# Talk:Dimensional analysis/Archive 1

## Derivation

"On the other hand, using length, velocity and time (L, V, T) as base dimensions will not work well (they do not form a set of fundamental dimensions), for two reasons:

Firstly, because there is no way to obtain mass — or anything derived from it, such as force — without introducing another base dimension (thus these do not span the space).

Secondly, because velocity, being derived from length and time (V = L / T), is redundant (the set is not linearly independent)."

Time gets derived from length and length so it´s also redundant.

"For example, it makes no sense to ask if 1 hour is more or less than 1 kilometer, as these have different dimensions, nor to add 1 hour to 1 kilometer. On the other hand, if one travels 100 km in 2 hours, one may divide these and conclude that one's average velocity was 50 km/hour."

time is length divided (not multiplied!) by length.

79.210.178.28 (talk) 12:02, 28 May 2009 (UTC)

## Dimensionless logs?

Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of 3 kg is undefined, but the logarithm of 3 is 0.477.

I am a 12th grade science student, and this struck me as a bit odd. Is there something wrong with the equation pH=-log10[H+]. Because [H+] is definitely in N.L-3 (where N is amount of substance - moles in other words) i.e. mol.dm-3. Although I admit that I've been wondering what the units of pH is (we're told it's dimensionless, but this seems wrong: or maybe the above theory that "dimensionless" quantities are not unitless has something to it) --Taejo 16:12, 8 August 2005 (UTC)

probably the better expression would be
${\displaystyle {\mbox{pH}}=-\log _{10}\left(H^{+}\cdot {\mbox{cm}}^{3}\cdot {\mbox{mole}}^{-1}\right)\ }$
actually, the log function could have a dimensional argument but the result would be a weird log(dimension) term.
${\displaystyle \log {(1{\mbox{ft}})}-\log {(1{\mbox{m}})}=\log {\left({\frac {1{\mbox{ft}}}{1{\mbox{m}}}}\right)}\ }$
is a real number. r b-j 18:07, 8 August 2005 (UTC)

${\displaystyle {\mbox{pH}}=-\log _{10}\left(H^{+}\cdot {\mbox{cm}}^{3}\cdot {\mbox{mole}}^{-1}\right)\ }$ may be a better expression (I dunno what you mean by better), but if your logs are base 10 (which I assume they are because they aren't natural or binary logarithms) then it isn't true. I'm pretty certain it's dm3. So anyway, can we say that log(3kg) is not undefined, it is log(3) + log(kg) = log(3) + log(1000) + log(g) = 3.477 + log(g) [where g is grams] --Taejo 21:53, 19 August 2005 (UTC)

## Huntley's refinement

I have made a large edit which is essentially including the section "Huntleys refinement" and removing the "worked example" which is essentially a derivation of the drag equation. I commented out the drag equation because I think the examples in the Huntley section are example enough. I have transferred this derivation to the drag equation article.

This whole idea of vectors having separate components for each dimension has further ramifications, including the fact that vector operators (tensors) also have differently dimensioned entries, and that is the motivation for showing that certain matrices can be squared without losing their dimension. PAR 06:24, 7 January 2006 (UTC)

Huntley's refinement is not sound, although it sometimes yields correct results, as shown in the examples you gave. I have made an extensive study of this proposal and published two papers relating to it in J. Franklin Institute 320, 267 (1985) and 320, 285 (1985). Huntley's refinement assumes that the dimensional symbols Lx and Ly are each elements of a group isomorphous with L; there is no relation like Lx Ly = Lz. Lz and Ly are independent. The fallacy in this system can even be seen in the projectile range problem. If you try to solve it using Range (in the x direction), initial velocity in the x direction, g, and the angle the projectile initially makes with the horizontal, θ. If you assume that the angle is dimensionless, then you would assume by the usual method that

${\displaystyle R=Cv_{x}^{a}g^{b}\theta ^{c}}$

so equating for Lx gives 1=a; equating for Ly gives 0=b; equating for T gives 0=-a-2b. These equations are inconsistent--they have no solution. If, on the other hand, you assume that angle has dimensions Ly/Lx then equating for Lx gives 1=a-c; equating for Ly gives 0=b+c; equating for T gives 0=-a -2b. These have a solution with a=2, b=-1 and c=1. This solution is troubling too--the power of θ is rather too definite. Actually the assumption that θ has dimensions Ly/Lx cannot be right, for then we could not take sin(θ) or cos(θ) because a series expansion of them would require adding unlike powers of Lx, which is not allowed.

The correct extension (given in my papers) is to introduce the idea that physical quantities such as lengths have orientations in space and have orientational symbols associated with this, analagous to the dimensional symbols such as L. The group that the orientational symbols belong to is not the same as the one that dimensionals symbols form. It is called the vierergruppe, and has only 4 elements (dimensional symbols form a group with an infinite number of elements). The orientational symbols have multiplication rules Lx Ly = Ly Lx = Lz, and Lx Lx=Ly Ly= Lz Lz =1, the identity element. These symbols are assigned to each of the physical quantities involved in the problem to be solved, resulting in a set of equations that supplement the dimensional equations, and sometimes provide a little more information for getting a more constrained solution than that obtained from dimensional analysis only. These symbols present no problems for transcendental functions of angles because of the multiplication rule that orientational symbols follow.

