My Number is Bigger! (Everymen Needed!)

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PsiSquared
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Fri Jan 15, 2016 9:48 am UTC

{3}15 = 15{2}15 = 14{2}491520 ~ 13{2}10147967.96 ~ 12{2}10^10^147967.43 ~ 11{2}10^10^10^147967.43 ~ ... ~ 147967.43F14 ~ 5.17F15

Notes:

For large n, {2}n = n*2^n ~ 2^n = 10^(n*log 2)
For very large n, {2}n ~ 10^(n*log 2) ~ 10^n

Also, notice that {3}15 can be roughly approximated by F15. This is a general result: {3}n is roughly equal to Fn. And the same is true for many other functions, such as Username's n|()() notation and the function a^^n (for any smallish value of a>2)

Of-course, this is only a rough approximation. {3}15 is actually closer to F16, 15|()() is actually a bit larger than F16, and 3^^15 is closer to F14 (about 1.099F14). But it is a good example of a general principle of googology: Once you repeat a process enough times, the details are less important than the type of recursion you're doing.

Another principle, which is similar but far more accurate, is that once x goes beyond F4 or so, it doesn't matter whether you do 10^x or 3^x or 2^x.

For example, once you've figured out that 3^^15 ~ 1.099F14, you can know - without doing any further calculations - that 3^^20 ~ 1.099F19 and 3^^1000 ~ 1.099F999 (just replace every 3 you're adding to the power tower with a 10).

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Fri Jan 15, 2016 10:52 am UTC

20|()()

Around 10^^21, though slightly larger actually.

Now, at some point even the numbers on top of the power tower don't matter that much...
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Fri Jan 15, 2016 11:32 am UTC

username5243 wrote:Now, at some point even the numbers on top of the power tower don't matter that much...


Indeed.

Once you have a power tower with "oh, about 10100 tens" then the top number becomes completely irrelevant. At least until you've calculated the height of the tower itself precisely.

But we haven't reached that point yet. So if we really want to be precise, we can still calculate a number like 20|()() to better precision:

20|()() = 20|()|...20 times...|() = 1.048576E26 | ()|...19 times...|() =
= E(1.048576E26 * log (1.048576E26))| ()|...18 times...|() ~ E(2.73E27)|()|..18 times...|() ~ (E(2.73E27))F18 = (2.73E27)F19 ~ 27.436F20 ~1.438F21.

BTW, I'm not complaining that you didn't do that. The exact calculation is time consuming and "10^^21" is perfectly fine for many purposes. I'm simply pointing out that it is still both possible and meaningful to do so, at these scales.

And I send:
<5,2,2> = [5,1,2 | 2] = [5, 5^2, 2 | 1] = [5,25,2] = [5,24,5^2] = [5,24,25] = [5,23,5^25] ~ [5,23,3E17] = [5,22, 5^(3E17)] =
= [5,22, 10^(3E17 * log 5)] ~ [5,22,E(2.1E17)] ~ (2.1E17)F23 ~ 17.319F24 ~ 1.239F25

(and if I wanted to go by the "quick-and-dirty" approach: <5,2,2> ~ F(5^2) = F25)

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Fri Jan 15, 2016 3:47 pm UTC

{3}{2}{1}{0}1 = {3}64 = 64{2}64 = 63{2}270 ~ 62{2}10^10^20.55 ~ 20.55F64

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Fri Jan 15, 2016 8:05 pm UTC

And now we reach:

10^^100 = F100

(A number that Jonathan Bowers calls a "giggol")
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sat Jan 16, 2016 3:56 pm UTC

<5,3,2> = [ 5,1,3 | 2 ] = [ 5, [5,1,3] , 3 ] = [ 5, 5^3, 3 ] = [5,125,3] = [5,124,125] = [5,123,5^125] ~ [5,123,2.35E87] =
= [5,122,5^(2.35E87)] ~ [5,122,EE87.22] ~ (EE87.22)F122 = 87.22F124 = 1.941F125

(or using the dirty-quick method: <5,3,2> ~ F(5^3) = F125)

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Sat Jan 16, 2016 4:47 pm UTC

Image
Alternatively:
Image

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sat Jan 16, 2016 5:51 pm UTC

<7,3,2> ~ 2.462F343
(for the calculation, see below)

This number can also be written as: 7^7^7^7...^7^7^7^3 with 7^3 (343) sevens.

