Talk:Tetration

(Redirected from Talk:Super-exponentiation)

People are using the tetration article and its talk page to document their own nomenclature, notations and ideas. I would like to see the tetration page focus on material that has references or at least some common consensus among the community of researchers that spend significant time studying it. I will discuss some of my concerns on because of the broader implications.

I have removed the link to the following:

Quintation

Can anyone cite a reference to this term in any journals? I’ve never seen it before anywhere, the only term I’ve seen published is pentation. How does the entry enhance the quality of the Wikipedia?

I have also taken the liberty of moving the entry Digits of 2@5 to Talk:Open problems in tetration because of the following message while editing this page - WARNING: This page is 41 kilobytes long. Please consider condensing the page and moving the detail to another article so it is not approaching or in excess of 32KB.

Note: Open problems in tetration and Talk:Open problems in tetrationwill likely be deleted soon.

Does anyone need the Digits of 2@5 or find it useful? In what manner does it enhance the quality of this entry? Daniel Geisler

Contents

1 Notation

2 Questions about super-exponentation

3 Conjecture about super-exponentiation and cosines

4 Inverse of super-exponentiation

5 True or false??

6 Categorizing this article

7 Soft hyphens in long numbers

8 Negative "super-exponents"?

9 Calculation errors?

10 Page title

11 Carets

12 Tetration number names

13 Anon minor formula edit

14 The other successor of exponentiation

Notation

> How many different names and notations are there for tetration? I've seen them called super-exponents and hyper-powers; the operation tetration and hyper-4, and at least three different symbolologies (Knuth's up arrows, the related ^^ notation, and the horrid left-superscript). Should we cross-reference any of these?

Questions about super-exponentation

1. Does anyone know how to define a@b when b is not an integer?? If a is not an integer, it is almost as easy as it is if both are integers. 1.5@2 = 1.5^1.5, but how about if b is not an integer??

> This is an open problem in mathematics. I've seen attempts to define them using fractals, combinatorics, and dynamics, but there's not that much progress.

2. Is e@2 known to be irrational?? The first few decimal places are 15.154262241479.

3. How about e@3, which is around 3814279.1047602??

I do not think the "subject" has been sufficiently studied to give an answer to those questions. Question 1 may either be senseless or be trivial via logarithms and exponentials. Just my 2c. Pfortuny 19:06, 7 Mar 2004 (UTC)

In one of Rudy Rucker's books, he calls this idea "tetration". -- Tarquin 19:11, 7 Mar 2004 (UTC)

But Wikipedia doesn't have a page titled tetration.

I don't understand your point. I said there are other names for this concept that we should perhaps mention in the article -- Tarquin 14:13, 8 Mar 2004 (UTC)

I finally made tetration a re-direct page. But, is there a corresponding "pentation"?? If so, what is its symbol??

Who wrote the last remark? Please sign your ideas so the conversation makes sense?

Theoretically, an unlimited number of binary operations can be methodically built upon one another via iteration. Practically, there is very little compelling justification to do so, however.

On rare occasion, you will discover an equation in an area of applied math which can be expressed more concisely via tetration than involution ("exponentiation"). With successively higher binary operations, though, I do not know of any more advantageous expressions of equations that exist.

This leaves only one practical usage for higher binary operations which I can think of: a more concise expression of extremely large, combinatoric values such as Graham's number. OmegaMan

Conjecture about super-exponentiation and cosines

Using Microsoft Works Spreadsheet, I found the following properties:

1. If n is even, the limit of x@n as x approaches 0 is 1.

2. If n is odd, the limit of x@n as x approaches 0 is 0.

From this, I have conjectured, but not proven, that defining a@b when b is not an integer can be done in terms of a function containing cosine in it, because these limits are the same as (cos (pi*(x/2)))^2. User 66.32.73.125

Inverse of super-exponentiation

I would like to create an article about the inverse operation of super-exponentiation, used by the same source that has the @ symbol for super-exponentiation using the & symbol. However, I don't know what to call it, and I feel afraid someone will very likely put it on vfd. Do you know?? 66.32.82.95 17:56, 2 Apr 2004 (UTC)

All inverse binary operations express what can also, of course, be expressed using ordinary, non-inverse or straightforward binary operations. So, they should only be used wherever comparatively convenient.