Huntley's addition often seems to require, to me, a non-intuitive assignment of orientations, especially to quantities that are scalars, such as viscosity. In orientational analysis (as I call it) viscosity is always orientationless (assigned the identity element). This idea also shows the distinction between pairs of things that are intuitively and physically distinct (as work and torque, numeric and angle, and so on). One of the pair is orientationless, the other has orientational character.

I think the Huntley addition should be removed.DonSiano 18:13, 7 January 2006 (UTC)

Ok, I have reverted it, but we need to figure this out. I will read your statement more closely and respond soon. Also, could you email me those papers? Thanks - PAR 21:26, 7 January 2006 (UTC)

## removal of dimensional analysis

The dimensional analysis article has been replaced by one on units. This is not proper, and there are a number of much better articles that covers the material on units besides:Unit of measurement, and Units conversion by factor-label. The article as it stands today should be reverted back to the old article on dimensional analysis. DonSiano 23:21, 24 February 2006 (UTC)

To Patdw - I agree with DonSiano - Please don't move this article until you discuss it with the people who are working on this article first. PAR 00:51, 25 February 2006 (UTC)

## Units vs. dimensions calculations

I think that the calculations in the introduction section seriously disrupts the flow of the article, and really don't belong here, but in the article on units and/or conversion factors. This calculation of feet and meters and adding seems to be out of place and should be replaced with a reference. This is an article about dimensional analysis, and once the distinction of dimensions from units is made, the latter should be dropped.DonSiano 22:52, 26 February 2006 (UTC)

well, i wasn't around or paying attention last February, but i fully disagree and i returned it to the article. speaking as an engineer and an educator, when engineers speak of "dimensional analysis" they are talking about whether or not they are comparing or adding quantities that are of the same species of animal. length measured in feet is the same species of animal as length measured in meters and, for the neophyte, this spells out exactly why it is meaningful to add feet to meters. or why it is meaningful to compare horsepower to kg-m2-s-3 but not to kg-m2-s-2. this dimensional arithmetic is what engineers do. Rbj 02:45, 1 May 2006 (UTC)

## There are no conversion factors between dimensional symbols???

This is mentioned in the introduction and this statement is obviously false: h-bar, c and G are conversions factors that allow you to convert any physical quantity into any other. If you want to do dimensional analysis pretending that L, T and M are incompatible, then you must forbid the use of these constants. This also explains why you have three "incompatible" dimensions: Precisely because you don't allow the use of three conversion factors. You can then assign different dimensions to L, T and M, which has the effect of making the three conversion factors dimensionful. Count Iblis 12:32, 2 May 2006 (UTC)

hi count, i think all was meant was that different power and product combinations of ${\displaystyle \hbar \ }$, ${\displaystyle c\ }$, and ${\displaystyle G\ }$ (along with ${\displaystyle \epsilon _{0}\ }$ in my opinion) can be used to convert from some dimensioned (or dimensionless) physical quantity and any other dimensioned (or dimensionless) physical quantity. is that not true? i am not saying the equation that does that conversion is meaningful from some theory of physics or not, but you can construct the conversion between - you name it - to any other amount of physical stuff. r b-j 01:46, 8 July 2006 (UTC)

## notation

Please, 217.84.175.39, stop messing around in the several "dimension" related articles. The square brackets mean "dimesion of" and in the articles where we have used italics, it is because they have been so used in the last 50 yrs. If there are "new" rules, show them . --Jclerman 11:00, 28 July 2006 (UTC)

did you even notice what I edited? The other ones were mistakes, which I accepted - no reason to pull out the lobe (is this the right expression in english?) -- 217.84.175.39 20:19, 28 July 2006 (UTC)

## edits of 07/28/06

hi Don,

"In mechanics, every dimension of physical quantity can be expressed in terms of distance (which physicists often call "length"), time, and mass..." - in this context "physical quantity" means the set of all physical quantity. sorta like "reality"; in some contexts it is meaningful to talk about "a reality", the reality of something specific, but it is also meaningful to speak of "reality" which means all reality. it is incorrect to say "In mechanics, every dimension of a physical quantity can be expressed in terms of distance (which physicists often call "length"), time, and mass..." because "a physical quantity" has only one dimension (assuming that the "identity dimension" is what we call the dimension of pure or dimensionless numbers). anyway, i reworded it.

"commensurate" is precisely (or concisely) the right word here. things (of which physical quantities are) must be commensurate for there to be any meaning of comparing them quantitatively (or adding or subtracting). this is really the fundamental thing.