And the above value can be calculated (using WolframAlpha for the first few iterations) like this:
<7,3,2> = [7,343,3] = [7,339,7^7^7^7^3] = [7,339,289.80F3] ~ 289.80F(3+339) = 289.80F342 ~ 2.462F343

(or by the quick-and-dirty method: <7,3,2> ~ F(7^3) = F343)

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Sat Jan 16, 2016 6:45 pm UTC

10^^500
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sat Jan 16, 2016 6:47 pm UTC

And that's 9.

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Re: My Number is Bigger! (Everymen Needed!)

Postby Sabrar » Sat Jan 16, 2016 8:09 pm UTC

<9,3,2> ~ F(9^3) = F729

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Sat Jan 16, 2016 10:02 pm UTC

10^^1000
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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Sun Jan 17, 2016 2:07 am UTC

F2016 = 10↑↑2016

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sun Jan 17, 2016 8:26 am UTC

Sabrar, is everything we are posting clear so far? If not, don't hesitate to say so.
(It seems that all the everymen except Sabrar has left. A dangerous omen?)

Anyway:

{4}2 = {3}{3}2 = {3}{2}{2}2 = {3}{2}8 = {3}2048 = 2048{2}2048 =
= 2047{2}2048x22048 ~ 2047{2}6.62x10619 ~ 2046{2}EE619.3 ~ 619.3F2048 ~ 2.792F2049

(and the rough calculation: {4}2 = {3}2048 ~ F2048)

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Sun Jan 17, 2016 11:32 am UTC

10^^5000
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Re: My Number is Bigger! (Everymen Needed!)

Postby Keyboard masher » Sun Jan 17, 2016 5:56 pm UTC

1+2{3}4=1+{3}{3}4>1+{3}2^^4=1+{3}65536>1+2^^65536>2^^65536.

Clearly, the 1+ was the key.

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Sun Jan 17, 2016 6:28 pm UTC

My hope is that this isn't fundamentally flawed - at least when doing it through a forum. Then again, that's pessimistic.

(From user's definition:)

n>0 = 10^n
0>1 = 10>0 = 10^10
(n+1)>1 = (n>1)>0 = 10^(n>1)

1>1 = 10^(0>1) = 10^^3
2>1 = 10^(1>1) = 10^^4

5678>1 = 10^^(5678+2) = 10^^5680 = F5680

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sun Jan 17, 2016 7:22 pm UTC

emlightened wrote:My hope is that this isn't fundamentally flawed - at least when doing it through a forum. Then again, that's pessimistic.


I don't understand what you've just said. What is the "this" you're referring to? And what's pessimistic?

If you are referring to the situation I've raised:

It seems that most math-oriented "everymen" are simply not interested enough in googology to closely follow a thread like this one for a long period of time. This actually makes sense, if you think about it: Any person with enough enthusiasm for the subject would already be a googologist himself.

I think that what this thread really needs is an active, committed "everyman" co-moderator. Someone whose very job is to give constant feedback and telling us how well we're doing. Unfortunately, I'm not very optimistic about finding volunteers for this job...

Anyway:
FE4 = F10000 = 10^^10000 = 10^^10^4

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Sun Jan 17, 2016 8:20 pm UTC

I was referring to that situation, and saying that my opinion was pessimistic, to clarify.

Finding a moderator for this sort of game would probably be best done beforehand, if we do it again.


7^^7^7 ~ 10^^10^6

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Sun Jan 17, 2016 8:28 pm UTC

For those interested in more accurate approximations:
7^^7^7 ~ 5.843F(823 542)

And I'll post:
8^^8^8 ~ 7.180F(16 777 215)
(or in very rough terms: FE7 = 10^^10^7)



ׂ

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Re: My Number is Bigger! (Everymen Needed!)

Postby Sabrar » Sun Jan 17, 2016 8:43 pm UTC

PsiSquared wrote:Sabrar, is everything we are posting clear so far? If not, don't hesitate to say so.

No problems so far with the notations. I generally like my math to be precise so it irks me a little to see the approximations described in this post but I understand why they are necessary. Ask me about 3 pages later whether I'm still comfortable with it. :)

Send: FE8

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Sun Jan 17, 2016 10:35 pm UTC

Unfortunately, for virtually any two notations in googology, we can only ever use approximations, and never exact values. There are some exceptions, like 3-element chained arrows being the same as up arrows, but not many.


