Universe Sandbox
General Category => Astronomy & Science => Topic started by: vh on May 21, 2012, 02:58:24 PM
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So this post is about 'Knuths Up Arrow Notation and Hyperoperation', which is basically a fancy way of writing big numbers so they look small. Although this is an Astronomy and Science forum, maybe we could make it a forum for all sorts of academiology.
There is a wikipedia article on this, but i feel i can explain it better...i think
so 3*3 is equal to 3+3+3
and 3^3 is equal to 3*3*3 which is equal to (3+3+3)+(3+3+3)+(3+3+3)
3↑3 is equal to 3^3 and a↑b is equal to a^b
3↑↑3 is equal to 3 to the power of 3, 3 times. So 3^3^3
3↑↑↑3 is equal to 3 to the power 3↑↑3 times. So 3^3^3^3^....... 3↑↑3 times
They're not that big of a deal, for example, 3↑↑↑3 has only around 3.6 trillion digits.
3→3→3=3↑↑↑3
a→b→c=a↑↑↑c amount of arrows↑↑↑ b
2→3→4→5 = (2→3→4)→(2→3→4)→(2→3→4)→(2→3→4)→(2→3→4)
so
3→3→3→3=(3→3→3)→(3→3→3)→(3→3→3)=(3↑↑↑3)→(3↑↑↑3)→(3↑↑↑3)=(3↑↑↑3)↑↑↑↑ 3↑↑↑3 arrows ↑↑↑ (3↑↑↑3)
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if you want to talk about big #'s, than: (googleplex)(googleplex)>ur arrow thingies.
(i tink)
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2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2>googleplex→googleplex→googleplex→googleplex
and easier to write out too
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(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)
>2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2
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The googleplex thing is quite miniscule :)
In fact, i can write a number that is
(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex) times bigger than the number you just wrote like this: 10↑↑4
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nah. how about
12↑1008
that's pretty close to graham's number, amirite?
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that equals to 12↑(8↑↑100)
and it is bigger than my previous number, but not this one:
3→3→3→3
not sure about GN, i'll check tommorow
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no the ↑100 is equivalent to ↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑
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oh, i thought you meant as in 8 stacked 100 times
well then, my number is still bigger because the number of ↑ arrows in my number is more than the stars in this galaxy :P
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ok fine.
18↑8↑↑↑8M↑100↑M↑M80
M = Moser's number
welp, still not one trillionth to finite numbers.
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moser < 3→3→4→2
your number is then a bit less than
(18→100→8→(8↑(3→3→4→2)))→80((3→3→4→2)→(3→3→4→2))
i think...
a bigger number: G→G→G→G→G→G→G→G
i think..
which means (G→G→G→G→G→G→G)→ itself G times
and then a single G→G→G→G→G→G→G equals
(G→G→G→G→G→G)→ itself G times
and so on
G is grahams number
Grahams number: 3↑↑↑3 is the first tower G1
3→3→(3↑↑↑3) is the second tower G2 this also equals
3→3→3→3→3
G3 is 3→3→3→3→3→3→3
and so on...
Grahams number, G64 is
3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3
which alone is probably bigger than your number :P
Mosers # is miniscule in comparison
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42.
pwn'd.
i has got the largest number. :D
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I am sorry if this is considered necromancy, but I found this thread on a google search and I would like to clear up some misunderstandings.
A googol is just 10^100, multiplying them together does not give much to size on this scale. A hundred googols multiplied together is just (10^100)^100 = 10^10000, very small compared to what the up-arrows make.
The OP misunderstands the way chained arrows work. They are much more powerful that he writes.
It is right that n→m→p means n↑pm, but longer chains are evaluated according to this rule (where X is a subchain):
X→n→m=X→(X→n-1→m)→m-1
And if any chain has a one it is simply cut there: X→1→Y = X
Since 2↑↑↑↑↑↑2 = 4 no matter the number of up-arrows, the long chain above claimed to be bigger than the googol mess is actually just a four.
On the other hand 3→3→3→3 turns out to be bigger than Grahams number g64 even though g64 has a completely uncountable number of up-arrows. (Try to expand the chained arrow according to the rules above). So Grahams number is not 3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3→3
it is way way way way way less.
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woah did you type that or copy that?
and also what's the diff from
->
and
^
|
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woah did you type that or copy that?
and also what's the diff from
->
and
^
|
I typed it.
If there is just one arrow then they are the same (and the same as exponentation ^). The difference is in how long chains are expanded.
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no i mean the up arrow and the side arrow
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does 3→3 = 3↑↑↑3?
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does 3→3 = 3↑↑↑3?
No, just 3↑3 = 3^3 = 27 actually. It only becomes powerful with longer chains.
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ok so 3→3→3 is 3↑↑↑↑↑↑↑↑↑3 then? or is it 3↑↑↑3↑↑↑3?
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ok so 3→3→3 is 3↑↑↑↑↑↑↑↑↑3 then? or is it 3↑↑↑3↑↑↑3?
3→3→3 is 3↑↑↑3
You cannot just put paranteses and for example switch 3→3→3 with 3→(3→3) = 3→27 or (3→3)→3 = 27→3
Chains are analyzed according to the rules I wrote above.
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Interesting...
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tass is nao the expert on knuts up arrow notation! applus