The Math Factor Podcast

4 Comments »

1. stampy said,

   March 24, 2007 at 9:03 pm

   Hi there, well, obviously a googol ^ googol is not a googol multiplied by itself 100 times!! And 10^500 raised to the 10^500th power is not 10^500 multiplied by itself 500 times!!

   What are the other two mistakes? Great podcast, thanks!

2. Sye Heinlein said,

   March 26, 2007 at 5:57 am

   would another error be that a googol ^ googol is larger then the 10^500. You stated that the later was larger?

   Didn’t really think this response through but i guess it is.

   I could only find the mistake that was previously mentioned and the one I just pointed out. What is the other??

3. strauss said,

   April 2, 2007 at 7:37 am

   Hi there; I was counting these as the same error; to be more precise, each particular moment of spacing out was counted as one error!

4. rmjarvis said,

   March 26, 2007 at 9:52 am

   I only spotted those two. But (10^500)! is definitely larger than googol^googol.

   Take the natural log of each:

   ln(g^g) = g ln (g) = 10^100 ln(10^100) = 10^100 * 100 ln(10)

   For the second one, use Sterling’s formula for factorials of very large numbers:

   ln(x!) ~= (x/e) ln(x)

   ln(10^500 !) = 10^500/e ln(10^500) = 10^500 * 500 ln(10)/e

   So basically, as long as there were more than around 100 9’s on that page, then that entry beats the googol^googol entry. But it doesn’t beat googol plex ^ googol plex.

   In fact, this kind of analysis makes this week’s puzzle quite tenable. Think about taking n logs of the nth entry in each series.

   That is, compare the series:

   ln(1) ln(ln(2^2)) ln(ln(ln(3^3^3))) …

   and

   ln(1) ln(ln(2!)) ln(ln(ln(3!!))) ….

   I won’t go the next step and give the answer here, but suffice to say, one series gets way way bigger than the other. It’s no contest.

   Peace,
   Mike

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