24 Hours in a Day

Date: 08/15/2001 at 17:11:43
From: Elisabeth
Subject: Factors

When deciding to divide the day in equal parts it was decided to 
divide it into 24 hours. Why not 23 or 25? What other number could I 
use to represent the hours of the day? Why?

Date: 08/16/2001 at 13:49:06
From: Doctor Ian
Subject: Re: Factors

Hi Elisabeth,

One nice thing about 24 is that it has lots of integer divisors:

  24 = 1 * 24

     = 2 * 12

     = 3 * 8

     = 4 * 6

This means that you can talk about lots of different fractions of a 
day:

  1/24 of a day = 1 hour

  1/12 of a day = 2 hours

   1/8 of a day = 3 hours

   1/6 of a day = 4 hours
 
   1/4 of a day = 6 hours

   1/3 of a day = 8 hours

   1/2 of a day = 12 hours

Try doing that with 23 or 25.  

Now, the _reason_ that 24 has so many integer divisors is that it has 
a lot of prime factors. You can generate the factors of a number by 
looking at all the subsets of its prime factors:
    
  24 = 2 * 2 * 2 * 3     =>  2
                             2 * 2
                             2 * 3
                             2 * 2 * 2
                             2 * 2 * 3
                             2 * 2 * 2 * 3
                             
Contrast that with 25:

  25 = 5 * 5             =>  5
                             5 * 5
  
In fact, let's look at the prime factors of the numbers from 2 to 25:

  Number   Prime factors     
    2       2                 
    3       3                 
    4       2, 2              
    5       5                 
    6       2, 3              
    7       7                 
    8       2, 2, 2           
    9       3, 3              
   10       2, 5              
   11       11                
   12       2, 2, 3           
   13       13                
   14       2, 7              
   15       3, 5              
   16       2, 2, 2, 2        
   17       17                
   18       2, 3, 3      
   19       19
   20       2, 2, 5
   21       3, 7
   22       2, 11
   23       23
   24       2, 2, 2, 3
   25       5, 5

As you can see, 16 and 24 have more integer divisors than any other 
numbers under 25. So, why not divide the day into 16 hours?  Maybe 
that makes an hour too big, or maybe it's convenient to be able to 
divide a day into thirds. There's no way to know for sure. 

You might consider looking at all the numbers up to 100, to see what 
other numbers might make even better choices. For example, 

  36 = 2 * 2 * 3 * 3

so it might be at least as good a choice as 16 or 24.  Also,

  72 = 2 * 2 * 2 * 3 * 3

so that would give us even more flexibility. 

Can you find other numbers with even more prime factors than this? Do 
you see any problems that would arise from having a 72-hour day, or a 
36-hour day? How about a 10-hour day? 

If you're interested in this kind of thing, you might also think about 
why we divide a circle into 360 degrees. 

For some history on the hours in the day, minutes in the hour, seconds 
in a minute, see hours at How Many? A Dictionary of Units of 
Measurement, by Russ Rowlett, the University of North Carolina at 
Chapel Hill:

   http://www.unc.edu/~rowlett/units/dictH.html   

or visit the MacTutor Math History archives:

   History Topics: Babylonian mathematics
   (select #1. An overview of Babylonian mathematics) 
   http://www-history.mcs.st-and.ac.uk/history/Indexes/Babylonians.html   

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