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### Counting to One Billion

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Date: 12/7/95 at 18:0:26
From: Vicki Shaffer-White
Subject: Counting to one billion

How long does it take to count to one billion counting one number per
second?

V. L. Shaffer-White
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```
Date: 3/11/96 at 21:37:2
From: Doctor Patrick
Subject: Re: Counting to one billion

Hi -

Counting non-stop, at one number a second, it would take you 31 years,
251 days, 7 hours, 46 minutes, and 39 seconds to count to 1 billion.
Here's how I came up with that figure:

To count to 1 billion would take 999,999,999 seconds, because the
timer would start as soon as the person says "one", one second would
have passed when the person says "two," and so on.  [Thanks to Jacques
de Wet for this observation.]

Now, there are 31557600 seconds in one year
(seconds*minutes*hours*days). Divide 999,999,999 by that figure for
the number of years.  Then divide the remainder (which has to be less
then a year) by the number of seconds in a day for the number of days.
If you keep doing this, eventually you get down to the number of
minutes, and the remainder of that will give you the number of
seconds.

Doctors Patrick and Ken, The Math Forum
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```
Date: 02/28/2001 at 23:29:00
From: Bob Giesen
Subject: Archive solution error

The above answer claims, "To count to 1 billion would take 999,999,999
seconds, because the timer would start as soon as the person says
'one', one second would have passed when the person says 'two', and so
on."

This runs contrary to the problem as it was posed. If only one second
has passed when the person says "two," then either (A) two numbers
have been counted in the first second or (B) "two" is just beginning
to be verbalized at the top of the second second - which means that
"one bilion" will begin to be verbalized at the top of the one
billionth second and not completed until *after* 999,999,999 seconds
have passed (meaning that we need that billionth second).

(A)
"one"     "two"   "three"       ... "one billion"
|_________|_________|_______..._____________|
0         1         2       ...        999,999,999

(B)
"one"     "two"   "three"    ...    "one billion"
|_________|_______|_________ ... ___|______________|
0         1       2          ... 999,999,999  1,000,000,000

This isn't the same as measuring the distance between the
infinitessimally small points on a number line. In order to count one
number per second, one second must be allotted to each number
-including the first and last ones.

The incorrect fraction of a minute in your solution should be

Bob
```

```
Date: 10/05/2001 at 08:50:17
From: Doctor Peterson
Subject: Re: Archive solution error

Hi, Bob.

My first reaction was that a one-second error is trivial, considering
both the magnitude of the time involved, and the more real error
introduced by merely supposing that you could say "999,999,999" in one
second.

But certainly there is a principle involved; we want to avoid
"off-by-one" errors in other counting problems, so we shouldn't teach
the wrong method here.

But I think the real issue is one of how we interpret the
question. You are taking it to mean, Supposing it takes one second to
pronounce each number, how long is it from the time you start saying
"one" until you finish saying "one billion"? Taken that way, you are
of course right.

But that interpretation really doesn't make any sense, because in fact
it will take a different time to say each number. Is there another way
to read the question? Yes. Suppose that by "count" we mean something
like "press a button on a counter that many times." It is not
necessary that each count take a full second to accomplish; the
counting can be essentially instantaneous, with a one-second interval
between counts. I think that's the only way to make this question at
all realistic. And, taken this way, the original answer is correct.
To count to two, we press the button once, wait a second, press it
again, and we're done; it took only one second, not two. The same is
true of siimilar questions involving, say, how long it takes a clock
to strike twelve.

Note that, if the question had supposed we counted one per HOUR, you
would certainly not give the same answer you gave!

So there's definitely some truth to your objection, but the real
problem is simply that Dr. Patrick neglected to state his assumptions
at the start, something I always recommend when a question might be
taken different ways.

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