Talk:Order of magnitude

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12/09/06 - Changed th example given in the first paragraph into something more neutral.

Contents

[edit] Factor vs. Power

The first sentence of this entry is wrong. An order of magnitude is in no sense a factor of ten. In this context, the only way factor can be interpreted is as a synonym of divisor. For example, "2 and 3 are factors of 6" is correct, while "1000 is a factor of 10" is not, because no-one in their right mind would want to express 10 as 1000*(something).

I edited this entry previously to change "factor of ten" to "power of ten". I realize that I should have left a comment on this change, because it was later reverted (still incorrectly I must say). So unless there are substanciated objections, I would like to change "factor" to "power" again. If others feel that "powers of ten" is too precise, perhaps something more winded like "an order of magnitude is a range of numbers that differ from a power of 10 by a factor of order 1 (a single digit number)" would be better.

-- Igor

[Peak:] Whoever first wrote "factor" is correct. 6 has two prime factors, 2 and 3. 100 can be factored as 10*10, and one can therefore say that 100 has two factors of 10. It would be incorrect to say that 100 has two powers of 10.

Consider also the following from MSN Encarta:

If a quantity were 10 times greater than another, it would be an order of magnitude greater; if 100 times greater, it would be two orders of magnitude greater.

If you substitute "ten times" for "an order of magnitude", the meaning is unchanged. "Two orders of magnitude" means "two factors of 10", i.e. 10*10 as previously discussed.

In any case, by breaking up the preamble into separate sections about "OOM", "OOM of a number" and "OOM estimate", there is not much possibility of confusion. Peak 00:08, 28 Jan 2004 (UTC)

Then I suggest a slight modification to make this meaning more clear:
For example, two numbers are said to differ by "three orders of magnitude" if they differ by three factors of ten, that is one is approximately 1000 times larger than other. --Igor Jan 28 14:20:22 UTC 2004

[edit] Remove orders of magnitude?

Following the changes by User:Onebyone to Orders_of_magnitude_(length) and Orders of magnitude (mass) it looks like the series being removed. Is this intended? -- User:Docu

I'm not removing the series, and I don't intend to propose any files for deletion. However, the current situation is that large sections of some of the series (and indeed all of power before I modified it) would be a lot more usable on fewer pages. I'm in the process of modifying length, in which several of the subpages have quite a few entries on them, and it's a close call whether those should be separate lists. If the main articles become unwieldy, the more densely-populated parts can always be returned to subpages, but I strongly dislike cases where there are only 0-3 entries on a page. Onebyone 00:33, 26 Jan 2004 (UTC)
Then add more entries. But leave the stubs. --mav 06:39, 26 Jan 2004 (UTC)
It's difficult to name more than 3 things which are about the size of an atom, without just listing lots of different atoms. It wouldn't help people to get a better idea of how big atoms are if you did - I believe that's the purpose of these lists, rather than "list of all things mentioned on Wikipedia of size order an angstrom", or else they're woefully incomplete. If you can find so many atom-sized things that the length page becomes unusable as it is, you can easily move the list back to the subpage.
I guess what I'm saying is that these lists should have started on a single page each and migrated out. What they're actually doing is starting on a squillion empty pages and very slowly bulking up, and in the mean time they're a bit horrible. Onebyone 18:17, 26 Jan 2004 (UTC)
It would show the relative sizes of different atoms! This is very important in some areas, oh like chemistry. Wny not have summary/overview pages that have a whole bunch of different orders of mag on it (what you want) that only has a few notable items under each order of mag. Then have links to the individual orders of mag articles that will have a much longer listing (like every element that is in a certain order of mag). Those individual articles will in turn have back-links to the summary/overview articles. --mav
I agree with that principle. It is worthwhile having atomic radii in Wikipedia for comparison, certainly, but unless the purpose of these lists really is "list of all sizes mentioned on Wikipedia" (in which case I'll not touch them again, because I don't think that's worth doing), then I do not think that 1 E-10 m and 1 E-9 m are the right places for all 115 (and rising) elements. Maybe List of elements in order of atomic size would be appropriate? We could then link to that from the relevant OoM pages and from each element's page. Similarly, an article on a building will mention its height, but if there is to be a "list of the heights of every tall building in the world", then putting it in 1 E2 m would bury everything else in that list, thereby destroying its usefulness for anything except buildings. Onebyone 12:18, 27 Jan 2004 (UTC)
Where there are only 1-3 entries on a page, the sequence probably needs work and may be better on the main page, such as Orders_of_magnitude_(mass). I'm less sure about Orders of magnitude (length). An advantage of the series is that one finds comparable units easily. Something that is more difficulte, if the are all on the same page, or, grouped on another magnitude e.g. 1_E12_m� (includes E14). Whatlinkshere can help expand the articles, but it doesn't work today. -- User:Docu
An advantage of the series is that one finds comparable units easily. What's difficult about a list in order of size? The advantage of having things on one page is that one finds somewhat bigger and smaller things at a glance, rather than following a link. Onebyone 18:17, 26 Jan 2004 (UTC)
Then have both. --mav
OK. Since there doesn't seem to be anyone arguing that the near-empty subarticles are vital, I'll restart doing some merging, but I'll keep in mind that the longer subarticles should be kept as they are. If there are cases where I've already merged when I shouldn't, just change things around. One important thing is to sort out the mess where (IIRC) Orders of magnitude (length) and 1 E2 m gave different heights for the Empire State Building. I assume that one includes a mast and the other doesn't, but it needs checking. Onebyone 12:18, 27 Jan 2004 (UTC)
I am arguing for both. Keep the stubs! I'm going to revert the redirections. --mav
Sorry, I misunderstood, by stubs I thought you meant those with a reasonable amount of content already. I think the compromise you've established is good. I'll still be taking a critical look at cases like 1 E13 m, 1 E14 m (1 entry each). Onebyone 21:29, 27 Jan 2004 (UTC)

