Number Of Digits; I Hate To Ask

storm5510 · 2009-09-06, 19:17

Is there a formula for calculating the number of digits in a large number?

An example would be 2¹⁸⁴⁷⁷

frmky · 2009-09-06, 19:35

Yes, take the base-10 log, and round up.

log (2^18477) = 18477 log 2 = 18477 (0.30103) = 5562.1

so 2^18477 has 5563 digits.

storm5510 · 2009-09-06, 19:54

Quote:

Originally Posted by frmky

18477 log 2 = 18477 (0.30103) = 5562.1

Okay, thanks. I'm trying to do this on my TI hand-held calculator. It has "LOG" and "LN" I can't seem to replicate the example. "Log 2" above is base 2 logarithm?

Mini-Geek · 2009-09-06, 20:15

Quote:

Originally Posted by storm5510

Okay, thanks. I'm trying to do this on my TI hand-held calculator. It has "LOG" and "LN" I can't seem to replicate the example. "Log 2" above is base 2 logarithm?

In this case, frmky is only using the base 10 logarithm. When he said "log 2", he was referring to the base 10 logarithm of 2, which is about 0.30103. (keep in mind that log(y^x)=x*log(y), hence the importance of knowing what log(2) is)
The "LOG" button on your calculator is (almost certainly, anyway - just try log(1000) or something to check) the base 10 logarithm.

I doubt you'll be able to replicate the first step on your calculator, since if it's anything like my TI, it has a limit of 100 digits. You can, however, replicate the rest. Note that 0.30103 is not exactly log_10(2), but is close enough for this example.

If you did need the base x logarithm of y, you could use log(y)/log(x). (with log being the logarithm of any base, surprising as that may sound

)

storm5510 · 2009-09-06, 20:31

On my calculator if I enter 1000 and press Log, it returns 3, meaning 10³

Entering 2 and pressing Log gives me .301029996. This is the rounded number in his example.

Multiply that times 18477 and i get 5562.1.

It seems to work.

On the GIMPS home page is 2^42643801. If I do this like above, then I get 12,837,063.2

Mini-Geek · 2009-09-06, 20:35

Yep, you've got it.

Dougy · 2009-09-06, 21:52

Quote:

Originally Posted by frmky

Yes, take the base-10 log, and round up.

log (2^18477) = 18477 log 2 = 18477 (0.30103) = 5562.1

so 2^18477 has 5563 digits.

Although this method does not work if your number is a power of ten. For example $\log_{10} 100=2$ whereas 100 has three digits.

CRGreathouse · 2009-09-06, 22:09

Yes, the 'right' way is to round down and add one. But usually you'd know if you were dealing with a power of 10.

Dougy · 2009-09-06, 23:24

Hmmm... this should be on the Mersenne Wiki.

Batalov · 2009-09-07, 00:16

http://mersenneforum.org/showthread.php?t=3321

storm5510 · 2009-09-07, 01:23

http://www.mersenne.org/bench.htm

This link is broken. It is in the last post of the thread above. Understandable. All of this is from 2004.

2009-09-06, 19:17	#1
storm5510 Random Account Aug 2009 19×101 Posts	Number Of Digits; I Hate To Ask Is there a formula for calculating the number of digits in a large number? An example would be 2¹⁸⁴⁷⁷

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2009-09-06, 19:35	#2
frmky Jul 2003 So Cal 2·3·347 Posts	Yes, take the base-10 log, and round up. log (2^18477) = 18477 log 2 = 18477 (0.30103) = 5562.1 so 2^18477 has 5563 digits.

2009-09-06, 20:31	#5
storm5510 Random Account Aug 2009 3577₈ Posts	On my calculator if I enter 1000 and press Log, it returns 3, meaning 10³ Entering 2 and pressing Log gives me .301029996. This is the rounded number in his example. Multiply that times 18477 and i get 5562.1. It seems to work. On the GIMPS home page is 2^42643801. If I do this like above, then I get 12,837,063.2

2009-09-06, 20:35	#6
Mini-Geek Account Deleted "Tim Sorbera" Aug 2006 San Antonio, TX USA 17×251 Posts	Yep, you've got it.

2009-09-06, 22:09	#8
CRGreathouse Aug 2006 1011101100001₂ Posts	Yes, the 'right' way is to round down and add one. But usually you'd know if you were dealing with a power of 10.

2009-09-06, 23:24	#9
Dougy Aug 2004 Melbourne, Australia 2³·19 Posts	Hmmm... this should be on the Mersenne Wiki.

2009-09-07, 00:16	#10
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 10010010101101₂ Posts	http://mersenneforum.org/showthread.php?t=3321

2009-09-07, 01:23	#11
storm5510 Random Account Aug 2009 19×101 Posts	http://www.mersenne.org/bench.htm This link is broken. It is in the last post of the thread above. Understandable. All of this is from 2004.