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Author Topic: Knuths Up Arrow Notation and Conway Chained Arrow Notation  (Read 25434 times)

vh

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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)
« Last Edit: May 21, 2012, 03:53:44 PM by mudkipz »

blotz

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #1 on: May 21, 2012, 04:30:23 PM »
if you want to talk about big #'s, than: (googleplex)(googleplex)>ur arrow thingies.
(i tink)

vh

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #2 on: May 21, 2012, 04:38:21 PM »
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

blotz

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #3 on: May 21, 2012, 05:25:02 PM »
(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

vh

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #4 on: May 21, 2012, 05:59:00 PM »
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

Darvince

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #5 on: May 21, 2012, 08:06:41 PM »
nah. how about
12↑1008

that's pretty close to graham's number, amirite?

vh

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #6 on: May 21, 2012, 08:12:29 PM »
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

Darvince

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #7 on: May 21, 2012, 08:14:35 PM »
no the ↑100 is equivalent to ↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑

vh

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #8 on: May 21, 2012, 08:16:46 PM »
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

Darvince

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #9 on: May 21, 2012, 09:00:16 PM »
ok fine.

18↑8↑↑↑8M↑100↑M↑M80
M = Moser's number

welp, still not one trillionth to finite numbers.

vh

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #10 on: May 22, 2012, 03:14:57 AM »
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

blotz

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #11 on: May 24, 2012, 06:06:38 PM »
42.
pwn'd.

i has got the largest number. :D

Tass

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #12 on: January 22, 2013, 04:44:18 PM »
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.

blotz

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #13 on: January 22, 2013, 05:16:16 PM »
woah did you type that or copy that?
and also what's the diff from

->
and
^
|

Tass

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #14 on: January 23, 2013, 05:34:59 AM »
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.

blotz

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #15 on: January 23, 2013, 03:03:31 PM »
no i mean the up arrow and the side arrow

Darvince

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #16 on: January 23, 2013, 03:21:39 PM »
does 3→3 = 3↑↑↑3?

Tass

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #17 on: January 24, 2013, 08:05:59 AM »
does 3→3 = 3↑↑↑3?

No, just 3↑3 = 3^3 = 27 actually. It only becomes powerful with longer chains.

Darvince

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #18 on: January 24, 2013, 11:13:56 AM »
ok so 3→3→3 is 3↑↑↑↑↑↑↑↑↑3 then? or is it 3↑↑↑3↑↑↑3?

Tass

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #19 on: January 24, 2013, 12:47:02 PM »
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.

tuto99

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #20 on: January 24, 2013, 03:27:12 PM »
Interesting...

blotz

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Re: Knuths Up Arrow Notation and Conway Chained Arrow Notation
« Reply #21 on: January 24, 2013, 04:53:32 PM »
tass is nao the expert on knuts up arrow notation! applus