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### Number of Days in a Month

Date: 10/21/2000 at 21:10:00
From: Barbara Reyes
Subject: Calendar

Why can't all months of the year be the same number of days? In the
information I find about calendars, the reason for the difference in
the number of days is difficult for my 7- to 9-year-olds to
understand.

Date: 10/23/2000 at 11:59:17
From: Doctor TWE
Subject: Re: Calendar

Hi Barbara - thanks for writing to Dr. Math.

As long as we're sticking with 12 months in a year, the months can't
all be the same number of days because 365 isn't divisible by 12. So
the months will average 30 5/12 days (ignoring leap years). That means
that the best we could do is to have 5 months of 31 days and 7 months
of 30 days.

Leap years, of course, complicate this. Because the time it takes the
Earth to rotate around the sun is not an even number of days (it's
4 years to keep our calendar in sync with the seasons. Otherwise,
over the course of about 750 years, summers and winters would do a
year" (without the quotes) and click on "that exact phrase" in our Ask
Dr. Math search engine at:

http://mathforum.org/mathgrepform.html

Another complication deals with the human ego. July was named for
Julius Caesar. Some accounts say that July was supposed to be a 30-day
month, but when Julius Caesar adopted the calendar (called the Julian
calendar because he introduced it to the Roman Empire), he did not
want any other month being grander (i.e. longer) than the one named
after him. So he took a day away from February (making it 29 days,
30 days on leap years) and added it to July, making July 31 days.

Augustus Caesar (Julius Caesar's successor, for whom August was named)
did not want his month to be inferior to Julius', so he took another
day away from February (making it 28 days instead of 29) and added it
to August to make it 31 to match July.

That leaves us with a mess - 7 months of 31 days, 4 months of 30 days,
and one month with either 28 or 29 days. For more information on
calendars, check out the following websites:

The Julian and the Gregorian Calendars - Peter Meyer
http://www.hermetic.ch/cal_stud/cal_art.htm

Calendar, A History - Timekeepers
http://www.ernie.cummings.net/calendar.htm

Calendars for Earth - Martian Time
http://pweb.jps.net/~tgangale/mars/earthcal_new.htm

Perhaps we'd be better off just starting a new calendar and making up
our own. If we wanted months with all the same number of days
(ignoring leap years), we'd have to make either 5 months of 73 days
each or 73 months of 5 days each, because 5 and 73 are the only
factors of 365. We could take the opposite approach and make all the
months the same number of days on leap years, then "take away" a day
on non-leap years. With 366 we have a few choices: 2 * 183, 3 * 122,
or 6 * 61. Perhaps we could have 6 months of 61 days and take away one
day on non-leap years.

I realize I haven't made it any simpler, but perhaps explaining why
our calendar is the way it is can help you to understand the
complications. If you have any more questions, write back.

- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/

Associated Topics:
Middle School Calendars/Dates/Time
Middle School History/Biography

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