(12!)^! = 479001600^! ~ (5.26300)F(479001597) ~~ F479001600

"Therefore it is in the interests not only of public safety but also public sanity if the buttered toast on cats idea is scrapped, to be replaced by a monorail powered by cats smeared with chicken tikka masala floating above a rail made from white shag pile carpet."

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Mon Jan 18, 2016 12:20 am UTC

emlightened wrote:Unfortunately, for virtually any two notations in googology, we can only ever use approximations, and never exact values.


That's true. But in most notations, you can get as close as you like to any number you wish. So it really isn't that different from the way that we are forced to round numbers when we use scientific notation (say, by writing 7^7^7 ~ 3.7598x10695 974).

This, by the way, is why I think it is vital to agree on some kind of standard notation (similar to scientific notation) for our numbers on this thread. We've been using my "F-notation" for this up until now and it worked well... But we are getting close to the limit of this system already. So we'll need something else, soon.

Sabrar wrote:I generally like my math to be precise so it irks me a little to see the approximations described in this post but I understand why they are necessary.


They're not really necessary. You can calculate any googological number to arbitrary precision, even though it is impractical to write its decimal expansion in full.

The rough-and-quick approximations just make things easier to follow. Think of how easy it was for you to submit <9,3,2> and be sure that it is in the ballpark of F729. But you could have also done it the hard way, and gotten a nice accurate figure: 2.8423735885 F 728.

But there's a deeper reason for learning to use these "rough-and-quick" methods: Once the argument gets large enough, these rough-and-quick approximations actually give you very precise answers.

For example, let's say that I use WolframAlpha to calculate 7^^7:

7^^7 = 7^7^7^7^7^7^7 ~ 10^10^10^10^10^10^5.842593328962333 = 5.842593328962333 F 6

In this case WA gives the answer with a precision of 16 digits. Not bad, right?

Now, I'll use my rough rule of "a^x ~ 10^x when a is small and x is larger" to deduce 7^^8:

7^^8 = 7^(7^^7) ~ 7^(5.842593328962333 F 6) ~ 10^(5.842593328962333 F 6) = 5.842593328962333 F 7

This may seem like a "rough approximation", but the truth is that it is far more accurate than the original result given by WolframAlpha. In fact:
7^(5.84259332896233 F 6) > 5.84259332896232999999...9999 F 7 (with more 9's then there are atoms in the entire observable universe).

So if you feel okay with your calculator rounding things to 16 (or 100 or a million) digits, you should have no qualms about using these methods (as long as you're dealing with numbers larger than about F4).

Anyway:
<5,13,2> ~ 8.9310685950807299484102 F 1 220 703 125

The calculation (all the given digits are correct):
Spoiler:
<5,13,2> = 5^5^5^....[5^13 fives]...^5^5^5^13 = 5^5^5^....[1 220 703 125 fives]...^5^5^5^13
= 5^5^5^....[1 220 703 124 fives]...^5^5^5^1 220 703 125
~ 5^5^5^....[1 220 703 123 fives]...^5^5^5^(3.7518175023025795968973416370036x10853 234 868)
~ 5^5^5^....[1 220 703 122 fives]...^5^5^5^10^(2.622407895852385302953541490022x10853 234 868)
~ 5^5^5^....[1 220 703 121 fives]...^5^5^5^10^10^(2.622407895852385302953541490022x10853 234 868)
~ 5^5^5^....[1 220 703 120 fives]...^5^5^5^10^10^10^(2.622407895852385302953541490022x10853 234 868)
= 5^5^5^....[1 220 703 120 fives]...^5^5^5^(10^10^10^(2.622407895852385302953541490022x10853 234 868))
~ 10^10^10^....[1 220 703 120 tens]...^10^10^10^(10^10^10^(2.622407895852385302953541490022x10853 234 868))
= 10^10^10^....[1 220 703 123 tens]...^10^10^10^(2.622407895852385302953541490022x10853 234 868))
~ 10^10^10^....[1 220 703 124 tens]...^10^10^10^10^(853 234 868.418700243853009601566)
~ 10^10^10^....[1 220 703 125 tens]...^10^10^10^10^10^8.9310685950807299484102
= 8.9310685950807299484102 F 1 220 703 125

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Mon Jan 18, 2016 12:44 am UTC

And I'll post:

FE10 = 10^^10^10
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Mon Jan 18, 2016 1:12 am UTC

3^^^3 = 3^^3^^3 = 3^^(3^3^3) = 3^^(3^27) = 3^^(7 625 597 484 987) ~ 1.0990208494370288685629175 F 7 625 597 484 986
(a number that Jonathan Bowers calls a tritri. It is a power tower of 7 625 597 484 987 threes).

