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.  

Please help me!

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 
about 365.24 days), we have to add an extra day about once every 
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 
complete reversal. For more information on leap years, type in "leap 
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/