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12/09/06 - Changed th example given in the first paragraph into something more neutral.
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
Consider also the following from MSN Encarta:
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)
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
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 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)
See discussion at 1 E9 m². -- User:Docu
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
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)
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)
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.
The following table contains no useful information. --[[User:Eequor|
ᓛᖁ
ᑐ]] 18:14, 4 Dec 2004 (UTC)
In the following table the different quantities are lined up so that the following are in the same row:
See also the separate tables for time, length, area, volume, mass, energy, power, temperature and dimensionless numbers.
* 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.
The table uses units and prefixes that are commonly recognized:
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)
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:
The super logarithm yields the categories:
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).
The scale s of the pure double logarithm would have been as
x > 10: corresponding to ,
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: corresponding to
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: corresponding to
x < 1: corresponding to
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 |
![]() | This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
12/09/06 - Changed th example given in the first paragraph into something more neutral.
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
Consider also the following from MSN Encarta:
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)
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
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 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)
See discussion at 1 E9 m². -- User:Docu
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
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)
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)
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.
The following table contains no useful information. --[[User:Eequor|
ᓛᖁ
ᑐ]] 18:14, 4 Dec 2004 (UTC)
In the following table the different quantities are lined up so that the following are in the same row:
See also the separate tables for time, length, area, volume, mass, energy, power, temperature and dimensionless numbers.
* 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.
The table uses units and prefixes that are commonly recognized:
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)
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:
The super logarithm yields the categories:
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).
The scale s of the pure double logarithm would have been as
x > 10: corresponding to ,
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: corresponding to
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: corresponding to
x < 1: corresponding to
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 |