Really Counting to One Billion

Date: 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/