"Only like dimensioned quantities may be added, subtracted, compared, or equated. When unlike dimensioned quantities appear opposite of the "+" or "−" or "=" sign, that physical equation is not plausible, which might prompt one to correct errors before proceeding to use it." - there is a conceptual difference between "plausibility" and "correctness", the latter being the stronger condition. saying that kinetic energy of an object of mass m and velocity v is

${\displaystyle T=mv^{2}\ }$

is plausible, but not correct (it's off by a factor of 2). saying that kinetic energy is

${\displaystyle T=mv\ }$

is not even plausible. if one were to compare mv to mgh (h for height) it could not be meaningfully done. to compare mv2 to mgh can be meaningfully done but would yield an incorrect answer. r b-j 17:29, 28 July 2006 (UTC)

Regarding my edit which was reverted, [1]:

I found the discussion of L, M, and T rather hard to follow:

• There was intermittent use of "L" for "distance" instead of "L" for length. It would be easier to understand if there were only one term used throughout, and alliteration makes it easier to remember which quantity is which.
• Division, which is normally represented by a vertical line, is represented by a diagonal line, and the symbols are serialized, rather than in their normal formation. Dimensional analysis is often presented as a cancelling operation, which is easy to visualize in an above/below fashion. In fact, the above/below arrangement is used later in the article; it would be better to be consistent.
• For elementary and high school students that are learning about dimensional analysis for the first time, I'm not sure all of them would know what the 2 means, so I added an explanation about squaring.

I think making the article presentable for readers is more important than simplifying the wikitext. (Though I agree the LaTeX is rather annoying to deal with. But hey, I managed to learn enough on the spot to do what I needed to do.)

I was rather surprised that the clarifying changes were reverted, so I guess I'm asking for reconsideration and perhaps a third opinion. -- Beland 01:52, 30 September 2006 (UTC)

I have to say that I agree with the reversions. This article is not the place to explain the mathematical phrase "raising to a power" or "squaring". Division is very commonly specified with a backslash "/" and is usually preferable for in-line equations. The "simple example" is really a comparatively simple example, and should not be labelled "complex". As Don Siano says, the use of M, L, and T as three of the most fundamental dimensions of physics is really not arguable. I do agree with Beland, however, that a consistent use of the words "length" or "distance" is preferable, and one might argue that they specify slightly different concepts. Any article has to assume a certain amount of background. The usual idea is that the first paragraph should be readable by a high school student. I think this article is good (perhaps not perfect) in this regard. PAR 14:08, 30 September 2006 (UTC)
I do actually disagree with the statement that "the use of M, L, and T as three of the most fundamental dimensions of physics is really not arguable". It's just a convention; you don't need dimensonful quantities at all in physics, as explained by Michael Duff here. Count Iblis 14:44, 30 September 2006 (UTC)
The article said "The dimensions of a physical quantity are associated with symbols, such as M, L, and T[citation needed]". It does not say that these three are the "most fundamental" nor does it say that these are the only possibilities (it uses the phrase "such as" to make the idea of dimensions rest on familiar ground. Further on, it says that quantities such as electrical charge with a dimension labled "Q" are often introduced. I think perhaps a whole paragraph or section discussing the choice of dimensions for different fields of physics (esp thermodynamics and E&M) should be discussed, as well as the posibility of working in an area of physics in which all the physical quantities are made dimensionless. A case can be made that the choice is partly made for convenience) I am going it make a pass at this.DonSiano 17:13, 30 September 2006 (UTC)
there are multiple issues that Beland brought up that i just don't get. first, "Division, which is normally represented by a vertical line, is represented by a diagonal line..." ??? Do you mean "N divided by D" is "N|D"?? i have never, ever seen such notation for division. it's always been N/D. i have never seen the use of backslash for division except in MATLAB for matrix equations. when A = BC, where A and C are both column vectors, B is a square matrix, and A and B are known, then C is solved conceptually by "dividing" both sides by B and you get C = B\A. other than that, i have never seen backslash to mean any kind of "division".
i also don't understand the conceptual problem with powers attached to dimensions. L2 is area if L is length. we say the floor space of our apartment is 90 m2. big deal. how is it that this is beyond high school students? now, certainly, the Buckingham π theorem is likely beyond high school students, but the introductory part should not be beyond high school juniors and seniors taking physics and/or chemistry or any other physical science. it is just an extension of keeping your units straight and understanding that you can't add, subtract, or "compare apples to oranges".
about which "fundamental" dimensions to count, i think that this article, Fundamental units, Physical quantity, Physical constants, and perhaps Planck units or Natural units all have something to do with each other and we should try to have both conceptual and notational consistancy between the articles. r b-j 18:53, 30 September 2006 (UTC)

## Dimension of interest on money

Years ago I read in some kind of economics encyclopaedia published by Palgrave about the dimension of interest on money. As far as I recal it was 1/t but I could be wrong. It would be nice to have something added about this and how it was derived. I do not have enough know-how to do it myself. And does money have a dimension or is it dimensionless, or could it fruitfully be used as a dimension in economics etc.? Thanks. —The preceding unsigned comment was added by 80.0.123.238 (talk) 21:03, 31 December 2006 (UTC).

