Knuths Up Arrow Notation and Conway Chained Arrow Notation

[vh]

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)

[blotz]

if you want to talk about big #'s, than: (googleplex)(googleplex)>ur arrow thingies.
(i tink)

[vh]

2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→2→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]

(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(googleplex)(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]

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]

nah. how about
12↑¹⁰⁰8

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

[vh]

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]

no the ↑¹⁰⁰ is equivalent to ↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑

[vh]

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

[Darvince]

ok fine.

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

welp, still not one trillionth to finite numbers.

[vh]

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

Mosers # is miniscule in comparison

[blotz]

42.
pwn'd.

i has got the largest number.

[Tass]

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]

woah did you type that or copy that?
and also what's the diff from

->
and
^
|

[Tass]

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]

no i mean the up arrow and the side arrow

[Darvince]

does 3→3 = 3↑↑↑3?

[Tass]

does 3→3 = 3↑↑↑3?

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

[Darvince]

ok so 3→3→3 is 3↑↑↑↑↑↑↑↑↑3 then? or is it 3↑↑↑3↑↑↑3?

[Tass]

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]

Interesting...

[blotz]

tass is nao the expert on knuts up arrow notation! applus