OK, for length I've left the very small and very large as redirects but restored the other articles. In the future I will be less conservative in what I put in the individual OoM articles due to the fact that the summary pages exist. If somebody wants an overview, they should go there, if they want the detail they should go to an individual OoM page. I will continue to help maintain the individual entries, and others can work on the summary OoM articles. They can then concentrate on only having notable examples. Both are important and both should be kept. --mav

I strongly disagree that the individual subarticles should be kept. If people link to them, they can redirect to the relevant Oom page. The situation as it stands now is unacceptable because:
  1. Contrary to what they claim, the individual pages do not facilitate comparison of different orders of magnitude because one has to flip between several different pages to make such a comparison.
  2. Jumping "at random" between the pages is impossible because (to my knowledge) there's no central list of all the individual pages unnecessarily difficult (and would be rendered pointless by the existence of the full list at the main Oom page, anyway). (This could be changed using categories, but I'm not recommending that.)
  3. It encourages duplication of effort (in keeping the same information at two different locations).
As I see it, there's only one legitimate reason to have multiple pages for information like this, and that's if the main page gets too long. But as already discussed, this can be done later as needed. See also my other comment below. - dcljr 09:22, 29 Aug 2004 (UTC)

I changed my comment above because there is, of course, a central list since one appears in this very article! However, the individual Oom pages are inconsistent in linking to the subpages: see, for example, Orders of magnitude (mass) and Orders of magnitude (power). - dcljr 09:45, 29 Aug 2004 (UTC)

[edit] Page title and article text

See discussion at 1 E9 m². -- User:Docu

[edit] WikiProject Orders of Magnitude

To update this project the current pratices, we could move the templates to a new page WikiProject Orders of Magnitude. The templates may be easier to find where there are now. -- User:Docu

[edit] Moved discussion from Talk:1 E5 m*

* May I suggest adding a line to the top of all "chain page" talk pages referring to this page for the general discussions? --J.Rohrer 22:17, 13 Jul 2004 (UTC)

