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May 15

Graham's number

Quoting from Graham's number —

Indeed, the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies at least one Planck volume. Even power towers of the form $\scriptstyle a^{b^{c^{\cdot ^{\cdot ^{\cdot }}}}}$ are useless for this purpose, although it can be easily described by recursive formulas using Knuth's up-arrow notation or the equivalent, as was done by Graham.

I'm very confused. Why couldn't we simply do $\scriptstyle 10^{googol^{googol^{googol^{\cdot ^{\cdot ^{\cdot }}}}}}$ until we get approximately the right number of digits? Nyttend (talk) 02:24, 15 May 2012 (UTC)[reply]

Because the result won't be nearly big enough, for any reasonable number of googols in the tower. --Trovatore (talk) 02:28, 15 May 2012 (UTC)[reply]

Indeed. When our article says it's "unimaginably large", it ain't kidding. FiggyBee (talk) 04:34, 15 May 2012 (UTC)[reply]

If it's that big, how was Graham able to calculate the values of the digits that appear in "Rightmost decimal digits of Graham's number"? I'm sorry if the result should be obvious; the article really is inaccessible without a mathematical background. Nyttend (talk) 05:16, 15 May 2012 (UTC)[reply]

Well, he sure didn't do it by computing the number and then throwing the rest of it away :-). I don't actually know the details, but see modular arithmetic for a bare-bones start to the ideas involved. For one step further, see exponentiation by squaring. That won't get you remotely close by itself to computing the digits, but it might illustrate the kind of thing that's involved.

I haven't tried to work it out myself, so I couldn't tell you whether Graham's method was a reasonably routine elaboration of these ideas, maybe adding applications of Fermat's little theorem and routine generalizations of it, or if something seriously deep and novel was needed. --Trovatore (talk) 06:35, 15 May 2012 (UTC)[reply]

An example that may help:

\scriptstyle 5^{5^{5^{5^{\cdot ^{\cdot ^{\cdot }}}}}}

. You can see from5*5*5*5*.. giving 25; 125; 625; 3125; 15625 that the last three numbers alternate between 125 (for odd number of 5's) and 625 (for even number). You also know that all the powers of 5 are odd numbers, so you can conclude that

\scriptstyle 5^{5^{5^{5^{\cdot ^{\cdot ^{\cdot }}}}}}

will end in 125. The method for the Graham numbers will be more complex I imagine. Ssscienccce (talk) 11:33, 15 May 2012 (UTC)[reply]

New equation without sources

Is it possible to make an article without sources even though simple math can back it up? It is a very, very new equation, and is EXTREMELY reliable. Is it possible to write an article like that? — Preceding unsigned comment added by 76.123.97.247 (talk) 14:53, 15 May 2012 (UTC)[reply]

The equation would have to be notable to be worthy of an article in any case. If there are no sources talking about it - if it's just something you came up with, for example - then it is not worthy of an article regardless of any other considerations. -Elmer Clark (talk) 16:22, 15 May 2012 (UTC)[reply]

You can feel free to post it here on the reference desk if it's something you came up with and you want to get feedback on it or discuss it, though. -Elmer Clark (talk) 16:23, 15 May 2012 (UTC)[reply]

We would be VERY interested in seeing this new equation you have to show us. --COVIZAPIBETEFOKY (talk) 17:06, 15 May 2012 (UTC)[reply]

No, it is not within our mandate to host such articles. That violates the policy prohibiting original research. Sławomir Biały (talk) 19:15, 15 May 2012 (UTC)[reply]

I suppose it depends on how simple and obvious the formula is. Let's say you were editing the article on dogs, and it contained a sourced table with the number of dogs per state and total number of pets per state. If you wanted to apply the formula DOG_PERCENTAGE = 100*DOGS/PETS, to each state, and add that column to the table, that would be fine. StuRat (talk) 22:07, 15 May 2012 (UTC)[reply]