I'm just guessing here, but I would think that money does have a dimension. Its units would be dollars, pounds, yen, whatever, just like dimensions of length has units of meters, feet, whatever. The interest rate would be dimensionless since it would be the ratio of two units of money. PAR 01:54, 1 January 2007 (UTC)
Sorry about that - interest rate would have dimensions of 1/time, because its a ratio of two units of money per unit time. PAR 22:59, 1 January 2007 (UTC)

## Concern about "a more complex example" -- Non-expert comment

I don't know much about Wikipedia etiquette, so I apologize if I shouldn't be posting questions here since I am someone who is only using this page to learn...

I'm a bit confused about the "more complex example" of dimensional analysis that talks about the energy in a vibrating string. Following the dimensional analysis, the author identified 4 important variables for solving the problem; l, A, s and E eliminating density (which I agree with). For Buckingham's Pi analysis, this gives n=4 variables with m=3 fundamental dimensions (L,M and T). So shouldn't this equation be solvable using only 1 dimensionless group? Why does the author attempt to use 2 dimensionless groups?

I think this should at least be explained.

ArcticFlamesFan (talk) 18:15, 10 April 2008 (UTC)

## Buckinghams pi theorm

In dimensional analysis i think there are two methods to obtain the dimensions of any given quantity one being Reyleigh's method and the other Buckinghams pi theorm can anyone add information regarding the same to make the article complete and informative. Kalivd (talk) 15:41, 24 September 2008 (UTC)

## Reversion

I have reverted the edits of 79.210.178.28 for the following reasons:

• The logarithm of 3 is not 0.477121255. This is only an approximation
• A corollary is not the same as a summary
• The statement "is to some extent arbitrary" is much more understandable than "can be expanded and/or delimited (more generally: can be modified)"
• consistency is not the same as redundancy.

PAR (talk) 16:22, 28 May 2009 (UTC)

## Equilibrium constants in chemistry

It is a strange situation that I think might be worth mentioning, as chemists appear at first glance to be using equations that are dimensionally inconsistent! For example, the dissociation constant of an acid HA in water is Ka = [H+][A-]/[HA] , which should have dimensions of moles/liter. Yet they plug it into the equation delta G = -RT ln (K), that is, they are taking the natural log of a quantity which is not dimensionless. Shouldn't we somehow explain how this can actually "work"? 69.140.12.180 (talk) 14:12, 5 June 2009 (UTC)Nightvid

The concentrations in the equations are actually concentrations relative to a standard concentration of 1 mol dm^-3. So really all the concentrations in the expression are dimensionless quantities. —Preceding unsigned comment added by 82.6.96.22 (talk) 22:25, 30 November 2010 (UTC)

## Definition

The definition (with a little box one) is overlength on my browser (firefox 3.5.2). I am just a newbie to Wiki. Can anyone fix it so that it is more readable? 121.203.38.158 (talk) 17:18, 12 September 2009 (UTC)

Fixed by removing leading space. Vsmith (talk) 18:29, 12 September 2009 (UTC)

## Historical Reference and Biological Examples

I have added a reference to Newton taken, as indicated, from the first of two long articles by Walter Stahl back in 1961 about dimensional analysis in mathematical biology. I would like to add some more brief historical references from that paper and also a brief reference to dimensional analysis in biology. Alanfmcculloch (talk) 11:52, 19 November 2009 (UTC)

## Polynomials of mixed degree

Similarly, while one can evaluate monomials (xn) of dimensional quantities, one cannot evaluate polynomials of mixed degree on dimensional quantities: for x2, the expression (3 m)2 = 9 m2 makes sense (as an area), while for x2 + x, the expression (3 m)2 + 3 m = 9 m2 + 3 m does not make sense.

I changed it to this:

Similarly, while one can evaluate monomials (xn) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for x2, the expression (3 m)2 = 9 m2 makes sense (as an area), while for x2 + x, the expression (3 m)2 + 3 m = 9 m2 + 3 m does not make sense.

The difference is that it now says "with dimensionless coefficients". Consider this polynomial of mixed degree:

${\displaystyle x^{2}+(3{\text{ meters}}\cdot x),\,}$

where x is a distance. That does make sense. Now consider a more commonplace example:

${\displaystyle {\frac {-1}{2}}\cdot \left(32{\frac {\text{foot}}{{\text{second}}^{2}}}\right)\cdot t^{2}+\left(500{\frac {\text{foot}}{\text{second}}}\right)\cdot t.}$

This is the height to which an object rises in time t if the acceleration of gravity is 32 feet per second per second and the initial upward speed is 500 feet per second. It's a polynomial of mixed degree and it makes perfect dimensional sense. Michael Hardy (talk) 05:15, 9 February 2010 (UTC)

...and now I've added that example to the article. Michael Hardy (talk) 05:25, 9 February 2010 (UTC)

## Can we get consistent regarding the style of symbols for basic physical dimensions?

Bold or no-bold? Italics or not? Serif or sans? What's it gonna be? 70.109.186.166 (talk) 02:07, 12 October 2010 (UTC)

## Temperature: how many dimensions to consider? and, Rigor?

In dimensional analysis, temperature is usually treated as a distinct dimension from energy. But temperature is just the average of many particular bits of energy. (This is a problem that potentially occurs in applying dimensional analysis to classical physics, and also obviously to thermodynamics, but is also closely analogous to the problem of applying it to fundamental physics where mass is energy and time periods are lengths. It is completely different from the problem of which basis of the dimensions to treat as fundamental; rather the problem is what number of dimensions to include in the first place.) Knowing that the choice will affect the results of the dimensional analysis, how does one decide whether to (for example) treat temperature as independent of energy or not?

