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Working with Huge Numbers

Date: 05/15/99 at 11:12:55
From: James Sherrington-Kendall
Subject: Is there anywhere on the Internet I can work out EXTREMELY 
         big numbers?

Dear Sir,

I am trying to find either a good program or a Web site where I can 
work out rather large numbers. If not, I would be grateful if you 
could tell me a method for working out a number such as 4 to the 
power 300 billion.

Thanks ever so much,

Date: 05/17/99 at 13:37:39
From: Doctor Peterson
Subject: Re: Is there anywhere on the Internet I can work out 
             EXTREMELY big numbers?


What you need depends on what you need it for. If you want a program 
to calculate large numbers precisely, there are arbitrary precision 
calculators that will give hundreds of digits, such as this:   

Unfortunately, it seems to make my computer slow down horribly, so I 
don't recommend it. You might find more by searching the Web for 
"arbitrary precision."

However, you may not need all that precision (that is, all the digits 
of your answer), and if you don't you should use a different method, 
because working with many digits puts a big load on your computer. In 
your example, 4^300,000,000,000 would take 180,617,997,399 digits just 
to write out, or something like 37,000,000 pages. You won't find a 
calculator that can handle this exactly.

How did I figure that out? Many large calculations can be done easily 
(but only approximately) using logarithms. We can first find the 
logarithm of your number this way:

    log(4^300,000,000,000) = log(4) * 300,000,000,000
                           = 0.602059991328 * 300,000,000,000
                           = 180617997398.4

then undo the log by raising 10 to this power. The calculator will 
complain if you do this last step directly, because it can't handle 
such large numbers; but we can break it down:

    4^300,000,000,000 = 10^180617997398.4
                      = 10^180617997398 * 10^0.4
                      = 2.45 * 10^180,617,997,398

That's the 180,617,997,399-digit number I mentioned!

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Calculators, Computers

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