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Re: My Number is Bigger! (Everymen Needed!)

Postby SirGabriel » Mon Jan 18, 2016 1:35 am UTC

PsiSquared wrote:This, by the way, is why I think it is vital to agree on some kind of standard notation (similar to scientific notation) for our numbers on this thread.

I agree, I think that was the problem with the last "everyman version".

10^^10^17

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Mon Jan 18, 2016 2:28 am UTC

10^^10^20
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Re: My Number is Bigger! (Everymen Needed!)

Postby Sabrar » Mon Jan 18, 2016 9:25 am UTC

Thanks PsiSquared, your explanation was very useful!

10^^10^5^2

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Mon Jan 18, 2016 9:25 am UTC

SirGabriel wrote:
PsiSquared wrote:This, by the way, is why I think it is vital to agree on some kind of standard notation (similar to scientific notation) for our numbers on this thread.

I agree, I think that was the problem with the last "everyman version".


Well... technically we did have a standard notation on that thread (User's extended arrows). But:

1. We didn't emphasize the fact that it was standard.
2. We didn't adequately explain how to compare numbers in that notation.
3. We didn't adequately explain how to perform calculations on number in that notation.

I hope we'll do better on this thread.

Anyway:

{3}{2}100 ~ 1.499434257056938095374045418838222832 F 126 765 060 022 822 940 149 670 320 537 602 ~ F(1.26765x1032)
(note that the "rough-and-quick" method would give us {3}{2}100 ~ F(100x2100) = F 126 765 060 022 822 940 149 670 320 537 600, which is correct to 32 significant digits!)

And I hereby propose that once the number after the 'F' becomes too large to write in full, we'll adapt "Fn" as our standard notation. To know which number is larger, simply compare the n's (eventually n itself will need to be written in F-notation, but we already know how to compare those, so this shouldn't be a problem).

EDIT: I've changed the F-exponent from ...600 to ...602 to correct a counting-error on my part.

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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Mon Jan 18, 2016 10:17 am UTC

{3}{2}{2}{1}3 = {3}{2}384 = {3}(2391×3) = (2391×3){2}(2391×3) ~ 2.0706231572578724 F (2391×3+2) > FE118

We're getting near to that point. I'm fine with it.

Spoiler:
2391×3+2 = 15 130 370 379 415 480 017 515 151 398 455 147 701 150 619 879 858 731 520 492 144 667 230 357 160 254 928 874 783 078 241 875 807 606 069 745 148 277 817 346


Edit: Corrected approximation. Thanks, Psi.
Last edited by emlightened on Mon Jan 18, 2016 11:52 am UTC, edited 1 time in total.

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Mon Jan 18, 2016 11:07 am UTC

10^^10^100 = F(10^100) = FE100 = FEE2

I'd say we've hit the limit about now; I've always considered a googol about the limit for easy to write out.

Though I guess the hard limit is around 10^^10^60000 since 60000 is how many characters you can put into your post.

Also, if you wondered how we're faring on the |...

2|()()() is about 10^^257, but 3|()()() requires TWO tetrations! (That is it's 10^^10^^X.) I know such numbers probably sound scary, but come to think of it, 10^^10^100 would sound pretty scary to someone who only knows standard exponents...
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Mon Jan 18, 2016 11:14 am UTC

User, your number is smaller than Emlightened's (because 2391>10100)

emlightened wrote:2.0706231572578724 F (2391×3+1)


I think that should be 2.0706231572578724 F (2391×3+2), since {2}384 is already 2391×3 ~E118 ~ EE2.07.

Anyway, I'll try to introduce my G-notation here. If anybody finds this too confusing, please say so:

First, for non-integer x I'll define:
Fx = (10^<the fractional part of x>) F (<the integer part of x> + 1) <- Ignore this please.
Fx = <the fractional part of x> F (<the integer part of x> + 1)

For example:
F2 = 0F3 = 10^10^10^0 = 1010
F2.1 = 0.1F3 = 10^10^10^0.1 ~ 10^10^1.259 ~ 1018

Got that?