Consequently, subtraction is used less often than addition; evolution is used less often than involution (often awkwardly called "exponentiation"). Division is the only inverse binary operation which rivals multiplication in its commonplace usage.

In my opinion, there is no hope for an inverse binary operation of tetration (often awkwardly called "super-exponentiation") being used at all since it could only be communicated clearly in terms of tetration. Moreover, agreed standards in mathematical language and notation would become a problem. OmegaMan

> Remember that there isn't an inverse for tetration; there are two inverses. Addition and multiplication are commutative, so they have only one inverse each -- but exponentiation* has two, roots and logs. In the same way, there are 'hyperlogs' and 'hyperroots' that undo tetration.

I couldn't find a dictionary entry that links "involution" with exponentiation, although it does have a different mathematical meaning. I'm not sure it's wise to use such a term.

True or false??

True or false: there are plenty of Wikipedia links that change in the following category:

Originally, the link at Article A was a direct link to Article C, but later, someone modifies it and makes Article A link to Article B where Article B re-directs to Article C.

There appear to be plenty in the case of this article being C, but I want to know if there are plenty with no particular Article C. 66.245.23.108 22:56, 12 Jul 2004 (UTC)

Categorizing this article

Can anyone think of a category for this article to go into?? 66.245.77.90 00:36, 26 Aug 2004 (UTC)

Soft hyphens in long numbers

I've placed soft hyphens () into the 155-digit (206-character) value of 4↑↑3 because it was making the page super-wide in my browser. - dcljr 05:08, 29 Aug 2004 (UTC)

Negative "super-exponents"?

The last three entries on the page read:

n↑↑(-1) = 0 for all real numbers not equal to 1

n↑↑(-2) = negative infinity for all real numbers greater than 1

n↑↑(-2) = infinity for all real numbers between 0 and 1

Are these by definition? (Whose?) Can someone explain to me (and in the article itself) what a negative "super-exponent" (or whatever you'd call it) would even mean? - dcljr 05:22, 29 Aug 2004 (UTC)

Take the sequence negative infinity, 0, 1, 2, 4, 16, 65536. Does it make sense?? What would make more sense to you?? 66.245.127.199 21:19, 30 Aug 2004 (UTC)

What would make more sense to me -- and probably to dcljr -- is what I've replaced that list of identities with. Even if my TeX skills leave much to be desired. --Aponar Kestrel (talk) 20:24, 2004 Sep 15 (UTC)

Calculation errors?

I could be wrong, but the values I get for numbers as simple as $3\uparrow\uparrow3$ differ from those on the page. The values I get would be:

$1\uparrow\uparrow3$ = $\,\!1^{1^1}$ = 1
$2\uparrow\uparrow3$ = $\,\!2^{2^2}$ = 16
$3\uparrow\uparrow3$ = $\,\!3^{3^3}$ = 19,683
$4\uparrow\uparrow3$ = $\,\!4^{4^4}$ = 4,294,967,296
$5\uparrow\uparrow3$ = $\,\!5^{5^5}$ = 298,023,223,876,953,125
$6\uparrow\uparrow3$ = $\,\!6^{6^6}$ = 10,314,424,798,490,535,546,171,949,056

... with similar divergences for the next rows. Have I missed something? These values are easily checkable with a pocket calculator, if anyone would care to back me up. Aydee

Hmm. You're doing (3^3)^3 (= 19683), the page does 3^(3^3) (= 7.62559748×10¹²).

If I go to a linear calculator (like Google Calculator) and ask it for "3^3^3", it converts it into 3^(3^3).

But the definition of this function on the page is $x \uparrow\uparrow y = x \mbox{ raised to its own power }y\mbox{ times}$ . Does that mean x^x, then the result of that raised to x, etc.? Or does it mean the resolution of the symbol $\,\!x^{x^x}$ ?

