Counting to One Billion
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
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
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 adjusted upwards, to 40 seconds. 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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