Really Counting to One BillionDate: 06/08/2001 at 11:04:01 From: Mr. Short's Class Subject: "Really" Counting to one billion We want to know approximately how long it will take to count to one billion. In the archived answer Counting to One Billion http://mathforum.org/dr.math/problems/shaffer_white12.7.html the question and answer assumed that each number took one second to say. But a number like 777,777,777 clearly takes more than one second to say. How can we find how long it would really take to count to one billion? Thank you. We look forward to your reply. Lunenburg High School Integrated II Honors Math Date: 06/08/2001 at 11:49:02 From: Doctor Rick Subject: Re: "Really" Counting to one billion Hi, class. Thanks for writing with an interesting question. I don't know of a formula that would tell me how long it takes to say a particular number. Let's just take one step closer to reality than the archived answer, and suppose that it takes 1 second to say one DIGIT of a number. (If you think this is too long, you can just scale the answer by the fraction of a second that you think the average syllable takes. For instance, it would take half as long if each digit took half a second.) The question now is, how many digits are there in the numbers from 1 to a billion (10^9, using the American system)? There are 9 1-digit numbers. There are 99-9 = 90 2-digit numbers, for a total of 90*2 = 180 digits. There are 999-99 = 900 3-digit numbers, for a total of 900*3 = 2700 digits. You see the pattern: going up to 9 digits, we have the sum 1*9 + 2*90 + 3*900 + 4*9000 + 5*90000 + 6*900000 + 7*9000000 + 8*90000000 + 9*900000000 Factoring out a 9 from each term (and reversing the order), we have 9(900000000+80000000+7000000+600000+50000+4000+300+20+1) = 9(987654321) Multiply that out to get 8,888,888,889 digits in 1 billion numbers. (Hmm, that's an interesting number, very close to 8/9 of 10 billion. Maybe you could study this some more; is there another way to derive the number?) All that's left is to multiply the time it takes to say a billion digits, one per second (the answer in the archives) by 8 8/9. That's approximately 281 years. To be more precise (years, months, days, etc.), you'll have to follow the procedure outlined in the Archives all over again. Counting to One Billion http://mathforum.org/dr.math/problems/shaffer_white12.7.html You might consider making my estimate more accurate. For instance, though 10 is a two-digit number, the time it takes to say it puts it in the "1-digit" category. Perhaps we should only count the NON-ZERO digits?? - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 06/08/2001 at 12:09:36 From: Doctor Rob Subject: Re: "Really" Counting to one billion Thanks for writing to Ask Dr. Math, class. Because there are 9 numbers with 1 digit, 90 with 2 digits, 900 with 3 digits, and so on, there are 9*1 + 90*2 + 900*3 + 9000*4 + ... + 900000000*9 + 1*10 = 9*(900000000+80000000+7000000+600000+50000+4000+300+20+1) + 10, = 9*987654321 + 10, = 8,888,888,899 digits to be counted. Actually, it takes me about 4.5 seconds to say a 9-digit number like "nine hundred ninety-nine million, nine hundred ninety-nine thousand, nine hundred ninety-nine," so I would multiply this by 4.5/9 = 1/2 to get the approximate actual time in seconds. Divide by 86400 to convert this to days, and then by 365.25 to convert to years (and round off to two significant figures). - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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