The definition at Knuth's up-arrow notation indicates that it is the latter. But if so, then the definition of "iterated exponentiation" is not accurate. It would seem more accurate to say something like "the xth power of x, y times".

- KeithTyler 21:33, Sep 15, 2004 (UTC)

Good point. It'd probably be best to use the definition on Knuth's up-arrow notation in some form to clarify this, but I'm not sure what the best policy in these cases may be; my gut feeling is that this article needs the definition and Knuth's up-arrow notation should refer here, as the up-arrow notation is merely a method of representing tetration. On the other hand, I could be wrong. Any other ideas? Aydee 01:41, 2004 Sep 16 (UTC)

I just added a quick example of iterated expectation to the page. Perhaps that's sufficient to clear up any misunderstandings? - dcljr 06:37, 10 Oct 2004 (UTC)

Page title

On July 4, 2004, User:Gdr asked this page to be renamed, but it hasn't been renamed. What happened?? 66.245.71.98 15:45, 16 Sep 2004 (UTC)

Your helpful wizards have finally woken up and noticed! Noel 20:01, 17 Sep 2004 (UTC)

Carets

Given that, as pointed out in the article, the up-arrow symbol (when present on a computer keyboard) is used similarly to the caret symbol (indicating superscripts), couldn't we just replace all the double up-arrows on the page (except the ones next to Knuth's name) to double carets (^^)? This would make the page readable in all browsers. Just a suggestion... - dcljr 06:56, 10 Oct 2004 (UTC)

Tetration number names

Please read very slowly and carefully:

Can anyone come up with a way to name numbers that tetration can help visualize the magnitude of?? One way that I came up with is a building with 103 floors, each of which is for numbers of various sizes; the higher the floor, the larger the numbers. The Tetrational System (the numbering system that this uses) has the following number names:

For numbers <= 10^30, whose magnitudes are easy to visualize without tetration, it uses the same number names as Rowlett. On the first 2 floors, we have:

First floor: Two, Three, Six, Nine, Twelve, Fifteen, Eighteen, Twenty-one, Twenty-four, Twenty-seven, Thirty

Second floor: Hundred, Thousand, Million, Gillion, Tetrillion, Pentillion, Hexillion, Heptillion, Oktillion, Ennillion, Dekillion

Now, let's go onto the third floor. This is where the numbers get large enough that tetration is a useful way; each term is 10 to the power of the previous term:

Third floor: Googol, Froogol; the remaining words on this floor are the same as those on the second floor only that they use -illoogol instead of -illion (that is, Milloogol to Dekilloogol are 10^(10^6) through 10^(10^30.) (Froogol is a back-formation on Froogle on the model of Googol/Google.)

Fourth throught 103rd floors: Simply take the number names on the third floor and add "-plex" for those on the fourth floor, "-duplex" for the fifth floor, "-triplex" for the sixth floor, and so on all the way to "-centuplex" for the 103rd floor. This makes the largest number in the building (dekilloogolcentuplex) 10^10^10^...10^10^10^30 with a total of 103 numbers (102 tens and a 30) between exponent signs.

Any numbers too large for this?? At this moment, I know of only one number too large for this building, Graham's number. 66.245.98.219 23:12, 16 Nov 2004 (UTC)

Anon minor formula edit

Line 33, was: $2 \uparrow\uparrow (n-3)$ − 3

Now is: $2 \uparrow\uparrow (n+3)$ − 3

As I am no expert, please check if this revision is valid. -- AllyUnion (talk) 10:33, 10 Dec 2004 (UTC)

(The following discussion was movied from the Super-exponentiation entry on Wikipedia:Pages needing attention/Mathematics. Paul August ☎ 20:06, Feb 7, 2005 (UTC)")

Super-exponentiation -- is this notation genuine? The source given is a mathforum link to where the notation is "invented" by a enthusiastic student. It seems like a dup of Knuth's up-arrow notation. Motor 19:48, 18 Mar 2004 (UTC)