Alright, well here goes: 1.41421356. This is the number needed to find the hypotenuse of a right triangle. The "L" sides MUST be even for the equation to work. All you must do is multiply one of the equal sides by the (Bermudez) equation (1.41421356) and you will get the length of the hypotenuse of the right triangle.What are your thoughts? — Preceding unsigned comment added by 76.123.97.247 (talk) 13:29, 16 May 2012 (UTC)[reply]

This number is already well-known as the square root of 2. For a right triangle whose legs are equal, the allegedly "new equation" is a trivial consequence of the Pythagorean theorem, and has been known for thousands of years. (The "new equation" is, of course, false for all other right triangles.) Sławomir Biały (talk) 13:41, 16 May 2012 (UTC)[reply]

I think this is a great observation for you to make. Our article Special right triangles covers this briefly, at Special_right_triangles#45.E2.80.9345.E2.80.9390_triangle (We use

{\sqrt {2}}

instead of the decimal approximation in the figure, because it is exact). I suppose a little more detail could be added; feel free be bold and improve it! Sorry, you don't get to name this number though, it is already known as Pythagoras'_constant, and this property has been known for ~2000 years. :) SemanticMantis (talk) 13:44, 16 May 2012 (UTC)[reply]

Obvious troll is obvious. --COVIZAPIBETEFOKY (talk) 18:42, 16 May 2012 (UTC)[reply]

If you thought this was a troll, then your reply up-thread is an obvious feeding. I was merely assuming good faith ;) SemanticMantis (talk) 21:52, 16 May 2012 (UTC)[reply]

Then I plead guilty. Of course it was obvious before, but it's undeniable now. No one could possibly discover that many digits of the square root of two without already knowing what they're looking for. --COVIZAPIBETEFOKY (talk) 02:30, 17 May 2012 (UTC)[reply]

What does "Bermudez" mean here? —Tamfang (talk) 20:41, 18 May 2012 (UTC)[reply]

Probably his or her name; 76.123.97.247 Bermudez. FiggyBee (talk) 06:31, 19 May 2012 (UTC)[reply]

Aren't mixed number useless?

What's the purpose of mixed numbers like $2{\tfrac {3}{4}}$ ? It gets confused with implicit multiplication, and it's nothing more than $2+{\tfrac {3}{4}}$ . — Preceding unsigned comment added by OsmanRF34 (talk • contribs) 23:16, 15 May 2012 (UTC)[reply]

It's shorter than

2+{\tfrac {3}{4}}

and doesn't require doing any math to figure out it's between 2 and 3 (assuming you aren't going to include weird fractions like

2{\tfrac {5}{4}}

). StuRat (talk) 23:23, 15 May 2012 (UTC)[reply]

At least for me, they were introduced in gradeschool before we learned that multiplication was written using juxtaposition. I don't think I saw mixed fractions much once we started learning algebra. --COVIZAPIBETEFOKY (talk) 02:17, 16 May 2012 (UTC)[reply]

It's a weird pattern. For several years, we hammer into students that they need to always reduce their fractions, and then we spend several years trying to get them to stop.--130.195.2.100 (talk) 03:28, 16 May 2012 (UTC)[reply]

I don't see why anyone should think this is weird. Implicit addition has been used for thousands of years, and seems more logical to me than the "modern" weird notion of juxtaposition being interpreted as multiplication (which I apply only in algebra). After all, our decimal notation uses juxtaposition implicitly (2.75 means 2 whole units plus 75 hundredths), so why shouldn't

2{\tfrac {3}{4}}

be interpreted similarly? Dbfirs 07:30, 16 May 2012 (UTC)[reply]

Well, they are not very useful in mathematics — it's much easier to manipulate

{\frac {11}{4}}

. But sure, for everyday stuff (distances on trail markers, say, if tenths of a mile is misleading precision), they can be a reasonable choice. --Trovatore (talk) 08:07, 16 May 2012 (UTC)[reply]

