How is the best way to compare big numbers? They are result of two functions with different asymptotic growth. For example:
Googleplex which is $10^{{10}^{100}}$ to $1000!$
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You might look at http://math.stackexchange.com/questions/72646/help-me-put-these-enormous-numbers-in-order-googol-googol-plex-bang-googol-s – Ross Millikan Jul 22 '12 at 01:07
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Take logarithms. Use Stirling's formula. – Qiaochu Yuan Jul 22 '12 at 01:32
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Apropos to Ross's comment, see this article as well. – J. M. ain't a mathematician Jul 22 '12 at 03:55
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$10^{googol}$ compared to $1000!$
$1000!=1000\times999\times998...<1000^{1000}$
$1000^{1000}=(10^3)^{1000}=10^{3000}$
since a googol is drastically larger than $3000$, the first number is much, much greater.
In general logarithms (equivalently, converting to a base and comparing exponents) are a great way for comparing large numbers. For example: whether $2^{523} <^? 3^{228}$ may not be obvious, but even knowing very rounded values for $\log(2)$ and $\log(3)$ will let you compare $523\log(2)$ and $228\log(3)$ quite easily, which is an equivalent problem.
Robert Mastragostino
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