No, Knuth's up-arrow notation is for all operations from super-exponentiation upwards, and super-exponentiation is just for that operation itself. 66.32.89.242 23:33, 2 Apr 2004 (UTC)

I rewrote this page using Knuth's up-arrow notation and the standard term "tetration". Really this page should be moved and redirected to Tetration. Gdr 13:23, 2004 Jul 4 (UTC)

Done. Noel 20:06, 17 Sep 2004 (UTC)

I currently put this article in Category:Mathematics, but I want to see if anyone can give a more specific category for this article. 66.245.77.90 00:44, 26 Aug 2004 (UTC)

http://mathworld.wolfram.com/PowerTower.html This is topic is discussed at Mathworld under the heading "Power Tower" GulDan 17:48, Sep 16, 2004 (UTC)

I have added information about the standard notation with supporting references. I strongly recommend using the published notation and nomenclature. MathWorld’s entry on Power Tower is great, but they do have a sub page for tetration that directs to their Power Tower entry. Technically these are not the same thing; tetration is much more comprehensive than Power Tower, but almost all published research on tetration has been confined to the Power Tower due to the profound difficulty of the subject. I have changed the category to arithmetic. User:Daniel Geisler 12:56, Dec 28, 2004 (UTC)

(The following discussion was movied from the Super-exponentiation entry on Wikipedia:Pages needing attention/Mathematics. Paul August ☎ 20:09, Feb 7, 2005 (UTC)")

Super-exponentiation -- is this notation genuine? The source given is a mathforum link to where the notation is "invented" by a enthusiastic student. It seems like a dup of Knuth's up-arrow notation. Motor 19:48, 18 Mar 2004 (UTC)

No, Knuth's up-arrow notation is for all operations from super-exponentiation upwards, and super-exponentiation is just for that operation itself. 66.32.89.242 23:33, 2 Apr 2004 (UTC)

I rewrote this page using Knuth's up-arrow notation and the standard term "tetration". Really this page should be moved and redirected to Tetration. Gdr 13:23, 2004 Jul 4 (UTC)

Done. Noel 20:06, 17 Sep 2004 (UTC)

I currently put this article in Category:Mathematics, but I want to see if anyone can give a more specific category for this article. 66.245.77.90 00:44, 26 Aug 2004 (UTC)

http://mathworld.wolfram.com/PowerTower.html This is topic is discussed at Mathworld under the heading "Power Tower" GulDan 17:48, Sep 16, 2004 (UTC)

The other successor of exponentiation

The article seems to suggest that tetration is the logical extension of the sequence of addition, multiplication, and exponentiation. However, what about ((nⁿ)^n...) for m occurences of n? It seems to me this operation is equally an extension: Since exponentiation is non-commutative, the sequence bifurcates at this point. This "inferior super-exponentiation", if you will, has interesting properties of its own. For instance, a negative value for m corresponds to taking the n^th root m times (since it can remove exponents), so m=0 yields $\sqrt[n]n$ .

Anyway, shouldn't there be an article about this operation also? Who here knows its standard name(s) and notation(s), or does it have any? (I thought "iterated exponentiation" was reasonable, but the article's (and discussion's) usage conflicts.) --Ddawson 12:08, 20 Mar 2005 (UTC)

This is simply ~~n^{n^m}~~ $n^{n^m}$ (NESTED SUBS/SUPS ARE BROKEN ON WIKIPEDIA >:() , there is nothing especially "new" about it relative to the sense that tetration is some new operation. Dysprosia 22:25, 20 Mar 2005 (UTC)

599

Good point. (That should be $n^{n^{m-1}}$ , though, since I count the base n.) I see what you mean: this can be defined by a fixed expression, whereas tetration must be defined recursively in a formal sense. Ddawson 15:06, 21 Mar 2005 (UTC)

Maybe there could be a quick mention of this, somewhere soon after the definition of tetration, with the explanation of why it's not nearly as interesting. My reasoning is that it's still a valid way of iterating exponentiation. Ddawson 15:25, 21 Mar 2005 (UTC)