20041121, 19:45  #1 
2,039 Posts 
how do you find number of digits of a 2^n number?
hi all,
I found this program yesterday and i was thinking about how to tell how many digits are in a 2^n number if i knew n. I made a Microsoft Excel spreadsheet and calculated out about 40 sequences, and eventually found a pattern relating the n's that end in 0 (multiples of 10). my pattern was ((3/10)*n) + 1. for example, using 0 for n, the result (number of digits in 2^n) is 1, as it should be. using 10, 20, 30, and 40 for n, i get 4, 7, 10, 13 respectively. i checked the actual digits by doing the math on the calculator and counting digits, and the pattern works. i found this page http://www.utm.edu/research/primes/mersenne/index.html which showed a table of known mersenne primes. i tested my equation on the table, and got good results until n=1279. according to the chart, 2^n should have 386 digits. however, when i plugged it into my equation, i came up with 384 digits. each n after that also gave incorrect answers, and they get continually worse. for example, the 40th mersenne prime, 2^20996011 1 has 6,320,430 digits according to the chart, but only 6,298,804 digits according to my equation (i used n=20996010 to get the answer, because i found a pattern that shows that after a multiple of 10, the next 3 have the same amount of digits, followed by 3 with 1 more digit, another 3 with 2 more digits, and then the next multiple of 10 is reached). So, i was wondering how you guys got the number of digits in a 2^n number, and why my equation and the "known" digits are so far off. thanks in advance, michael p.s. if you don't understand what i wrote, say so and i'll try to clarify. 
20041121, 19:53  #2 
Jun 2004
UK
139 Posts 
floor(log 2 * n) + 1
Where log is base 10. 
20041121, 19:55  #3 
P90 years forever!
Aug 2002
Yeehaw, FL
1101111110110_{2} Posts 
There are ceiling (n * log10(2)) digits. The base 10 log of 2 is .30103

20041121, 20:01  #4 
5×359 Posts 
ah, that makes sense. now i see why mine are so close at the beginning and then so far off later on.
but then, another quick questionwhat do floor and ceiling mean? 
20041121, 20:08  #5 
P90 years forever!
Aug 2002
Yeehaw, FL
2×3×1,193 Posts 
floor is round down. ceiling is round up.

20041121, 20:10  #6 
2984_{10} Posts 
ah, got it. thanks

20041122, 16:31  #7 
Sep 2002
3×257 Posts 
Are floor and ceiling terms simply the same thing as the boundaries of the logarithmic functions?

20041122, 19:39  #8 
Jun 2004
UK
139 Posts 
They're just "functions".
floor refers to the integer that is closest to a number without being higher. ceiling refers to the integer that is closest to a number without being lower. They're just fancy names for rounding up and rounding down. 
20041123, 05:30  #9 
Oct 2003
Australia, Brisbane
111010110_{2} Posts 
I don't know about other languages.. but I know that I use floor and ceiling a lot in both excel and matlab. Both floor and ceiling are acual "functions" in both programs. However, I haven't actually ever seen floor or ceiling written in any maths textbooks as far as I can remember... I could be wrong though.

20041123, 09:08  #10  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
2·3·1,699 Posts 
Quote:
floor(x) is represented by _ x _ and ceiling(x) has the horizontal lines at the top. Paul 

20041123, 18:02  #11 
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×599 Posts 
See
http://mathworld.wolfram.com/FloorFunction.html and http://mathworld.wolfram.com/CeilingFunction.html for clear examples and fuller explanation. 
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