This page (and I guess the related ones as well) is wrongly named: "1 E5 m" means "15 m", which of course equals "1 m", and not the intended "100 km". The name should be "10 E5 m" = "105 m". Besides, it is clearer to not separate the notation of the exponent from the number it affects, thus "10E5 m" or "10e5 m" rather than "10 E5 m"; another option is to use a caret: "10^5 m". OTOH, I think that for numbers up to a million or a billion, writing them out in full would be the clearest: "100,000 m", or even better "100 km". Using the exponential notation makes full sense when dealing with really huge zillions with dozens of zeroes; but when dealing with numbers in the commonly used range, that notation is counterintuitive because numbers like 100 or 10,000 aren't commonly spelled 10E2 or 10E4 beyond some scientific writings, and so "10E2 m" or "10E4 m" look arcane to most people, while "100 m" and "10 km" are instantly understandable for everyone (and what most people would naturally query in the Wikipedia search). Uaxuctum 13:55, 10 Jul 2004 (UTC)

While your interpretation of the "E" as an equivalent of "to the power of" (or the caret sign ^) is suggestive, this is not common terminology. Instead, the notation "E..." is to be read as "times 10 to the ...-th", see for example scientific notation, or the syntax of floating point literals in most programming languages.
The reason for the rather cryptical naming of these articles is (as far as I understand) ease of linking: the articles seem to be intended to be linked from, ideally, every occurence of a quantity of the given order of magnitude to assist in "getting a feeling" for it, especially for those not familiar with the unit system (being more accustomed to anglo-saxon measures, for example). As all these articles use a consistent naming scheme now, you do not have to remember any individual article names to implement this. You may consider creating appropriate redirects, however. --J.Rohrer 20:22, 10 Jul 2004 (UTC)
It's not "an interpretation of mine". Do a Google search for "10E5" and you'll get scores of technical and university pages where "10E5" stands clearly for "105" - which is not suggestive of it not being common terminology. Using "1 E5" to mean "105" is an idiosyncratic, not an intuitive choice, leading to confusion of interpretation and hindering the representation of other powers, because if E means "times 10 to the power of", how on earth does one represent "205", "15", "25", etc.?
The notation is not intended for this, but for representation of floating point numbers in a fixed-base numeral system (usually base 10, of course), so, for instance, 205 would be 32 E5. For general powers, one would use another notation, for example using the caret.
The immense majority of people are not naturally going to search for "1 E5" (nor link to "1 E5") when they mean "100,000". So, even if one chooses to name all the articles with a consistent but idiosyncratic, cryptic and confusing terminology, one isn't helping to make the Wikipedia any user-friendly. Names of articles should not be cryptic, but clear and free from possible confusions of interpretation. The E method is by all parameters not the appropriate one. The two options that do not lead to confusion are the caret method ("10^5 m") and the more user-friendly "100,000 m" (or even clearer "100 km"). Uaxuctum 17:21, 13 Jul 2004 (UTC)
Who would search for such an article at all? You will usually come to these order-of-magnitude articles by following wikilinks, so one may argue that, from the readers' point of view, the naming is secondary. Personally, I do not particularly like the "E" notation, so I do not intend to defend the naming scheme any further, but in my opinion it is certainly not wrong and probably not even particularly misleading.
Whatever, if you wish to continue this discussion I suggest moving it on the parent article talk page. I will copy the above text there. Notice however that the topic seems to have been discussed before, see archive. --J.Rohrer 22:13, 13 Jul 2004 (UTC)
The archive mentioned above is currently at Talk:Order of magnitude/chain page names. -- Paddu 19:47, 13 April 2006 (UTC)

[edit] "1 E* J" series and others...