I assume you mean False precision. Dru of Id (talk) 10:03, 17 May 2012 (UTC)[reply]

130.195.2.100 didn't say that the notation was weird; he/she said that the teaching pattern was weird. --COVIZAPIBETEFOKY (talk) 18:45, 16 May 2012 (UTC)[reply]

The old convention for juxtaposition, ab=a+b if a>b and ab=b-a if a<b, is used for roman numerals: V=5, I=1, VI=6, IV=4. And also for time:

2

is two o'clock,

{\tfrac {1}{4}}

is a quarter,

2{\tfrac {1}{4}}

is a quarter past two, and

{\tfrac {1}{4}}2

is a quarter to two. Bo Jacoby (talk) 12:43, 16 May 2012 (UTC).[reply]

This is an example of something that occurs frequently with any form of representation, especially in maths. Very few notations always mean exactly one thing; one needs context to disambiguate the notation. — Quondum ^☏ 12:55, 16 May 2012 (UTC)[reply]

I am not familiar with this notation

{\tfrac {1}{4}}2

for a quarter to two. (For that matter, even

2{\tfrac {1}{4}}

for a quarter past two is unfamiliar, but I suppose I would be able to figure it out if someone left me a note to meet at

2{\tfrac {1}{4}}

; the other notation I would not have figured out.

Bo, can you say in what context this notation is/was used? --Trovatore (talk) 08:10, 17 May 2012 (UTC)[reply]

I guess it is more used in Danish than in English. (I am a Dane). The time "half past one" is in Danish pronounced "halv to", litterarily "half two", and earlier written ½2. Now it is written 13:30 but still pronounced "halv to". Bo Jacoby (talk) 15:28, 17 May 2012 (UTC).[reply]

In England, apparently, "half two" means 2:30. Which startles at least some Americans. —Tamfang (talk) 20:54, 18 May 2012 (UTC)[reply]

The convention is also used in Danish numbers: 'halvanden' = ½2 = 1½ is still used while 'halvtredje' = ½3 = 2½ is rare. 'halvtreds' means 'halv-tred-sinds-tyve' = ½3×20 = 50. 'tres' means 'tre-sinds-tyve' = 3×20 = 60. 'halvfjerds' means 'halv-fjerd-sinds-tyve' = ½4×20 = 70. 'firs' means 'fir-sinds-tyve' = 4×20 = 80. 'halvfems' means 'halv-fem-sinds-tyve' = ½5×20 = 90. Bo Jacoby (talk) 21:09, 17 May 2012 (UTC).[reply]

You learn something new everyday - today, for me, the etymology of "halvannen". I've used "halvannen" (it's archaic but still in use in Norwegian) all my life but never really pondered the construction (preposition of numbers is rarely used elsewhere in Norwegian except for clock times) Jørgen (talk) 12:09, 18 May 2012 (UTC) [reply]

What's sinds? —Tamfang (talk) 20:54, 18 May 2012 (UTC)[reply]

Looks like "times". StuRat (talk) 23:45, 18 May 2012 (UTC)[reply]

Right! See [1] if you really want to know. Bo Jacoby (talk) 17:10, 19 May 2012 (UTC).[reply]

Same in Polish: still-in-use półtora is a slightly simplified 'half of the second' meaning 11⁄2 (see in Wiktionary pół-, wtóry and półtora), also archaic półtrzecia = 'half of the third' = 21⁄2 and półczwarta = 31⁄2. Similar in other slavic languages, e.g. Russian wikt:полтора = (pl) półtora = 11⁄2. --CiaPan (talk) 09:59, 21 May 2012 (UTC)[reply]

See also sestertius. —Tamfang (talk) 08:07, 22 May 2012 (UTC)[reply]

Wikipedia:Reference desk/Archives/Mathematics/2012 May 15

Contents

May 15

Graham's number

New equation without sources

Aren't mixed number useless?