how do you find number of digits of a 2^n number?

2004-11-21, 19:45

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.

marc · 2004-11-21, 19:53

floor(log 2 * n) + 1

Where log is base 10.

Prime95 · 2004-11-21, 19:55

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

2004-11-21, 20:01

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?

Prime95 · 2004-11-21, 20:08

floor is round down. ceiling is round up.

2004-11-21, 20:10

ah, got it. thanks

Jwb52z · 2004-11-22, 16:31

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

marc · 2004-11-22, 19:39

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.

dave_0273 · 2004-11-23, 05:30

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.

xilman · 2004-11-23, 09:08

Quote:

Originally Posted by 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.

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

cheesehead · 2004-11-23, 18:02

See

http://mathworld.wolfram.com/FloorFunction.html and

http://mathworld.wolfram.com/CeilingFunction.html

for clear examples and fuller explanation.

2004-11-21, 19:45	#1
Unregistered 2²×3²×7² 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.

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2004-11-21, 19:53	#2
marc Jun 2004 UK 139 Posts	floor(log 2 * n) + 1 Where log is base 10.

2004-11-21, 19:55	#3
Prime95 P90 years forever! Aug 2002 Yeehaw, FL 2×53×71 Posts	There are ceiling (n * log10(2)) digits. The base 10 log of 2 is .30103

2004-11-21, 20:01	#4
Unregistered 2⁷·3·17 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 question--what do floor and ceiling mean?

2004-11-21, 20:08	#5
Prime95 P90 years forever! Aug 2002 Yeehaw, FL 2×53×71 Posts	floor is round down. ceiling is round up.

2004-11-21, 20:10	#6
Unregistered 1C5₁₆ Posts	ah, got it. thanks

2004-11-22, 16:31	#7
Jwb52z Sep 2002 2·3·7·19 Posts	Are floor and ceiling terms simply the same thing as the boundaries of the logarithmic functions?

2004-11-22, 19:39	#8
marc Jun 2004 UK 8B₁₆ 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.

2004-11-23, 05:30	#9
dave_0273 Oct 2003 Australia, Brisbane 2×5×47 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.

2004-11-23, 18:02	#11
cheesehead "Richard B. Woods" Aug 2002 Wisconsin USA 2²·3·641 Posts	See http://mathworld.wolfram.com/FloorFunction.html and http://mathworld.wolfram.com/CeilingFunction.html for clear examples and fuller explanation.