Interested parties should see the List of energies in joules, which I just created before coming across this (these) page(s). (See also Talk:List of energies in joules.) I should have followed the "Orders of magnitude" link first! Anyway, the information there should probably be refactored into Orders of magnitude (energy). There are also collections of pages for length, area, volume, time, etc., as pointed out in earlier discussions. In my opinion, all of these should be redirected to the relevant "Order" pages to avoid future confusion and duplication of effort. - dcljr 09:02, 29 Aug 2004 (UTC)

I prefer the collections (or "chains") of pages 1 E6 m, 1 E7 m, etc. That is how this area of WP began originally. The "Orders of magnitude (foo)" pages began as either an overview or a botched attempt at simplifying the main "Orders of magnitude" page. I would suggest we look into restoring the "chain" pages, a number of which have been turned into redirects. -- Tarquin 12:06, 29 Aug 2004 (UTC)

[edit] Meanings other than "power of 10"?

The definitions given for "order of magnitude" on this page all relate to powers of 10. Which is fine, given that is what people often mean when they use the phrase, and this page is certainly useful.

However, I can think of one refinement and one addition that might be useful. It's my understanding that "order of magnitude" is dependent on the base in which one is counting. It means "raising the exponent by one" on whatever the base exponent is. For example, in base 2, one order of magnitude is one power of 2. So, 1000 binary (8) and 0010 binary (2) differ by 2 orders of magnitude. The "power of 10" definition is so often used because we (or most of us, anyway) count in base 10. But there is nothing inherently significant about powers of 10 in either mathematics or science. It's just notation. For a more general use of "order of magnitude", see here. I can't think of an easy way to work this into the current page, and given all the careful work done here, I don't want to mess things up!

My other point is that "order of magnitude" is sometimes used in mathematics to mean "asymptotic to". For a ref, see this link. Again, I don't want to screw things up, but I think this could be added as a disambiguation note at the end of the article. I don't think this usage is common, but it might be useful to include it. Gwimpey 05:50, Oct 21, 2004 (UTC)

[edit] factor thing

It is confusing. the factor thing. and it does not reflect the real feeling of the phrase. I tried to make it easier to understand and closer to the real easthetical feel. I see there had been some discussion on it. Please see that if some people here are saying it is confusing, there must be much more users out there who also find it confusing. I guess people will revert it because they seem to have been working on it for a long time, but please make it easier to understand even after that.

[edit] Order of magnitude table

The following table contains no useful information. --[[User:Eequor|ᓛᖁ♀ᑐ]] 18:14, 4 Dec 2004 (UTC)

[edit] Orders of magnitude of various quantities

In the following table the different quantities are lined up so that the following are in the same row:

  • length and the approximate time taken by light to cross that length
  • area of a square and the length of one side
  • volume of a cube and the area of one face
  • mass of some water and its volume at 4 degrees Celsius or 277.16 K

See also the separate tables for time, length, area, volume, mass, energy, power, temperature and dimensionless numbers.

Time Length Area Volume Mass Energy Temperature
(x 3)* (m) (m2) (m3) (kg) (J) (K)
(second) (metre) (square metre) (cubic metre) (kilogram) (joule) (kelvin)**
10-44 s 10-35 m          
...
10-28 s 100 zm        

1 pK
10-27 s 1 am        

1 nK
10-26 s 10 am       1 peV

 
 