Also, while the article explains that the potential units will form a vector space (for which the base dimensions will form a vector basis set), there is almost no explanation of why it is useful to conceptualise dimensional analysis in this way (except the small emphasis of LI alternative choices of base dimensions)?

Is there any proof that dimensional analysis is valid? Cesiumfrog (talk) 05:17, 3 November 2010 (UTC)

Cesiumfrog - check out Buckingham Pi theorem PAR (talk) 17:00, 3 November 2010 (UTC)
Usually, when dealing with thermodynamics, temperature is never alone, it is always in the form kT (k Boltzmann's constant) or RT (R, gas constant), so there is no problem: these expressions are energy and energy per mole, respectively.--GianniG46 (talk) 08:52, 3 November 2010 (UTC)
You don't mention it, but temperture needs mention in the "Position vs displacement" section, since it is perhaps the worst-behaved in that regard since negative temperature doesn't generally make sense, but negative temperature difference is perfectly fine. —Ben FrantzDale (talk) 12:43, 3 November 2010 (UTC)
FYI: Check out Negative temperature. I have no opinion one way or the other right now. Also, I cannot think of any case where temperature cannot be replaced by the energy per particle kT or energy per mole RT, even in Newtons law of cooling. PAR (talk) 16:55, 3 November 2010 (UTC)

From the point of view of fundamental physics where one does not distinguish between certain (or any) dimensions, argument based on dimensional analysis seem to be nonsense. However, if you try to define what "classical" is from within fundamenal physics, you see what is going on: you are looking at a certain scaling limit of the fundamental theory and the dimensional analysis argument within classical physics is a simple consequence of a scaling/renormalization argument within the fundamental theory where no dimensions are assumed to exist. John Cardy has also made this point in one of his books (I think the book scaling and renormalization in statistical physics). Count Iblis (talk) 17:01, 3 November 2010 (UTC)

I explain here how this works when deriving the classical limit from special relativity while sticking to natural units. Count Iblis (talk) 17:23, 3 November 2010 (UTC)

## well-established physics described by an editor as "confusing and obscure"

In this edit, the edit comment was "(Commensurability: Confusing and obscure, here. It can be told elsewhere:"If we put light speed =1, then...")".

The comment is wrong. From the POV of special relativity, well-established by e.g. the dangerous nature of nuclear bombs and the energy obtained from nuclear power stations, decay of muons falling towards the Earth, etc., the lengths of different space-time paths should all be measured in the same units. Measuring them in different units and imagining these as incompatible units is only correct in pre-relativistic thinking, or in the non-relativistic limit. From a knowledge POV, suggesting that time and space cannot be measured in the same units is wrong according to the huge majority of professional physicists.

Saying "If we put light speed = 1" does not reverse the error.

As i said in my edit comment, 1 hour is about 10^12 m = 10^9 km, so it makes perfect sense to ask which is greater. It makes sense to ask if 1 m is greater than 1 inch, and it makes sense to ask if 1 hour is greater than 1 km. The answer to both questions is yes. Choosing later on to set 1 inch approx 2.54 cm does not reverse the error of saying that m and inches are fundamentally different units.

For this reason, i am undoing the edit. Boud (talk) 20:53, 5 November 2010 (UTC)

This is where the extensions of Huntley and Siano come into play. In these extensions, lengths perpendicular to each other are not considered to have commensurate dimensions. You can obviously have a consistent theory by assuming that they are commensurate, but to assume they are not will also give a consistent theory which yields more information. That is Huntley and Siano's point. In special relativity, time and space are perpendicular to each other. Here too, you can have a consistent theory by assuming that space and time be measured in the same dimensions, but again, to assume they are not will give a consistent theory which yields more information. Before the theory of relativity ("Lorentz relativity") there was Galilean relativity, in which time and space were separate, and required separate dimensions. With a new understanding of relativity and the Huntley/Siano extensions, they remain separate. Lets not go backward unless it is helpful to do so. PAR (talk) 22:19, 5 November 2010 (UTC)

To be fair Boud, nobody is denying relativity, but it would be confusing if every explanation of basic dimensional analysis were prefaced with disclaiming "in pre-modern physics.." clauses. Because it implies that the explanation is wrong (thus undermining the content of this article) but does so obscurely (a link to the page on SR is certainly insufficient for the casual reader to deduce your intended point) and is even esoteric (since even among relativity experts the interpretation you assume is not universal: it is still conceivable to treat c as a dimensional constant in relativistic physics). Instead I suggest a new section to separately explain how dimensional analysis relates with those fundamental insights from modern physics. It appears Count Iblis has already prepared suitable material and references. Cesiumfrog (talk) 07:27, 6 November 2010 (UTC)

