Birthdays Every Year

Date: 05/14/99 at 19:14:48
From: Neil Clark
Subject: Birthdays every year

Hello -

My name is Neil. I am 8 years old and home schooled. I was thinking. 
Why does my birthday DAY of the week always go up each year? For 
example, if my birthday was on a Monday, next year it will be on a 
Tuesday. Why doesn't it go backwards instead of forwards?

We found out that each year is really 365 1/4 days but does that mean 
that the day goes up that many minutes so it is the next calendar day?

Date: 05/17/99 at 09:20:54
From: Doctor Peterson
Subject: Re: Birthdays every year

Hi, Neil. Great question!

The first thing to think about is, how long is a year? As you said, a 
year (as measured by the sun) is very close to 365 1/4 days long. But 
our calendar has to work with whole days, so most years are just 
counted as 365 days long. After four years, those extra 1/4 days we've 
ignored add up to an extra whole day, so we add it on - that's called 
the leap day, and the years when we add it are called leap years. 

There are other rules for leap years, but that's all that matters at 
the moment: each calendar year is either 365 or 366 days.

Now, how many weeks are there in a year? We can find that by dividing 
365 by 7:

      ___52_
    7 ) 365
        35
        --
         15
         14
         --
          1

This tells us that

    365 = 7 x 52 + 1

That is, a (non-leap) year consists of 52 weeks plus an extra day. For 
a leap year, there will be two extra days, since the remainder of 366 
divided by 7 is 2.

Now, if your birthday is on Monday this year, 364 days later (52 
weeks) it will be Monday again. But your birthday is not 364 days 
later, but 365 days later - on Tuesday. If it's a leap year and leap 
day falls between these two birthdays, then you'll actually have to 
wait two extra days, so your birthday will be on Wednesday.

So the simple answer to your question is that any day of the year 
moves one day later each year, and two days each leap year, because 
the remainder of 365 divided by 7 is 1. Can you figure out how long 
the year would have to be to make the day of the week move the other 
way?

Remainders can be very useful in many problems involving large 
numbers. If you're interested in how the calendar works, you may enjoy 
reading about how to find the day of the week for any day of any year. 
That, not surprisingly, again involves remainders. Here are some 
places you can look in our archives to learn about leap years and 
different ways to find the day of the week:

  http://mathforum.org/dr.math/problems/wong2.25.98.html   
  http://mathforum.org/dr.math/problems/chris10.21.98.html   
  http://mathforum.org/dr.math/problems/durling5.21.97.html   
  http://mathforum.org/dr.math/problems/simms12.4.96.html   

Have fun, and keep thinking about things like this!

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