1 µK
10-25 s 100 am        

1 mK
10-24 s 1 fm       0.001 meV

0.01 meV
0.1 meV

1 K
10-23 s 10 fm       1 meV

10 meV
100 meV

 10 K

100 K
1000 K
10-22 s 100 fm 10-28 m2     1 eV

10 eV
100 eV

 10,000 K

100,000 K
106 K
10-21 s 1 pm     10-33 kg

10-32 kg
10-31 kg

 1000 eV

104 eV
105 eV

109 K
10-20 s 10pm     10-30 kg

10-29 kg
10-28 kg

 1 MeV

10 MeV
100 MeV

109 K
10-20 s 10pm     10-30 kg

10-29 kg
10-28 kg

 1 MeV

10 MeV
100 MeV

1012 K
10-19 s 100 pm 10-20 m2

10-19 m2

   10-27 kg

10-26 kg
10-25 kg

 1 GeV

10 GeV
100 GeV

1015 K
10-18 s 1 nm 10-18 m2

10-17 m2

   10-24 kg

10-23 kg
10-22 kg

 1 TeV

10 TeV
100 TeV

1018 K
10-17 s 10 nm 10-16 m2

10-15 m2

   10-21 kg

10-20 kg
10-19 kg

 0.0001 J

0.001 J
0.01 J

1021 K
10-16 s 100 nm 10-14 m2

10-13 m2

 10-21 m3

10-20 m3
10-19 m3

 10-18 kg

10-17 kg
10-16 kg

 0.1 J

1 J
10 J

1024 K
1 fs 1 μm 10-12 m2

10-11 m2

 10-18 m3

10-17 m3
10-16 m3

 10-15 kg

10-14 kg
10-13 kg

 100 J

1000 J
10000 J

1027 K
10 fs 10 μm 10-10 m2

10-9 m2

 10-15 m3

10-14 m3
10-13 m3

 10-12 kg

10-11 kg
10-10 kg

 100000 J

0.001 kWh
0.01 kWh

1030 K
100 fs 100 μm 10-8 m2

10-7 m2

 10-12 m3

10-11 m3
10-10 m3

 10-9 kg

10-8 kg
10-7 kg

 0.1 kWh

1 kWh
10 kWh

  
1 ps 1 mm 10-6 m2

10-5 m2

 10-9 m3

10-8 m3
10-7 m3

 10-6 kg

10-5 kg
10-4 kg

 100 kWh

1000 kWh
10000 kWh

  
10 ps 1 cm 1 cm2

10 cm2

 1 ml

10 ml
100 ml

 1 g

10 g
100 g

 100000 kWh

1 GWh
10 GWh

  
100 ps 10 cm 0.01 m2

0.1 m2

 1 l

10 l
100 l

 1 kg

10 kg
100 kg

 100 GWh

1000 GWh
10000 GWh

  
1 ns 1 m 1 m2

10 m2

 1 m3

10 m3
100 m3

 1 t

10 t
100 t

 100000 GWh

106 GWh
107 GWh

  
10 ns 10 m 100 m2

1,000 m2

 1,000 m3

10,000 m3
105 m3

 106 kg

107 kg
108 kg

 108 GWh

109 GWh

  
100 ns 100 m 1 ha

10 ha

 106 m3

107 m3
108 m3

 109 kg

1010 kg
1011 kg
1012 GWh
  
1 μs 1 km 1 km2

10 km2

 1 km3

10 km3
100 km3

 1012 kg

1013 kg
1014 kg
1015 GWh
  
10 μs 10 km 108 m2

109 m2

 1012 m3

 1015 kg

1016 kg
1017 kg
1018 GWh
  
100 μs 100 km 1010 m2

1011 m2

 1015 m3

 1018 kg

1019 kg
1020 kg
1021 GWh
  
1 ms 1000 km 1012 m2

1013 m2

 1018 m3

 1021 kg

1022 kg
1023 kg
1024 GWh
  
10 ms 104 km 1014 m2

1015 m2

 1021 m3

 1024 kg
1027 GWh
  
100 ms 105 km 1016 m2

1017 m2

 1024 m3

 1027 kg

 
1030 GWh
  
1 s 106 km 1018 m2

1019 m2

 1027 m3

 1030 kg
1033 GWh
  
10 s 107 km 1020 m2

1021 m2

   1033 kg
1036 GWh
  
100 s 1 AU     1036 kg
1039 GWh
  
1 h 10 AU     1039 kg
1042 GWh
  
10 h 100 AU     1042 kg
1045 GWh
  
1 day 1000 AU     1045 kg
1048 GWh
  
10 day 104 AU     1048 kg
1051 GWh
  
1 yr 1 LY     1051 kg
1054 GWh
  
10 yr 10 LY          
100 yr 100 LY          
1000 yr 1000 LY          
104 yr 104 LY 1040 m2

1041 m2

        
105 yr 105 LY          
106 yr 106 LY          
107 yr 107 LY          
108 yr 108 LY          
109 yr 109 LY          
1010 yr 1010 LY          
1011 yr            
1012 yr
and more		            

* Each time shown is linked to that time. However, the time taken for light to cross the corresponding length is 3 times the time shown.

