How Long Does It Take to Count to a Billion?

Date: 03/28/2003 at 02:04:22
From: Kandi
Subject: How long does it take to count to a billion?

In the answer in the Dr. Math archives:

   Counting to One Billion
   http://mathforum.org/library/drmath/view/58739.html 

when you said to divide the number of seconds of days... divide it by 
what? And also the number of seconds in your problem is wrong; you 
said it was 31557600 seconds, when it should be 3153600.

Date: 03/28/2003 at 10:04:15
From: Doctor Peterson
Subject: Re: How long does it take to count to a billion?

Hi, Kandi.

First, the number of seconds in a year used there is based on a 
365.25 day year, which takes leap years (mostly) into account:

    60*60*24*365.25 = 31,557,600 seconds

Your number of 3153600 is presumably a typo for 31536000, which is 
the correct number based on 365 days per year.

Now, to find the number of years equal to a billion seconds (minus 
one), we divide

    999,999,999 / 31,557,600 = 31.688 years

Taking a whole number of years, the remainder will be

    999,999,999 - 31*31,557,600 = 21,714,399 seconds

Now you have to divide THIS by the number of seconds in a day to find 
the number of days BEYOND 31 years. Repeat the same sort of process 
to find the extra hours past a whole number of days, and then the 
minutes and seconds.

An alternative approach is to work up the scale rather than down. 
First divide 999,999,999 by 60 to find the number of whole minutes, 
leaving the remainder behind as the number of extra seconds; then 
divide the number of minutes by 60 to find the number of whole hours, 
and so on. You should get the same result, with smaller divisors at 
each step.

If you have any further questions, feel free to write back.

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

Date: 03/28/2003 at 12:10:46
From: Doctor Ian
Subject: Re: How long does it take to count to a billion?

Hi Kandi,

Just to add to what Dr. Peterson said, an even better approximation
for the number of days in a year is 365.2425, as explained here:

   Seconds in a Thousand Years
   http://mathforum.org/library/drmath/view/58477.html 

The difference is small, 

  60*60*24*365.25   = 31,557,600 seconds

  60*60*24*365.2425 = 31,556,952 seconds

just 648 seconds (about 11 minutes) over the course of a year.  

Is a difference like this significant? Consider that NASA currently
estimates that it would take about 11 years to send a spacecraft to
Pluto. A typical interplanetary probe travels at about 20 km/sec.  

So if we're trying to estimate where the spacecraft would be after 11
years, a difference of 648 seconds per year translates to an error of

  648 sec/yr * 11 yr * 20 km/sec = 142,560 km

which is more than 60 times the diameter of the planet!

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

Date: 03/28/2003 at 13:33:58
From: Doctor Peterson
Subject: Re: How long does it take to count to a billion?

Hi, Kandi.

I'll just add a bit to what Dr. Ian said, to add both to our 
preciseness and your confusion:

I deliberately chose not to make the length of a year more precise 
than 365.25 days for a reason: since we are talking about a 31-year 
span, I figured we are not going to experience any adjustments to the 
length of a year other than a leap year every four years during this 
interval; which adds exactly 1/4 day every year on average. So by 
using this length for the year, we are in step with the year we 
actually use to measure time year by year. If you were to calculate 
the date and time on which you would actually finish counting, 
however, you would have to determine _exactly_ how many of the 
intervening years would have been leap years, and count days 
accordingly, rather than using the average.

Dr. Ian's answer would give an answer that has nothing to do with the 
actual date on a calendar, but is astronomically precise. You can 
decide which makes more sense for the problem!

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

Date: 03/28/2003 at 15:46:49
From: Doctor Ian
Subject: Re: How long does it take to count to a billion?

Hi Kandi,

I hope you're getting a sense of why people who need to deal precisely
with long intervals (like astronomers and computer scientists) avoid
using 'years' as a unit of time, preferring to measure time directly
in seconds, or sometimes in Julian Days (which are exactly 86400
seconds long).  

Note that when using calendars, we don't deal with parts of days. In
leap years we add a whole day, and in other years we add nothing at
all. There are no calendar years that have 365.25 days, or 365.2425
days. There are only years with 365 days, and years with 366 days.  

Imagine a land where an 'inch' is well-defined, but a 'foot' 
corresponds to the length (rounded to the nearest number of inches) of
the actual foot of whoever happens to be king at the time. And
suppose your kitchen is 100 inches wide. 

Now, while John is king, a foot is 10 inches, so your kitchen is 10
feet wide. When he dies, and James becomes king, your kitchen is still 
100 inches wide, but now a foot is 11 inches, so your kitchen is a 
little over 9 feet wide. It's the same room, but the size of a foot
has changed.

Now, suppose someone asks you how wide your kitchen would have been a
hundred years ago, or how long it will be a hundred years from now. 
You can deal with the past by keeping track of how large each king's
foot was, when he became king, and when he died. But what about the
future? Well, you might do something like this: "Let's look at a bunch 
of recent kings, and average their foot sizes. Then we'll use the 
average as the definition of a foot."  

That's workable - and it even lets you use 'foot' as a unit of length
without knowing what year you're talking about - but only if everyone
who tries it uses the same average. Two people who choose different
sets of kings are likely to come up with different averages. 

This is more or less what happens when you try to define an 'average
year'.  Note that in the 31 years from 2000 to 2030 inclusive, there
are 11323 days, which makes each year in that interval an average of
365.258 days long. However, in the 31 years from 2001 to 2031 
inclusive, there are 11322 days, which makes each year in that 
interval an average of 365.226 days long.  

Even if you use 100 consecutive years, the length of an average day
will still depend on which years you pick. But if you pick _any_ 400
consecutive years, you'll always get the same answer: 365.2425.  

Does this mean you should use the 400-year average? Not at all! What 
all of this means is that it's fine to refer to a particular _year_ 
(like 1492 or 2006), but using 'years' as a measure of time in
calculations is a bad idea unless everyone you're dealing with has
agreed to use the same definition. An exception would be if it's very
clear that you're not trying to be precise, e.g., noting that a
president is elected for four years, or that your cousin is 7 years 
old.  

To avoid the problem altogether, when computing things like how long
it would take to count to a billion, it would be safest to use days as
your largest unit of time, and note that it would be 'more than 31
years', to help people visualize the answer. Otherwise to avoid
misunderstandings, you really need to know when you're going to start
counting, or state the length of the average year that you used as
part of your result, e.g., 

  31 years (of 365.25 days), 251 days, 7 hours, ...

I hope this helps. 

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