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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 math.stackexchange.com/questions/72646/… –  Ross Millikan Jul 22 '12 at 1:07
    
Take logarithms. Use Stirling's formula. –  Qiaochu Yuan Jul 22 '12 at 1:32
    
Apropos to Ross's comment, see this article as well. –  J. M. Jul 22 '12 at 3:55

1 Answer 1

up vote 5 down vote accepted

$10^{googol}$ 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.

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