I was looking at math pages on the web concerning really big numbers, and one of the ones they used for comparison before getting into the meaty stuff (rapidly growing functions) was a maximum instantanious number of parallel universes similar to our own; that is they have the same size and number of particles.
Anyway, which is bigger: (10^80)^(10^123) or 10^10^125? Or are they equal? Working with iterated scientific notation makes me brain hurt.
It seems intuative that the first should be larger, but the extra three orders of magnitude in the second make me nervous.
Big number fun
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(10^80)^(10^123) is larger, since you have 10^80 and then you this have this number raised 10^123.
In the second case you only have 10^(10^125), about 83 orders of magnitude less (I think, apologies if I'm wrong).
In the second case you only have 10^(10^125), about 83 orders of magnitude less (I think, apologies if I'm wrong).
[img=left]http://img.photobucket.com/albums/v206/ ... iggado.jpg[/img] "You know, it's odd; practically everything that's happened on any of the inhabited planets has happened on Terra before the first spaceship." -- Space Viking
(Too... many... zeroes... Thank you lord, for cut and paste)
Thanks Hethrir,but b= 10^100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
I guess without the parenthesies your calculator compiled it the wrong way out.
But I think I've got it now. I dug out my calculus book and went over the laws of exponentation (ick, is that right?), so here we go:
"b" we know pretty well, it's that icky thing up there. "a" is tricky because it's a power to a power, but we have a rule for that; (a^m)^n=a^mn.
a=(10^80)^(10^123) which by the rule above=10^(80*(10^123))=10^(8*10^124), which now that I have my wits about me, is obviously less than 10^10^125 by ~200 squillion orders of magnitude.
Looking at it another way, if I decide to divide both a and b by 10, b=10^(10^125-1) but a=(10^80)^(10^123-1/80), and if we divide by 10 quite a few more times we discover that 10^123 is less than one eightyth 10^125.
Which is a lesson for me; don't get stupid just because it's (late) Summer. Or maybe if I want something done right do it myself. Or something else entirely that I'm not thinking of right now. Anyway, thanks for givin' it a shot.
Thanks Hethrir,but b= 10^100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
I guess without the parenthesies your calculator compiled it the wrong way out.
But I think I've got it now. I dug out my calculus book and went over the laws of exponentation (ick, is that right?), so here we go:
"b" we know pretty well, it's that icky thing up there. "a" is tricky because it's a power to a power, but we have a rule for that; (a^m)^n=a^mn.
a=(10^80)^(10^123) which by the rule above=10^(80*(10^123))=10^(8*10^124), which now that I have my wits about me, is obviously less than 10^10^125 by ~200 squillion orders of magnitude.
Looking at it another way, if I decide to divide both a and b by 10, b=10^(10^125-1) but a=(10^80)^(10^123-1/80), and if we divide by 10 quite a few more times we discover that 10^123 is less than one eightyth 10^125.
Which is a lesson for me; don't get stupid just because it's (late) Summer. Or maybe if I want something done right do it myself. Or something else entirely that I'm not thinking of right now. Anyway, thanks for givin' it a shot.