excelguru wrote:

excelguru wrote:

excelguru wrote:

ev3coolexit987654 wrote:

More improvement:
**(n) = *(*(…(*(*(n)))…)) with n↑↑**(n-1) *'s
***(n) = **(**(…(**(**(*(n))))…)) with n↑↑↑***(n-1) **'s
****(n) = ***(***(…(***(***(**(*(n)))))…)) with n↑↑↑↑****(n-1) ***'s
…

See a pattern? My number is **…**(271828184161803331415)^31415926535897932238462643383279502884197169399375105820974944 where there are *************[***(*(100))](***(*****(*****************(*************(*(1357924680314159265358979))))))

What does n↑↑**(n-1) mean?

Oh. It means n↑↑(**(n-1)). I thought you invented a new function without explaining it.

I assume *[n](1) = 1, right?

By the way, this is my 100th forum post

*(n) = G*(n-1), *(n-1)↑[*(n-1)]*(n-1)(*(n-1)) (from the post that introduced *)
And then we have the *[n] posts.

Congrats on 100 posts!

I finished the K function.

Directory of my stuff
H function
I function (lol 10:33:37 EST)
J function
K function
* function

A directory of my stuff:

Excelguru's extra-strong chained arrow notation
Super up-arrow notation
Hash notation and Graham notation
Super up-arrow notation (extension)
Super Ackermann function
Hyper-operator names
Extended G function
Another extension to Ackermann
Weak arrow notation
How to easily make big numbers
An alternative extension to up-arrows
An extension to that alternate extension
An extension to my original up-arrow extensions
2-parenthesis Ackermann function
An extension to Graham notation
An extension to the extended G function
The X function
The big list of numbers
The X+ function
My up-arrow numbers
Improving the Ackermann function

A(16180314159,16180314159,16180314159,16180314159,16180314159,16180314159,16180314159,16180314159,16180314159,16180314159,16180314159,16180314159)

The X function

The X function is another function I made. Here are the rules:

X(a) = a
X(a,b) = a^b
X(a,b,…,y,z,1) = X(a,b,…,y,z)
X(a,b,…,x,y,z) = X(X(a,b,…,x,y,z-1),X(a,b,…,x,y,z-1),…,X(a,b,…,x,y,z-1),X(a,b,…,x,y,z-1),z-1)

For example:

X(3,3,3) = X(X(3,3,2),X(3,3,2),2) = X(X(X(3,3,1),X(3,3,1),1),X(3,3,2),2) = X(X(X(3,3),X(3,3)),X(3,3,2),2) =
X(X(27,27),X(3,3,2),2) = X(27^27,X(3,3,2),2) = X(27^27,X(X(3,3),X(3,3)),2) = X(27^27,27^27,2) =
X(X(27^27,27^27),X(27^27,27^27)) = X((27^27)^(27^27),(27^27)^(27^27)) = ((27^27)^(27^27))^((27^27)^(27^27)) = (27^(27*27^27))^(27^(27*27^27)) = (27^27^28)^(27^27^28) = 27^(27^28*27^27^28) = 27^27^(28+27^28) > 27^27^27^28 > 27^27^27^27 = 27↑↑4

X↑X(a,b) = X(a,a,…,a,a) with b a's

Other than that, X↑X works just like X.

X↑X↑X(a,b) = X↑X(a,a,…,a,a) with b a's
X↑↑X(a,b) = X↑X↑…↑X↑X(a,a) with b X's

Then:

X↑↑X↑X
X↑↑X↑X↑X
X↑↑X↑↑X
X↑↑X↑↑X↑↑X
X↑↑↑X
X↑↑↑X↑X
X↑↑↑X↑X↑X
X↑↑↑X↑↑X
X↑↑↑X↑↑X↑X
X↑↑↑X↑↑↑X
X↑↑↑↑X
X↑↑↑↑↑X

And finally:

Xfinal(a,b) = X↑↑…↑↑X(a,a) with b arrows

How big is the mega?

In my review on Steinhaus-Moser notation, I explained that the mega is “two in a circle”. How big is the mega? It can be shown that it is equal to 256 in 256 triangles. Lets try to see how big it is. Let's use f(n) = 256 in n triangles.

f(0) = 256
f(1) = 256^256
f(2) = (256^256)^(256^256) = 256^(256*256^256) = 256^256^257 > 256^256^256 = 256↑↑3
f(3) > (256^256^256)^(256^256^256) = 256^(256^256*256^256^256) = 256^256^(256+256^256) > 256^256^256^256 = 256↑↑4

These values are getting hard to compare. However, if b>1, (a↑↑b)↑↑2 > a↑↑(b+1).

