# 9^9^9. **URL:** \ **Category:** In My Humble Opinion\ **Created:** [October 29, 2017, 5:42pm UTC](https://boards.straightdope.com/t/9-9-9/800163 "2017-10-29T17:42:09Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Jim\_B](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jim_b/32/4338_2.png) [@Jim\_B](https://boards.straightdope.com/u/Jim_B)\ **Post date:** [October 29, 2017, 5:42pm UTC](https://boards.straightdope.com/t/9-9-9/800163/1 "2017-10-29T17:42:09Z")
Interesting thing that I read a child, in a children’s book, naturally. The largest 3-digit number isn’t 999. It’s 9^9^9. Or, nine taken to the ninth power taken to the ninth power. This number is apparently gargantuan. In ordinary type, it would reach from New York to Cleveland. Also, some mathematicians have speculated on what digits the behemoth number begins with, and ends with. I forget the exact details. Believe it or not, I may even still have the book in my collection (it was a very interesting book, well worth holding on to). I will check. In the meantime, does anyone else know what I am talking about? What else is known about this number, and what more has been learned in the 30+ years since I got the book? Please divulge. 🙂 🙂 🙂 ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [October 29, 2017, 5:48pm UTC](https://boards.straightdope.com/t/9-9-9/800163/2 "2017-10-29T17:48:03Z")
If you’re allowed an expression that contains only three integers but can have any other symbols, you can go bigger. 9!^9!^9! for starters. But it’s somewhat arbitrary, it depends on the conventions about what the symbols mean. ---
**Author:** ![What\_Exit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/what_exit/32/10652_2.png) [@What\_Exit](https://boards.straightdope.com/u/What_Exit)\ **Post date:** [October 29, 2017, 5:51pm UTC](https://boards.straightdope.com/t/9-9-9/800163/3 "2017-10-29T17:51:26Z")
Reminds me of Heinlein’s “Number of the Beast” which was 6^6^6. If we’re going to get technical and say 9^9^9 is the largest 3 digit, what restricts us to base 10 then? I mean in Hex, F^F^F is significantly larger then 9^9^. Please see this link: [https://www.quora.com/What-is-the-value-of-9-9-9](https://www.quora.com/What-is-the-value-of-9-9-9) on 9^9^9. Someone says it is 956192041778915424074542039160053096041449. ---
**Author:** ![Jim\_B](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jim_b/32/4338_2.png) [@Jim\_B](https://boards.straightdope.com/u/Jim_B)\ **Post date:** [October 29, 2017, 5:52pm UTC](https://boards.straightdope.com/t/9-9-9/800163/4 "2017-10-29T17:52:24Z")
> [@Riemann](#): > > If you’re allowed an expression that contains only three integers but can have any other symbols, you can go bigger. 9!^9!^9! for starters. But it’s somewhat arbitrary, it depends on the conventions about what the symbols mean. I’m just telling you what I read in the book. I should also tell you all, IANAMathematician. I do have some education in the matter though, mostly hs math, and maybe even a little exposure to calculus. But I do find the subject fascinating, FWIW. ---
**Author:** ![Jim\_B](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jim_b/32/4338_2.png) [@Jim\_B](https://boards.straightdope.com/u/Jim_B)\ **Post date:** [October 29, 2017, 5:55pm UTC](https://boards.straightdope.com/t/9-9-9/800163/5 "2017-10-29T17:55:21Z")
> [@What\_Exit](#): > > Reminds me of Heinlein’s “Number of the Beast” which was 6^6^6. > > If we’re going to get technical and say 9^9^9 is the largest 3 digit, what restricts us to base 10 then? I mean in Hex, F^F^F is significantly larger then 9^9^. > > Please see this link: [https://www.quora.com/What-is-the-value-of-9-9-9](https://www.quora.com/What-is-the-value-of-9-9-9) on 9^9^9. Someone says it is 956192041778915424074542039160053096041449. Maybe I should have said, 9^(9^9). In any event, please hold me blameless for any inaccuracy. I am only telling you what the book says :). ---
**Author:** ![Ludovic](https://avatars.discourse-cdn.com/v4/letter/l/7ab992/32.png) [@Ludovic](https://boards.straightdope.com/u/Ludovic)\ **Post date:** [October 29, 2017, 5:56pm UTC](https://boards.straightdope.com/t/9-9-9/800163/6 "2017-10-29T17:56:40Z")
Tree(tree(tree(9))) ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [October 29, 2017, 5:58pm UTC](https://boards.straightdope.com/t/9-9-9/800163/7 "2017-10-29T17:58:56Z")
