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 [url]http://www.utm.edu/research/primes/mersenne/index.html[/url] 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.

floor(log 2 * n) + 1

Where log is base 10.

There are ceiling (n * log10(2)) digits. The base 10 log of 2 is .30103

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 question--what do floor and ceiling mean?

floor is round down. ceiling is round up.

ah, got it. thanks

Are floor and ceiling terms simply the same thing as the boundaries of the logarithmic functions?

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.

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.

[QUOTE=dave_0273]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.[/QUOTE]
In printed text the floor and ceiling functions are generally written as partial square brackets. I don't know that I can include the characters here, but they look something like this:

floor(x) is represented by |_ x _| and ceiling(x) has the horizontal lines at the top.


Paul

See

[url="http://mathworld.wolfram.com/FloorFunction.html"]http://mathworld.wolfram.com/FloorFunction.html[/url] and

[url="http://mathworld.wolfram.com/CeilingFunction.html"]http://mathworld.wolfram.com/CeilingFunction.html[/url]

for clear examples and fuller explanation.