Proof:

(a↑↑b)↑↑2 = (a↑↑b)^(a↑↑b) = (a^a↑↑(b-1))^(a↑↑b) = a^(a↑↑(b-1)*a↑↑b) > a^a↑↑b = a↑↑(b+1)

This proof assumes a↑↑(b-1) > 1, which is true for all b > 1. Therefore, f(n) > 256↑↑(n+1), so the mega = f(256) > 256↑↑257. So that is how big the mega is.

excelguru wrote:

How big is the mega?

In my review on Steinhaus-Moser notation, I explained that the mega is “two in a circle”. How big is the mega? It can be shown that it is equal to 256 in 256 triangles. Lets try to see how big it is. Let's use f(n) = 256 in n triangles.

f(0) = 256
f(1) = 256^256
f(2) = (256^256)^(256^256) = 256^(256*256^256) = 256^256^257 > 256^256^256 = 256↑↑3
f(3) > (256^256^256)^(256^256^256) = 256^(256^256*256^256^256) = 256^256^(256+256^256) > 256^256^256^256 = 256↑↑4

These values are getting hard to compare. However, if b>1, (a↑↑b)↑↑2 > a↑↑(b+1).

Proof:

(a↑↑b)↑↑2 = (a↑↑b)^(a↑↑b) = (a^a↑↑(b-1))^(a↑↑b) = a^(a↑↑(b-1)*a↑↑b) > a^a↑↑b = a↑↑(b+1)

This proof assumes a↑↑(b-1) > 1, which is true for all b > 1. Therefore, f(n) > 256↑↑(n+1), so the mega = f(256) > 256↑↑257. So that is how big the mega is.

Oh man.

The big list of numbers

Welcome to the big list of numbers. Here you can find all sorts of numbers. This is sort of a work-in-progress.

Imaginary and complex numbers
Coming soon!

negative numbers (-∞ - 0)

Negative infinity (-∞)
This shouldn't really belong here.

Negative K(K(K(K(K(K(K(K(K(K(10000000))))))))))

Negative TREE(3)

Negative Graham's number

Negative megol

Negative googolplex

Negative googol

Negative duotrigintillion

Negative 1,000,000

Negative 1,000

Negative one

Negative 0.5

Negative 0.25

Negative 0.01

Negative 0.001

Negative 10^-100

Negative googolminex

Negative googolplexminex

Negative reciprocal of Graham's number

Negative reciprocal of TREE(3)

Negative reciprocal of K(K(K(K(K(K(K(K(K(K(10000000))))))))))

sub-one numbers (0 - 1)
Coming soon!

sub-million numbers (1 - 1,000,000)

1

2

3

pi rounded to the nearest 10^-1,000,000

pi / π

8
My favorite number.

10

11

12
Also called a dozen.

13
A baker's dozen. It is also an “unlucky number” in some countries.

17

20

30

50

100
The number of zeroes in a googol.

144
A gross (a dozen dozens) or 12^2.

500

666
The number of the beast from the bible.

1000

1001

1010

1110

1337

1728
A “great gross”, or 12^3.

1729
The smallest number that can be expressed as the sum of two positive cubes in two different ways.

2000

2048

2048 is the name of a game where you have to merge twos together to get higher numbers, and the goal is to get 2048. You can keep on going until you get 65,536.

3000

10,000

20,000

50,000

65,536

The largest number you can get in 2048.

100,000

142,857
This number has some interesting properties. Here are the first seven multiples of 142,857:

142,857x1 = 142,857
142,857x2 = 285,714
142,857x3 = 428,571
142,857x4 = 571,428
142,857x5 = 714,285
142,857x6 = 857,142

But…

142,857x7 = 999,999

200,000

500,000

999,999

999,999.99999

999,999.99999999999999999999

I just had to include it.

Sub-googol numbers (1,000,000 - 10^100)

1,000,000
Under my current definition, this is the smallest large number. Once it was 6000, and even before that it was 50 (which nobody really considers a big number!)

1,000,001
A number somewhat analogous to 1001.

1,001,000
Another number somewhat analogous to 1001.

1,111,111

1,333,337

2,000,000

5,000,000

10,000,000
The next order of magnitude from a million.

