Magical Realist
Valued Senior Member
Ok..you may need an aspirin after reading this.
http://listverse.com/2012/03/12/10-enormous-numbers/
http://listverse.com/2012/03/12/10-enormous-numbers/
I, too, question some of this linguistic analysis. One of these guys said that since our words "first" and "second" don't fit the format of "third, fourth," etc. (i.e., they are not formed by inflecting the word for the number itself), this implies that our ancestors could only count to two.There's a story (of disputed veracity) that some primitive cultures have limited counting ability, only having words for "one", "two", and "many."
((((A googolplex to the googolplex power) to the googolplex power) to the googolplex power) to the googolplex power) to the googolplex power . . . . . until 12 of my 15 seconds run out, giving me time to make sure the parentheses are paired.You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.
Agreed on the use of the word. My number was "googolplex(up-arrowed googolplex times)" but I didn't want to bother trying to figure out the tex to write it.((((A googolplex to the googolplex power) to the googolplex power) to the googolplex power) to the googolplex power) to the googolplex power . . . . . until 12 of my 15 seconds run out, giving me time to make sure the parentheses are paired.
As far as I know, "googolplex" -- 10^(10^100), ten to the googol power -- is the largest number for which there is an English word. So it's reasonable to start with that.
If I'm not mistaken, we don't need a word for a larger number because this one is very much larger than the total number of subatomic particles in the universe.
Did you mean $$\left( 10^{10^{10^2}} \right) \uparrow \uparrow \left( 10^{10^{10^2}} \right) = \left( 10^{10^{10^2}} \right) \uparrow \uparrow \uparrow 2$$ ?Agreed on the use of the word. My number was "googolplex(up-arrowed googolplex times)" but I didn't want to bother trying to figure out the tex to write it.
...actually I was thinking something like $$\left( 10^{10^{10^2}} \right) \uparrow^{\left( 10^{10^{10^2}} \right)}\left( 10^{10^{10^2}} \right)$$ but thanks for laying out the starting tex!Did you mean $$\left( 10^{10^{10^2}} \right) \uparrow \uparrow \left( 10^{10^{10^2}} \right) = \left( 10^{10^{10^2}} \right) \uparrow \uparrow \uparrow 2$$ ?
What you're saying here is that if Fraggle Rocker omitted the parentheses completely he would produce a much greater result..?Because of the laws of exponents and they way you have your parentheses, you miss out on getting to truly large numbers.
((googolplex ^ googolplex) ^ googolplex) ^ googolplex
= googolplex^(googolplex^3)
= (10^(10^(10^2)))^( (10^(10^(10^2)))^3 )
= (10^(10^(10^2)))^(10^(3×10^(10^2)))
= 10^( 10^(10^2) × 10^(3×10^(10^2)) )
= 10^( 10^( 3×10^(10^2) + 10^2 ) )
And if you have N googolplexes in your expression, FR(N) then log log log log FR(N) = log log ( (N-1)×10^(10^2) + 10^2 ) ≈ log ( log (N-1) + 10^2 ) ≈ 2
While
$$\HUGE 10^{10^{10^{10^{10^{10}}}}}$$ is much larger (and easier to write). log log log log 10^(10^(10^(10^(10^10)))) = 10,000,000,000
Knuth's uparrow notation $$10 \uparrow \uparrow 6$$ is the same number.
So we can write $$10 = 10\uparrow \uparrow 1 \lt 10 \uparrow \uparrow 2 \lt \textrm{googol} \lt 10 \uparrow \uparrow 3 \lt \textrm{googolplex} \lt 10 \uparrow \uparrow 4 \lt FR(2) \lt FR(10^9) \lt 10 \uparrow \uparrow 5 \lt 10 \uparrow \uparrow 6 \lt 10 \uparrow \uparrow 10 = 10 \uparrow \uparrow \uparrow 2$$
http://mathworld.wolfram.com/PowerTower.html
Maybe this number would be bigger:What you're saying here is that if Fraggle Rocker omitted the parentheses completely he would produce a much greater result..?
Hmm, what about $$G^{G^{G^{G^{G^{G^{G^{G^{G^{G^{G}}}}}}}}}}...$$?Maybe this number would be bigger:
Googleplex to the (googleplex to the (googleplex to the (googleplex to the (googleplex to the googleplex power) power) power) power) power.
I can't see any way to write this in notation without the parentheses. And I don't see any way at all to write it in strict prose.
But this way was much harder. I'm still not sure I've got everything paired correctly!
Maybe this number would be bigger:
Googleplex to the (googleplex to the (googleplex to the (googleplex to the (googleplex to the googleplex power) power) power) power) power.
Ahh, there ya go. Since we only have 15 seconds, say Graham's number = GBut I think both Graham's number and $$3 \rightarrow 3 \rightarrow 3 \rightarrow 3 $$ are larger still.
But you invite the seriesAnother interesting challenge would be to make the largest definite number possible using any operators but only, say, 5 unique digits (once each) and no variables.
Ah yes, unary operators break things.But you invite the series
$$5 \rightarrow 6 \rightarrow 7 \rightarrow 8 \rightarrow 9 \\ (5!) \rightarrow (6!) \rightarrow (7!) \rightarrow (8!) \rightarrow (9!) \\ ((5!)!) \rightarrow ((6!)!) \rightarrow ((7!)!) \rightarrow ((8!)!) \rightarrow ((9!)!) $$
etc.
All of which are (finite!) magnitudes too large to easily place in terms of mere power towers.
Maybe :
$$\nolimits 7\uparrow^{9!}8$$