A math question

post by Mycroft65536 · 2011-06-18T02:22:12.786Z · LW · GW · Legacy · 5 comments

Is (4^^^4)/(3^^((3^^3)^^3)) larger than one?

I need to know for a game of Nomic

5 comments

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comment by Manfred · 2011-06-18T02:56:10.902Z · LW(p) · GW(p)

Yes. 3^^^4 < 4^^^4, and the denominator has the correct number and type of operations to be 3^^^4, since 3^^^4 = 3^^3^^3^^3 - however it's actually smaller. Up arrow notation evaluates from right to left, which results in the largest possible number. The parentheses just muck things up and make the result smaller - so the answer is much much greater than 1.

Replies from: Normal_Anomaly
↑ comment by Normal_Anomaly · 2011-06-19T01:34:16.420Z · LW(p) · GW(p)

So 3^^^4 is an exponential tower of 3s 3^^4 levels tall?

Replies from: Manfred
↑ comment by Manfred · 2011-06-19T02:03:11.528Z · LW(p) · GW(p)

Hm, let's find out. (Open this if you want to have a go at finding that answer first)

3^^X is an exponential tower of 3s that is X high. 3^^^4 = 3^^3^^3^^3 (i.e. it's 3 ^^ itself, with four 3s, just like 3^^4 is 3 ^ itself with four 3s). So 3^^^4 is an exponential tower of 3s that is 3^^3^^3 high. 3^^3^^3 is 3 ^^ itself 3 times, so it's 3^^^3.

So 3^^^4 is an exponential tower of 3s 3^^^3 levels tall.

Replies from: Normal_Anomaly
↑ comment by Normal_Anomaly · 2011-06-19T02:26:12.823Z · LW(p) · GW(p)

Thanks for the explanation. It was worth the brain explosion.

comment by Normal_Anomaly · 2011-06-18T02:46:40.043Z · LW(p) · GW(p)

This is a rough guess, but since 3^^3 is a number I could type out in this comment box, and the ^^ operator is in general way less powerful than the ^^^ operator, I'd say yes.