20,000,000

50,000,000

70,000,000

100,000,000
The second order of magnitude from a million. It is also called a myllion.

1,000,000,000
The third order of magnitude from a million, and the first number after a million which gets a unique name.

4,294,967,295
The largest number that can be stored in a long.

1,111,111,111

7,777,777,777

9,999,999,999

10,000,000,000
The first order of magnitude from a billion, and the fourth from a million.

100,000,000,000

1,000,000,000,000
The next number that gets a unique name after a billion. A trillion is the largest number that ends with “illion” that is recognized by most people, but further illion numbers do exist.

quadrillion = 10^15

byllion = 10^16
In Knuth's yllion system, x-yllion = 10^2^(n+2). This can make way bigger numbers than the regular illions, but it is harder to work with.

quintillion = 10^18

number of positions on a Rubik's cube

sextillion = 10^21

Avagadro's number = 602,214,129,270,000,000,000,000

septillion = 10^24

octillion = 10^27

nonillion = 10^30

Belphegor's prime = 1,000,000,000,000,066,600,000,000,000,001

This strange palindromic prime combines 13 (bad luck, the number of zeroes on each side), and 666 (the number of the beast, in the middle).

tryllion = 10^32
Knuth's yllions are an example of double exponential numbers. These numbers can at least be approximated with a^b^c, where a, b, and c are reasonably small numbers.

decillion = 10^33

undecillion = 10^36

duodecillion = 10^39

tredecillion = 10^42

quattuordecillion = 10^45

quindecillion = 10^48

10^50
There are approximately 10^50 atoms in the earth. This is the square root of a googol - we're making some progress here!

sexdecillion = 10^51

septendecillion = 10^54

octodecillion = 10^57

novemdecillion = 10^60

vigintillion = 10^63

quadryllion = 10^64

unvigintillion = 10^66
Then there's duovigintillion, trevigintillion, quattuorvigintillion, and so on.

10^80
There are approximately 10^80 atoms in the observable universe. We haven't even got to a googol yet!

trigintillion = 10^93

untrigintillion = 10^96

thousandthoogol = 10^97

hundredthoogol = 10^98

duotrigintillion / tenthoogol = 10^99

10^100-10^-10
Lots of calculators overflow just under a googol.

Nested exponential numbers (10^100 - 10↑↑100)

googol = 10^100 = 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
The googol is one of the “popular” large numbers. It is very huge, and the numbers you find in science rarely get bigger. WAY bigger numbers exist in pure mathematics. A googol atoms would fill 100 quadrillion copies of the observable universe.

70!
The number of ways to arrange 70 objects.

75!

80!

10^200

10^300

10^500

10^1000

I call this number the great googol.

10^10,000

I call this number the myrett googol.

10^1,000,000

I call this number the squomillet googol.

10^1,000,000,000

I call this number the cuomillet googol, or the squobillet googol.

10^1,000,000,000,000

I call this number the teomillet or squotrillet googol.

10^10^15

I call this number the peomillet or squoquadrillet googol. My scientific notation calculator can manipulate numbers up to this number, but has problems beyond it due to scratch not being able to display values accurately beyond a quadrillion. Up here is another big jump:

10^10^50

10^10^80

googolplex = 10^10^100

Knuth arrow numbers (10↑↑100 - 10→10→(10^100))
Coming soon!

Exploding arrow range (10→10→(10^100) - 3→3→3→3)
Coming soon!

Ad infinitum range (3→3→3→3 - ∞)
Coming soon!

Add 1729 plz?

The X+ function is an improvement on the X function that uses a rather unusual idea. It has things like X+(7,7,(6,5,(3)),3,7,(6,(7),6)). I need to introduce a new symbol, ◊. ◊ is like # but it can only have numbers.

X+((#),#2) = X(#2)
X+(◊,a,(a),#) = X+(◊,a,a,#)
X+(◊,a,(b),#) = X+(◊,a,X+(◊,a,(b-1),#))
X+(a,b) = a^b
X+(a,b,…,x,y,z) = X+(a,b,…,x,(a,b,…,x,(…(a,b,…,x,(a,b,…,x,y,z-1),z-1)…),z-1),z-1) with y z-1's
X+(○(a,b,…,x,y,z)○) = X+(○(a,b,…,x,(a,b,…,x,(…(a,b,…,x,(a,b,…,x,y,z-1),z-1)…),z-1),z-1)○) - ○ is a special symbol that can contain single parentheses.
X+(○(a,b)○) = X+(○(a^b)○)
X+(a,b,…,y,z,1) = X+(a,b,…,y,z); X+(○(a,b,…,y,z,1)○) = X+(○(a,b,…,y,z)○)
X+(a,b,…,y,1,z) = X+(a,b,…,y); X+(○(a,b,…,y,1,z)○) = X+(○(a,b,…,y)○)