As long as length (or displacement) and time ain't the same exact thing, c is a dimensionful constant that relates the two. 75.32.144.218 (talk) 15:44, 6 November 2010 (UTC)
This is a subjective issue. One can assign incompatible dimensions to length and time intervals and then c becomes dimensionful, but you don't have to do that. You can also plug in a constant c in the energy conservation law: total energy = kinetic energy + c potential energy. Then the units we now use corresponds to putting c = 1 here. But then, since potential energy and kinetic energy are not the same thing, you can decide to measure them in different units, making c different from 1. If you make those units incompatible, c becomes dimensionful. Count Iblis (talk) 16:28, 6 November 2010 (UTC)
There is the question, though, of what is gained by doing that, which is an important consideration. I don't know if such a thing would be useful, the way that saying perpendicular directions have different units is helpful. It is dimensionally correct, however. PAR (talk) 14:19, 7 November 2010 (UTC)
Well, maybe this should go to sci.physics.foundations or sci.physics.research, but I have never been convinced that time is "just another dimension, qualitatively no different from the three spatial axes." In that Minkowski tensor, it gets a -1 and x, y, z all get +1. Outside of the event horizon of a black hole, there is no "arrow of space" in the sense of arrow of time. There are other speeds in reality besides c to consider. 75.32.144.218 (talk) 17:00, 7 November 2010 (UTC)
Thats kind of like having three spatial dimensions: forward, up and right, and then saying that you are not convinced that forward is "just another dimension", because it always seems to be the direction in which you are looking. But you know its not, because other people's forward direction are not the same as yours, yet their physics is the same as yours. True, the forward direction doesn't get a -1 in the metric tensor, so its not exactly the same, but the only reason time gets a -1 in the metric tensor is because its the way YOU are moving thru time. In somebody elses metric tensor, the -1's and +1's are all mixed together (according to you), and the whole point of relativity is that it doesn't matter whose metric tensor you use, your special one where time has -1 in it, or somebody else's that doesn't. PAR (talk) 19:08, 7 November 2010 (UTC)
I think Mr/Ms AnonIP75.32 is going a little too far and stating a shallowness of familiarity with SR. The crucial point PAR is hinting at is that two observers will differ over whether they think that a given pair of events occur at the same time (separated only by space) or that the pair are separated in time as well as space. There is no objective way to completely differentiate "distance" and "period of time". On the other hand, I think PAR's comment risks confusing "metric tensor" (which is the same for each observer, as any geometric thing must be) with the components thereof (in the subjective natural basis according to each observer's velocity and orientation). Cesiumfrog (talk) 23:37, 7 November 2010 (UTC)
I think there is a logical disconnect here. I am saying that "time is not precisely like the 3 spatial dimensions" and that is not comparable to saying that I am not convinced that forward is "just another dimension", because it always seems to be the direction in which [I am] looking. PAR is saying that A implies B and I see no logical imperative that it does. The point of differentiation continues to be that whether it's my textbook or the other observers' textbooks, it's still [-1, 1, 1, 1] along the diagonal. The somebody elses do not look into their textbooks and see the -1's and +1's coming out differently. t is still conceptually different than x, y, and z. Time is still a different species of animal, even if there is some relationship to space. To say that time is the same as space in every respect would be to say that such is so in the historical and our common experience. That is not so.
There is no difference between up, forward, and right. Sometimes our head is oriented toward the sky, sometimes east and sometimes north. It's all the same, except there happens to be a large mass close by with a center of mass along one of those axes. 75.32.144.218 (talk) 10:21, 8 November 2010 (UTC)
I think this is more of a philosphical issue, because the two views are equivalent from a mathematical point of view, as the speed of light is always available to convert time interval to distances in any setting. In classical physics there is no such universal conversion factor and then time and space are not equivalent in this mathematical sense. When doing dimensional analysis in classical physics, you can only convert time intervals to distances using some characteristic speed that appears in the problem at hand.
But then we don't do "classical physics" or "relativistic phyisics", we simply do physics and depending on the problem at hand, we can decide to analyze it using classical physics. So, the decision that time should not be converted to lengths using the conversion factor c in some problem, is something that comes out of the analysis of the problem, e.g. when it transpires that one can ignore relativistic effects. But then the classical treatment is only an approximation and relativistic corrections would still be found by using c as a conversion factor.
Only in the classical limit where c becomes infinite, you really can't use c anymore. So, formally, you recover exactly the same conclusion you would reach by starting out with natural units and by inserting a scaling parameter c in equations to compute an appropriate scaling limit by letting c tend to infinity. Count Iblis (talk) 15:30, 8 November 2010 (UTC)

## Trig functions

If you can't take irrational functions of a dimensional quantity, how can we take sines/cosines etc of angles? I sort of accept that a radian is not really a proper unit, because it's a "natural" measure of angle, but a degree definitely is a unit. Is the use of degrees just an abuse of notation, with the degree really being a number, not a unit, equal to the number of radians in a degree? —Preceding unsigned comment added by 82.6.96.22 (talk) 22:35, 30 November 2010 (UTC)

