One Million Seconds

Date: 09/10/2001 at 22:00:53
From: Alyson Erdman
Subject: 1 million seconds

What does 1 million seconds convert to in weeks, days, hours, minutes, 
and seconds? (Nanoseconds, etc. are not needed.)  

I have already found out at least 1 week and 12 hours, but I am stuck 
there. Please help!

Date: 09/11/2001 at 11:34:34
From: Doctor Ian
Subject: Re: 1 million seconds

Hi Alyson,

There are 60 seconds in a minute, 60 minutes in an hour, and 24 hours 
in a day, which means that there are 86,400 seconds in a day, and 
604,800 seconds in a week.  

You probably know all this, which is why you were able to figure out 
that a million seconds is more than a week, but less than two weeks. 

Let's say we subtract a week from the total.  That leaves us with 

  1,000,000 - (1 * 604,800) = 395,200

seconds to work with. The next smallest unit is a day, or 86400 
seconds. 86,400 goes into 395,200 more than 4 times, but less than 
5 times. So now we're down to 

  1,000,000 - (1 * 604,800) - (4 * 86,400) = 49,600

seconds.   

So all you have to do is keep going this way, for hours (3600 seconds) 
and minutes (60 seconds), and the remainder that you end up with will 
be the number of seconds. 

By the way, note that what you're doing here is converting from base 
10 to a funky kind of 'base' in which the places are not powers of 10, 
but products of 60, 24, and 7. That is, you'll end up with something 
that looks like this:

  1,000,000 =   (a * 604,800)             
              + (b * 86,400) 
              + (c * 3600) 
              + (d * 60)
              + (e * 1)

Here, a is the number of weeks (which we know to be 1), b is the 
number of days (which we know to be 4), c is the number of hours, d is 
the number of minutes, and e is the number of seconds.  
  
Note that this is very much like what we do when we write a number 
like '209,534':

  209,534 =   (2 * 100,000)
            + (0 * 10,000)
            + (9 * 1,000)
            + (5 * 100)
            + (3 * 10)
            + (4 * 1)

In each case, we have a series of 'places', each of which has some 
associated size.  In the case of base 10 numbers, each 'place value' 
is a power of 10:

  209,534 =   (2 * 10^5)
            + (0 * 10^4)
            + (9 * 10^3)
            + (5 * 10^2)
            + (3 * 10^1)
            + (4 * 10^0)

whereas in the case of weeks, etc., the 'place values' are

  1,000,000 =   (a * 60^2 * 24 * 7)             
              + (b * 60^2 * 24) 
              + (c * 60^2) 
              + (d * 60^1)
              + (e * 60^0)

The way the 'bases' are constructed is different, but it's really the 
same idea. 

Why do we care? Well, it's terrifically convenient to be able to write

  209,534

instead of 

  2*10^5 + 0*10^4 + 9*10^3 + 5*10^2 + 3*10^1 + 4*10^0

isn't it? We can do this because we've all agreed on what the various 
place values mean. Similarly, we've all agreed that
  
  12:31:46

means 

  (12 * 60^2 seconds) + (31 * 60^1 seconds) + (46 * 60^0 seconds)

All we're doing here is extending the notation to include days and 
weeks:

  a:b:c:d:e = (a * 60^2 * 24 * 7 seconds) + ... + (e * 60^0 seconds)

Note that it's easy to make up notations like this - it's one of the 
things that mathematicians do all the time. The trick is getting 
everyone to _agree_ to use the same notation. (For example, some 
people might look at 'a:b:c:d:e' and think that 'a' represents months 
instead of weeks, which could make for interesting misunderstandings.  
International travelers already run into problems like this because 
Americans think that 6/9/01 means '6 months, 9 days, and 1 year' while 
Europeans think that it means '6 days, 9 months, and 1 year'.)

Finally, note that if you r-e-a-l-l-y get what's going on here, then 
you already know how to convert between number bases, like base 8, 
base 16, base 5, and so on: You figure out what the place values are, 
then use repeated division to work your way down from the largest 
applicable place to the smallest place. 

Not coincidentally, this is the same process you would use to convert 
from a large number of ounces to gallons, quarts, pints, and cups, 
which form another non-uniform 'base':

  1,000,000 ounces =   (a * 1 gallon)
                     + (b * 1 quart)
                     + (c * 1 pint)
                     + (d * 1 cup)
                     + (e * 1 ounce)

                   =   (a * 2^7 ounces)       4 quarts in a gallon
                     + (b * 2^5 ounces)       2 pints in a quart
                     + (c * 2^4 ounces)       2 cups in a pint
                     + (d * 2^3 ounces)       8 ounces in a cup
                     + (e * 1 ounce)

This may seem like more than you wanted to know, but this kind of 
thing shows up all over the place in math and science, so it's very 
useful to understand that what _seems_ at first to be an arbitrary 
collection of systems of weights and measures and number bases is 
really just the repeated application of a single, very powerful idea.  

I hope this helps.  Let me know if you'd like to talk about this some 
more, or if you have any other questions. 

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