This is a really fast growing function. For example, lets take X+(2,2,2):

X+(2,2,2) = X+(2,(2,2,1),1) = X+(2,(2,2),1) = X+(2,(4),1) = X+(2,X+(2,(3),1),1) = X+(2,X+(2,(3))) = X+(2,X+(2,X+(2,(2))))
= X(2,X(2,X(2,X(2,(1))))) = X(2,X(2,X(2,X(2,2)))) = 2^2^2^2^2

While:

X(2,2,2) = X(X(2,2,1),X(2,2,1),1) = X(X(2,2),X(2,2)) = (2^2)^(2^2) = 2^(2*2^2) = 2^2^3

More numbers.

10↑100 = googol
10↑10↑100 = googolplex
10↑10↑10↑100 = googolplexian
10↑10↑10↑10↑100 = googolplexitrion
10↑10↑10↑10↑10↑100 = googolplexitetron

Googolplexipenton, googolplexihexon, googolplexihepton, googolplexiocton, googolplexiennon, googolplexidekon

10↑10↑…↑10↑100 (100 10's) = googolhect (can also be called googolplexiennacontaennon)

We need some name for 10↑↑100 - this is a new kind of googol - some sort of mega-googol - I'll call it the megol.

10↑↑100 = megol
10↑↑101 = megolplex
10↑↑10↑100 = googolmegex
10↑↑10↑↑100 = megolmegex
10↑↑10↑↑10↑↑100 = megolmegexian

Now what? Well mega- is the SI prefix for 1,000,000, so let's continue with SI prefixes.

10↑↑↑100 = gigol
10↑↑↑10↑↑↑100 = gigolgigex
10↑↑↑↑100 = terol
10↑↑↑↑10↑↑↑↑100 = terolterex
10↑↑↑↑↑100 = petol
10↑↑↑↑↑↑100 = exol
10↑↑↑↑↑↑↑100 = zettol
10↑↑↑↑↑↑↑↑100 = yottol

Now we have run out of SI prefixes. Let's continue by making up new prefixes!

10↑↑↑↑↑↑↑↑↑100 = ennattol
10↑↑↑↑↑↑↑↑↑↑100 = dekattol

Let's have a new notation. gol(x) = 10↑100

gol(11) = hendekattol
gol(12) = dodekattol
gol(13) = triadekattol

Continue using Greek roots…

gol(20) = icosattol
gol(30) = triacontattol
gol(40) = tetracontattol
gol(90) = ennacontattol
gol(99) = ennacontaennattol
gol(100) = hectattol
gol(1000) = chiliattol
gol(10000) = myriattol
gol(10^6) = ekatomyriattol
gol(10^10) = dekadisekatomyriattol
gol(10^30) = ennisekatomyriattol
gol(10^33) = dekisekatomyriattol

These are just some examples. There are lots and lots of other names possible.

4294967295:
Biggest number that can be stored in a long.

Improving the Ackermann function.

I recently came up with this improvement on the Ackermann function. It changes the second rule to:

A(x,0) = A(x-1,x-1) instead of A(x-1,1)

There. That's faster!

my favorite large number is belphegor's prime: 1000000000000066600000000000001

The big list of functions

I was going to make a function list, but it was unsuccessful.

InvK(x)

Which K are you referring to? Mine?

InvORCHARD(x)

I don't think you defined orchard anywhere…

ev3coolexit987654 wrote:

InvK(x)

Which K are you referring to? Mine?

InvORCHARD(x)

I don't think you defined orchard anywhere…

Yes, I am referring to your K function. The ORCHARD function is defined ORCHARD(x) = TREE(TREE(…(TREE(TREE(3)))…)) with x “TREE”s

excelguru wrote:

4,294,967,295
The largest number that can be stored in a long.

Sorry! It was supposed to be 9223372036854775807

excelguru wrote:

ev3coolexit987654 wrote:

excelguru wrote:

4,294,967,295
The largest number that can be stored in a long.

Sorry! It was supposed to be 9223372036854775807

WHAT THE HECK?

Yes, it is 92233(whatever). I checked.