It has 369,693,100 digits, which, if written out in type 3mm wide, would cover a length of about 1,100km, which seems like it would take you almost to Indianapolis. Its last ten digits are …2627177289, its first ten digits are 4281247731… ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [October 29, 2017, 6:00pm UTC](https://boards.straightdope.com/t/9-9-9/800163/8 "2017-10-29T18:00:54Z")
> [@Jim\_B](#): > > I’m just telling you what I read in the book. I’m not doubting you about that. Nevertheless, there is no meaningful sense in which that is “the largest three-digit number” if you are allowed to use symbols at will. Even just restricting yourself to using common standard symbols, you can express an arbitrarily large number, e.g. (((((((((9!)!)!)!)!)!)!)!)!)… ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [October 29, 2017, 6:05pm UTC](https://boards.straightdope.com/t/9-9-9/800163/9 "2017-10-29T18:05:45Z")
> [@Riemann](#): > > I’m not doubting you about that. Nevertheless, there is no meaningful sense in which that is “the largest three-digit number” if you are allowed to use symbols at will. Even just restricting yourself to using common standard symbols, you can express an arbitrarily large number, e.g. > > (((((((((9!)!)!)!)!)!)!)!)!)… But 9^9^9 can be written using only the symbol ‘9’, in the standard exponentiation notation as 9[sup]9[sup]9[/sup][/sup]; so you don’t actually need any other symbols—in that sense, it’s truly a three-_digit_ number, not a three-digit-plus-any-other-symbols-you-care-to-add number. ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [October 29, 2017, 6:18pm UTC](https://boards.straightdope.com/t/9-9-9/800163/10 "2017-10-29T18:18:30Z")
> [@Half\_Man\_Half\_Wit](#): > > But 9^9^9 can be written using only the symbol ‘9’, in the standard exponentiation notation as 9[sup]9[sup]9[/sup][/sup]; so you don’t actually need any other symbols—in that sense, it’s truly a three-_digit_ number, not a three-digit-plus-any-other-symbols-you-care-to-add number. Yes, expressed that way the constraint makes more sense. (Although the expression is ambiguous, as has been noted.) ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [October 29, 2017, 6:55pm UTC](https://boards.straightdope.com/t/9-9-9/800163/11 "2017-10-29T18:55:06Z")
> [@Riemann](#): > > Yes, expressed that way the constraint makes more sense. (Although the expression is ambiguous, as has been noted.) Yes. Without ambiguity, the whole thing wouldn’t even work—essentially, one expects ‘three-digit number’ to mean ‘three consecutive digits in base-10 place-value notation’, but then it’s pointed out that, technically, the exponential 9[sup]9[sup]9[/sup][/sup] also only consists of three digits, and thus, is a ‘three-digit number’; but of course, there’s any number of different readings of ‘three-digit number’ that one can use equally well—varying the base, adding symbols that aren’t digits, and so on. That sort of ambiguity also means that 9[sup]9[sup]9[/sup][/sup] isn’t of terribly much mathematical interest, since it’s only special in an ultimately arbitrary system of notation, whereas the properties of something like π or _e_ are independent of such arbitrary choices. ---
**Author:** ![Pantastic](https://avatars.discourse-cdn.com/v4/letter/p/3bc359/32.png) [@Pantastic](https://boards.straightdope.com/u/Pantastic)\ **Post date:** [October 29, 2017, 7:03pm UTC](https://boards.straightdope.com/t/9-9-9/800163/12 "2017-10-29T19:03:13Z")
> [@Ludovic](#): > > Tree(tree(tree(9))) Or TREE(TREE(TREE(9))) if you want to stop messing around, since the lower case tree function isn’t quite as insanely fast growing. For people who aren’t familiar, the TREE function is a function that grows absurdly fast; TREE(1) is 1, TREE(2) is 3, and TREE(3) is so vastly incredibly huge that you need special notation to even vaguely talk about how big it is. [TREE sequence | Googology Wiki | Fandom](http://googology.wikia.com/wiki/TREE_sequence) ---
**Author:** ![whitetho](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whitetho/32/1059_2.png) [@whitetho](https://boards.straightdope.com/u/whitetho)\ **Post date:** [October 29, 2017, 7:53pm UTC](https://boards.straightdope.com/t/9-9-9/800163/13 "2017-10-29T19:53:54Z")
> [@Half\_Man\_Half\_Wit](#): > > It has 369,693,100 digits, which, if written out in type 3mm wide, would cover a length of about 1,100km, which seems like it would take you almost to Indianapolis. Its last ten digits are …2627177289, its first ten digits are 4281247731… This is a “show all work” question. Unless you show the other 369,693,080 digits, you will only get partial credit. Sorry, I don’t make the rules… ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [October 29, 2017, 7:54pm UTC](https://boards.straightdope.com/t/9-9-9/800163/14 "2017-10-29T19:54:36Z")