If you implicitly convert measurements from units of degrees into fractions of a complete rotation, you get a ratio, which is dimensionless. Cesiumfrog (talk) 23:14, 30 November 2010 (UTC)
Or better yet, simply consider a degree an arbitrary, human-made unit that is 1/57.2958 of a radian. A radian is a ratio of the arc length swept by an angle to the radial arm (and so is dimensionless). 70.109.174.178 (talk) 23:49, 30 November 2010 (UTC)
According to the Huntley/Siano extensions of dimensional analysis, distances in different directions do not have the same units. The radian is a ratio of the arc length swept by an angle to the radial arm, but since these distances are not in the same direction, the radian is not dimensionless. It is without units, but it is oriented as a vector perpendicular to the plane in which the angle lies. The square of an angle is without orientation. This means that ${\displaystyle \sin(\theta )=\theta +\theta ^{3}/3!+...}$ is an oriented quantity, while ${\displaystyle \cos(\theta )=1+\theta ^{2}/2!+...}$ is without orientation. If we take θ to be dimensionless, then the expression ${\displaystyle q=\sin(\theta )+\cos(\theta )}$ will be considered dimensionally consistent, but if θ is taken to be oriented, then q is not dimensionally consistent. The second option yields more information, since, in fact, an expression like q is never found in nature. (Note: ${\displaystyle \sin(\theta +\pi /2)=\cos(\theta )}$ appears inconsistent, but should be interpreted as an instance of ${\displaystyle \sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)}$ which is consistent.) PAR (talk) 01:38, 1 December 2010 (UTC)

## Scalar multiplication in the space of dimensions

In the section on mathematical properties, where dimensional symbols are neatly described as elements of a vector space with the rational numbers as scalars, we read the following:

When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the vector space.

This is all well, but then:

When measurable quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication in the vector space.

Raising a dimensional symbol to a power does not yield a scalar (i.e., a rational number), so I don't see how it could correspond to a scalar product. Raising MiLjTk, also written as (i,j,k), to the power of n, gives us MniLnjTnk, also written as (ni,nj,nk), which is still a vector and not a scalar.

If the current article wording is actually correct, it seems a clarification is needed. —Bromskloss (talk) 16:52, 14 February 2011 (UTC)

Scalar multiplication does not yield a scalar.--Patrick (talk) 17:54, 14 February 2011 (UTC)
Ah! I realised my mistake too late, with no computers around. Of course "scalar multiplication" means multiplication with a scalar not scalar product. Sorry about all that! —Bromskloss (talk) 18:40, 14 February 2011 (UTC)

## Formatting of ML^2T^−2 under "Commensurability"

I know it isn't too much of an issue for the tech-savvy, but the units ML2T−2 looks quite confusing, with the minus sign in the exponent of T joined with the letter itself, which is difficult to make sense of at first glance. Is there another format we could display it in, or possibly use a different pair of non-commensurable units? —Preceding unsigned comment added by 130.56.71.50 (talk) 10:33, 9 May 2011 (UTC)

Prefer ML2/T2 or ML2T -2? Cesiumfrog (talk) 01:19, 10 May 2011 (UTC)
It is usual to use the minus sign so I'd go for the non-breaking space if there is a problem. With a minus sign instead of a hyphen that would be ML2T −2 Dmcq (talk) 08:26, 10 May 2011 (UTC)
Another option is to put a math template round them all and leave out the non-breaking space as in ML2T−2. I'm not sure that is an advantage in this article though, I only normally do that where I also have stand alone equations so the fonts are the same for the variables. Dmcq (talk) 08:32, 10 May 2011 (UTC)
Justr had a look at the article and it does have a lot of stand alone equations. I noticed another thing though - they have the MLT in italic and I think they should be roman as italic is usually used for variables. Dmcq (talk) 08:36, 10 May 2011 (UTC)

## What the hell?

What's with the sudden disappearance of the article? I can only see "t analizi]]", whatever that is. The page history shows that a random IP deleted everything, so I hope someone can restore the previous version. — Preceding unsigned comment added by 203.116.31.110 (talk) 03:44, 30 September 2011 (UTC)

I'm afraid it is part of this encyclopaedia that anyone can edit business. Any eejit can come along and vandalise things. They are normally removed after a couple of vandalisms and the article reverted to the version before they came along, in this case it took about 50 minutes before the article was fixed. Dmcq (talk) 09:52, 30 September 2011 (UTC)

## 6 Dimensions for Electrics

Quantities have scales, scales have units, theories connect scales, an algebra rides on the theory, and the dimensions ride on the algebra. It's because of this that we can 'see through' dimensional analysis to convert gaussian units and formulae to something like SI, which have entirely different dimensions and units.

The exact number of dimensions depends on how many variables are not difined in the theory, or how many are set to unity. For example, one can use something like these (as logs of dimension) L = -9, M=-27, T=-12, Q=-18, kelvins=1, gives pretty much the correct size for atomic constants.