** These are the standard units but this table uses a variety of units, which can make it harder to read.

[edit] Units used in the table

The table uses units and prefixes that are commonly recognized:

[edit] Large numbers

This section has no practical value and seems unrelated to real-world orders of magnitude. --[[User:Eequor|ᓛᖁ♀ᑐ]] 19:25, 4 Dec 2004 (UTC)

I agree. Let's keep discuss of extremely large numbers on Large numbers, and keep this entry about "order of magnitude". 24.63.232.5 15:20, 9 January 2006 (UTC)

[edit] Extremely large numbers

For extremely large numbers, a generalized order of magnitude can be based on their double logarithm or super-logarithm. Rounding these downward to an integer gives categories between very "round numbers", rounding them to the nearest integer and applying the inverse function gives the "nearest" round number.

The double logarithm yields the categories:

..., 1.0023-1.023, 1.023-1.26, 1.26-10, 10-1010, 1010-10100, 10100-101000, ...

The super logarithm yields the categories:

0-1, 1-10, 10-10^{10}, 10^{10}-10^{10^{10}}, 10^{10^{10}}-10^{10^{10^{10}}}, \dots

For numbers close to zero, neither method is suitable directly, but the order of magnitude may be generalized to reciprocals.

Similar to the logarithmic scale one can have a double logarithmic and super-logarithmic scale. Generally, a point midway between two points corresponds to the generalised f-mean with f(x) the corresponding function log log x or slog x. In the case of log log x, this mean of two numbers (e.g. 2 and 16 giving 4) does not depend on the base of the logarithm, just like in the case of log x (geometric mean, 2 and 8 giving 4), but unlike in the case of log log log x (4 and 65536 giving 16 if the base is 2, but different otherwise).

[edit] Double logarithmic scale

The scale s of the pure double logarithm would have been log10log10x as

x > 10: s = log10log10x corresponding to x = 10^{10^{s}},

but the negative outputs in input range 1 < x < 10 do not have a useful meaning, so we require that x > 10. Furthermore the input minimum is unfortunately only 1, not 0 as needed, so another kind of double-logarithmic scale is added to be used when x < 0.1.

x < 0.1: s = − log10( − log10x) corresponding to x = 10^{-10^{-s}}

These two functions give the value s = 0 when x = 10 and x = 0.1. Because of this a third function is needed that sets s to 0 when the range is 0.1 < x < 10. If it is not desired to collapse this range to a single point, another double-logarithmic scale with offset could be defined by dividing the first function with 10 and multiplying the second with 10. Then both function correspond to the same x = 1 for s = 0.

x > 1: s = log10(1 + log10x) corresponding to x = 10^{-1+10^{s}}

x < 1: s = − log10(1 − log10x) corresponding to x = 10^{ 1 - 10^{-s} }

Different types of order of magnitudes
output order magnitude "s" log10 of log10log10 of log10(1+log10) of -log10(1-log10) of combination with offset
2 100 102 10100 1099 100.99 = 9.77 1099
1 10 101 1010 109 100.9 = 7.94 109
0 1 100 101 100 100 100
-1 0.1 10-1 100.1 = 1.26 10-0.9 = 0.126 10-9 10-9
-2 0.01 10-2 100.01 = 1.02 10-0.99 = 0.102 10-99 10-99

This introduces an error which is clearly visible for inputs smaller than 1010 and might make the double-logarithmic scale with offset harder to read.

input number x log10 order of magnitude combination of log10log10 and -log10(-log10) order of magnitude offset combination of log10(1+log10) and -log10(1-log10) order of magnitude
10100 100 2 2.00
1010 10 1 1.04
102 2 0.30 0.48
101 1 0 0.30
100 0 undefined or forced to 0 0
10-1 -1 0 -0.30
10-2 -2 -0.30 -0.48
10-10 -10 -1 -1.04
10-100 -100 -2 -2.00

Najro 20:21, 8 October 2006 (UTC)

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