> [@Pantastic](#): > > [TREE sequence | Googology Wiki | Fandom](http://googology.wikia.com/wiki/TREE_sequence) And a [somewhat more accessible video explanation](https://www.youtube.com/watch?v=3P6DWAwwViU&t=4s). ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [October 29, 2017, 9:47pm UTC](https://boards.straightdope.com/t/9-9-9/800163/15 "2017-10-29T21:47:32Z")
Perhaps the OP is misremembering. It could be 9[sup]9[sup]9[/sup][/sup], technically no symbols. But then that is still a fail since [tetration](https://en.wikipedia.org/wiki/Tetration) notation wins by a long shot. [sup][sup]9[/sup]9[/sup]9 is a _vastly_ larger number. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [October 29, 2017, 11:09pm UTC](https://boards.straightdope.com/t/9-9-9/800163/16 "2017-10-29T23:09:56Z")
> [@Jim\_B](#): > > Interesting thing that I read a child, in a children’s book, naturally. > > The largest 3-digit number isn’t 999. It’s 9^9^9. Or, nine taken to the ninth power taken to the ninth power. > > This number is apparently gargantuan. In ordinary type, it would reach from New York to Cleveland. Also, some mathematicians have speculated on what digits the behemoth number begins with, and ends with. I forget the exact details. > > Believe it or not, I may even still have the book in my collection (it was a very interesting book, well worth holding on to). I will check. > > In the meantime, does anyone else know what I am talking about? What else is known about this number, and what more has been learned in the 30+ years since I got the book? > > Please divulge. > > 🙂 🙂 🙂 First thing is your description is wrong. 9 taken to the 9th power taken to the 9th power is (9^9)^9 which is 9^{81} and is probably the number cited above. There can be no doubt of its last digit since 9 to any odd power ends in a 9, including 9^{9^9} which is 9^{387420489} a truly gargantuan number, although it would have only about 350,000,000 million digits, about 350 miles at 10 digits to an inch. NY to Cleveland is about 460 miles, so maybe a larger font was used. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [October 29, 2017, 11:53pm UTC](https://boards.straightdope.com/t/9-9-9/800163/17 "2017-10-29T23:53:42Z")
> [@Ludovic](#): > > Tree(tree(tree(9))) Judging from what I’ve just googled (and if I understand it correctly) Tree is worse than Graham (which had been my previous go-to fast growing function. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 30, 2017, 12:03am UTC](https://boards.straightdope.com/t/9-9-9/800163/18 "2017-10-30T00:03:45Z")
I’m not sure what the standard notation would be for it, but the fastest-growing function I’ve ever encountered “in the wild”, so to speak, is defined by the recurrence relation D(n) = D(n-1)+2^(D(n-1)) . ---
**Author:** ![Ludovic](https://avatars.discourse-cdn.com/v4/letter/l/7ab992/32.png) [@Ludovic](https://boards.straightdope.com/u/Ludovic)\ **Post date:** [October 30, 2017, 12:14am UTC](https://boards.straightdope.com/t/9-9-9/800163/19 "2017-10-30T00:14:33Z")
> [@Andy\_L](#): > > Judging from what I’ve just googled (and if I understand it correctly) Tree is worse than Graham (which had been my previous go-to fast growing function. I was debating putting TREE(g(A(9,9)) but then decided to go with just TREEs, but vB “helpfully” uncapped my post, and not wanting to spend the time adding a hidden character to preserve the caps, I left in the lesser function (although I didn’t know how much less it was at the time!) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [October 30, 2017, 12:21am UTC](https://boards.straightdope.com/t/9-9-9/800163/20 "2017-10-30T00:21:26Z")
> [@Andy\_L](#): > > Judging from what I’ve just googled (and if I understand it correctly) Tree is worse than Graham (which had been my previous go-to fast growing function. You probably mean (some variant of) Ackermann’s function, or some other fast-growing, yet computable, function. Graham’s number is just a number, remarkable as an explicit huge number that arises “in the wild.” [Next page](https://boards.straightdope.com/t/9-9-9/800163.md?page=2)