two calendars one
  was 260 days long and the other 365 days. The
  Jewish Year is 354 days long.

Both the Jewish and Incan calendars are based on
lunar cycles.
4
The Lunar Cycle

• A calendar based on the lunar cycle has months
  are usually 30 days long
• These months mark a full cycle of the moon.
• A full cycle of the moon is the time it takes for
  the moon to go from new moon to new moon again.

5
Is Our Calendar Right?

• Is a year exactly 365 days?

• No!
• A year is slightly longer than 365 days
  (365.24219 days is the more exact duration of one
  earth orbit around the sun).
• If we do not compensate for this underestimation,
  we would lose 0.24219 days a year. While this may
  not seem like a big deal, we would lose 484 days
  every 2000 years!

6
Leap Years

• Their have been many different ways to compensate
  for the loss of days in our calendar. Today we
  have leap years every four years. Since we have a
  leap year every year, one year is equal to 365.25
  days.

7
Reviewing Decimals

• When we write 365.25 we are using decimal places.
  The numbers to the right of the decimal hold
  places just as the positions to the left of the
  decimal spot do.

3 6 5 . 2 5
Hundreds Ones Tenths Tens Decimal
Hundredths
8
Interpreting Decimals

• The number 0.25 means 25 hundredths. This can be
  written as . So if we gain 0.25 days per year,
  how many days would we gain in a 1,000 years?
• If there are really 365.242 ( 0.242) days
  per year, how many days would we lose in 1,000
  years with only years that are 365 day long?

25 100
242 1,000
9
Leap Year Problems

• Because of leap years, we gain 0.25 days ever
  year or 250 days every 1,000 years as we learned
  on the last slide. We also learned that we lose
  242 days every 1,000 years by the exact number of
  days in a year. So how many days ahead are we
  after 1,000 years?

10
Compensating for Leap Years

• Every four years, we gain ONE whole day on
  February 29th. This means that we average a gain
  of 0.25 or 1/4th day for the past four years.
• However, earths orbit is 365.242 days, not
  365.25 days. Leap years are overestimating the
  time we actually lose!
• How do we lose the 0.008 days that we gain
  annually due to leap years?

11
Our Best Answer So Far

• Skip some leap years!
• Specifically, skip leap years every so often so
  that our calendar becomes synchronized to Earths
  orbit again.
• But how many leap years should be skipped??

12
Determine the Leap Years

• If a year were 365.240 days how many leap years
  should there be in 1,000 years?
• What if there were 365.245 days in a year?
• Using the two above answers how many leap years
  would have to be skipped in that 1,000 years if
  there is one every four years?

13
Incas, Mayans, and Aztecs

• The Mayans, Aztecs, and Incas all had 12 months
  every year in their calendar. The Mayan and Aztec
  months were 30 days in length exactly. The Incas
  based their months on the lunar cycle so most of
  their months were 30 days. If their years had 12
  months with 30 days each, how many days would
  there be in each year?

14
How Would You Fix the Problem?

• If their year had 360 days per year and we know a
  year is really 365 days long, how many days would
  they lose each year?
• If you were an Inca how would you propose to fix
  this problem?

15
What the Incas Did

• The Incan calendar, like the Jewish calendar, was
  based on the lunar cycle. So even though we do
  not know the exact length of an Incan year, lets
  assume that it was 355 days long. It was probably
  not 360 days long, because that would not have
  coincided with the lunar cycle.
• How many days short of a full year was the Incan
  year if it was 355 days long?

16
How the Incas Compensated

• If their year was 355 days long, the Incas would
  have lost 10 days a year. How could they make up
  those 10 days and still have each month start on
  the new moon?