And now:

xGn = FFF...FFFx (with n F's)
(notice that 1Gn = 10^^^n exactly)

Just like the F-notation, any number greater than 1 can be written in the form of xGn with 1<=x<10 and n being some positive integer:

7 = 7G0

2016 ~ 10^3.30449 = 3.30449F1 ~ F1.519 = 1.519G1

10^100 = 10^10^2 = 2F2 ~ F2.3010 = 2.3010G1

3F7 = F7.4772 = 7.4772G1

9F9 = F9.9542 = 9.9542G1

1F10 = F10 = F(F1) = 1.0000G2

2F10 ~ F10.30103 ~ F(10^1.01288) = F(1.01288F1) ~ F(F1.0056) = 1.0056G2

7.77F777 ~ F777.89042 ~ F(10^2.89092) = F(2.89092F1) ~ F(F1.4610) = 1.4610G2

1F(10 000 000 000) = F(F2) = 2.0000G2

And so on.

So this "G-scientific-notation" can be used as another "standard notation" for large numbers, at least in theory. Whether it is simple enough to be used in this manner on this thread, though, is remained to be seen (everyman feedback on this would be greatly appreciated).

Anyway, I submit:

2.4G2 = F(F2.4) = F(10^10^10^0.4) ~ F(1.0053049x10325)

And for those who are interested in the full 326-digit F-exponent:
Spoiler:
2.4G2 ~
7.2935189 F 10 053 049 235 514 790 040 057 589 654 645 625 230 831 832 179
310 415 488 225 995 890 922 664 857 535 237 990 111 384 922 354 137 439 704
325 507 094 409 296 908 302 596 355 517 599 521 281 900 594 030 804 522 894
300 098 336 974 767 699 610 580 269 490 740 973 412 812 795 557 735 944 867
609 540 074 577 592 182 117 771 837 705 510 652 750 049 445 951 486 171 090
020 305 134 607 022 797 043 241 783 645 632 620 485 918 407 760 858
Last edited by PsiSquared on Mon Jan 18, 2016 12:23 pm UTC, edited 4 times in total.

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Mon Jan 18, 2016 11:26 am UTC

10^^10^500
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Re: My Number is Bigger! (Everymen Needed!)

Postby emlightened » Mon Jan 18, 2016 12:00 pm UTC

Psi, as far as I can tell, F2.4 = 100.4F3 = 10^10^10^100.4 ~ 10^10^325, which is much bigger than your value. I think you want to remove the +1 in the definition.

FEE 2.999 ~ F(5.012608919418243 × 10997)

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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Mon Jan 18, 2016 12:21 pm UTC

emlightened wrote:Psi, as far as I can tell, F2.4 = 100.4F3 = 10^10^10^100.4 ~ 10^10^325, which is much bigger than your value. I think you want to remove the +1 in the definition.


Oops...

It was (10^frac(x))F(int(x)) at first, and then I decided to change it to (the equivalent) frac(x)F(int(x)+1) for better readability. Unfortunately, I forgot to remove the "10^" when I did that.

I'll fix it in a moment.

Meanwhile:

2.4774G2 = FF2.4774 ~ F(2.7855x101004)

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Mon Jan 18, 2016 12:28 pm UTC

10^^10^2000
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Mon Jan 18, 2016 12:33 pm UTC

Is that 7 or 8 consecutive posts?
(depends on whether we count the entry of 10^^10^100 - which wasn't the largest number when it was posted)

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Re: My Number is Bigger! (Everymen Needed!)

Postby username5243 » Mon Jan 18, 2016 12:47 pm UTC

It wasn't the largest, so it doesn't count as valid.
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Re: My Number is Bigger! (Everymen Needed!)

Postby PsiSquared » Mon Jan 18, 2016 1:11 pm UTC

Very well, so the consecutive-post-count was 7 and this is the 8th:

Define:
[|a,b,c|] = [a,c,b]

[|a,b,c,d|] = [ a,c,b | d]

(I'm doing this so the most significant arguments will always be on the right)

And submit:
[|5,6,2,2|] = [|5,6, [|5,6,2|] |] = [|5,6,5^5^6|] ~ [|5,6, 2.5487x1010921|] ~ F(2.5487x1010921)

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Re: My Number is Bigger! (Everymen Needed!)

Postby Sabrar » Mon Jan 18, 2016 1:41 pm UTC

@PsiSquared: could you please parse your latest submission in more detail? I just don't see what you do in the first step based on the definition.

FEE5 = F(10^10^5)


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