9^9^9

Date: 09/10/97 at 18:53:43
From: Sean H
Subject: 9 ^ 9 ^ 9

Dr. Math,

In my Algebra II class, my teacher asked us what the largest number we
can write is, only with a limit of 3 digits. I answered 9 ^ 9 ^ 9 
(nine to the ninth to the ninth) and was correct.

Getting home, and on my computer however, I set out to find the answer
to that question. 9 ^ 9 was easy:  387420489. But now, how in the 
world to do 9 ^  387420489?

I tried writing a C++ program to do it, only the largest data type you
can use, an unsigned long, only supports up to 10 digits. That wasn't
getting me anywhere. On the Internet, I saw one person say that the
answer would have 300,000,000 digits! Is this true?

So my question: Is it really that large an answer? And is there a way 
I could find the answer? If it really is that big, how long would it 
take to answer on a large mainframe? (I'm talking Cray here). Would it 
be hours, days, months, or years?

Thanks a lot,
Sean

Date: 09/25/97 at 15:47:28
From: Doctor Ken
Subject: Re: 9 ^ 9 ^ 9

Hi!

To find out how many digits some number has, the best way is to take 
the log base 10 of that number and round down to the nearest integer, 
and add 1.  

For instance, 42 has 2 digits, and the log base 10 of 42 is about 
1.62325: round that down and you get 1, add 1 and you get 2. 34578 has 
5 digits, and its log base 10 is 4.5388: round it down and you get 4, 
add 1 to get 5. 1000 has 4 digits and its log base 10 is 3. Round down 
to get 3, add 1 to get 4.

So we want to find the log base 10 of your number, 9^387420489. Let's 
do it:

 Log (9^387420489) = 387420489 * Log(9)       (Pull out the power)
                   = 387420489 * 0.954243     (do the log)
                   = 3.6969309963 x 10^8      (by calculator)
                   = 369693099.63

So your number has 369,693,100 digits. That's a big number!

By the way, we have a section of our site devoted to large numbers.  
See  http://mathforum.org/dr.math/faq/    for more info.

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/