Leo Young wrote a book on the subject (Systems of Units in Electricity and Magnetism), which shows that the principle systems of electromagnetism (with the exception of the Hansen or cgs-practical), are all coherent to a single theory, for which there are six dimensions. In essence, along with L,M,T and Q, one adds two new ones, S (which deals with the 4-pi) and U (which deals with the factor c). The various quantities then split over six dimensions.

For example, charge is Q. The produced flux is QS. The legitimate natural ways of defining S is via gauss's divergence theorm, or by way of the radiation (flux = S.source / 4.pi r²). The latter is much favoured by those who deal with point sources, it is still in the SI under gravity and light. The former brings into coincidence many values (electric polarisation vs induced field E in a dielectric etc). It is this convergence of names that is the rationalisation, not the change of formulae.

When you use Young's dimensional system, you can readily convert between non-rationalised cgs formulae and si formulae, based purely on some log-system of powers, eg M = Q = 1, L = T = 1, S = -1, [One sets S to 4pi or 1/4pi depending on direction. eg

  Coulomb's law     F = Q Q / 4pe r²    gives  ->  1 = 2  This means the rhs needs a -1, eg S.
F = S Q Q / 4pi e r²      Use S = 4pi to go to non-rationalised
F = Q Q / e r²


Wendy.krieger (talk) 09:05, 1 October 2011 (UTC)

## An equation with torque on one side and energy on the other would be dimensionally correct, but cannot be physically correct!

Why not? just interpret the constant of proportionality as an (effective) angle! Actually, this example shows that incompatible units don't really exist. As soon as you define a few, you are forced to define more and more ad infinitum to keep the system consistent. You needed to define a unit for angles to rescue the system, but it doesn't end there. You can now consider a two dimensional space of points (alpha1, alpha2) where alpha1 and alpha2 are angles. You can define angles in this space which you now have to assign a dimension that is different from radians.

At the fundamental level there is no difference between Length, Time and Mass. They can all be converted into each other using combinations of G, h-bar and c. These constants are nothing more than conversion factors and have no physical significance whatsoever.

Count Iblis 23:47, 9 August 2005 (UTC)

wow! i have to confess that the plethora of misspellings ("quantites", "quatities", etc.) were originally mine. holy crap! how did that get out?!
anyway, that particular statement precedes my contribution even though i ran with it a bit.
i tried to point out that a torque quantity can be converted to an energy quentity by use of an angle measured with the mathematically natural units, in radians, and that radians are dimensionless because they are a ratio of length over length (or a measure of swept area in the unit circle for circular trig and in the unit hyperbole for hyperbolic trig),
fix it how you see fit, Count. i dunno. r b-j 01:08, 10 August 2005 (UTC)

Rbj, the misspellings were corrected by someone else :). I'm not sure if it is a good idea to do something about this torque problem. This is a weakness of the idea that incompatible units/dimensions exist. But this idea is, unfortunately, the view of a very large part of the scientific community. Only some people who work in fundamental physics know better.
Dimensional analysis is actually nothing more than demanding that equations be nonsingular when taking the limits c --> Infinity, h-bar --> 0 and G --> 0. So, you pretend to live in a classical world, infinitely far removed from the Planck scale. All connections between Length, Time and Mass are thus lost.Count Iblis 12:55, 10 August 2005 (UTC)
i think i've heard some guys on sci.physics.research (say Jan Lodder or John Baez) try to tell me the same thing. i don't think that the fundamental physics community is the one intended to be serve by an article like this. it's sorta like the Dirac delta "function". we bone-head Neanderthal engineers need to think about that function (or non-function) as a limiting spike of unity area. but it ain't perfectly mathematically correct.
i do not understand why any differentiation of Length, Time and Mass (and Charge IMO) is lost, and it certainly is needed in what we do at our human scale.
if you have ideas, i'm happy to read them. r b-j 16:12, 10 August 2005 (UTC)

Let me give an example. I think you agree with me that mass and energy are the same things, but (usually) expressed in different units.

i kinda agree, but only if we say that length and time are the same things, but (usually) expressed in different units. if 299792458 meters are exactly the same thing as 1 second, then i agree with you that mass and energy are the same things. but i am not sure that squares with the premise of dimensional analysis. i think, when we do dimensional analysis, that time, length, mass, and electrical charge are different classes of "stuff" and all of the other physical "stuff" that we quantify (like force, energy, voltage, temperature, etc.) can be expressed in terms of time, length, mass, and charge.

If you consider the kinetic energy of a particle:

${\displaystyle E(v)={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}E(0)\ }$ (1)

boy! i dunno if i agree with that. isn't the kinetic energy of a particle
${\displaystyle T=m_{0}c^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}$  ?

Let's consider the 'nonrelativistic' limit. Let's expand (1) in powers of ${\displaystyle 1/c\ }$:

${\displaystyle E(v)=E(0)+{\frac {1}{2}}E(0)v^{2}/c^{2}+...\ }$

The energy difference ${\displaystyle E(v)-E(0)\ }$ is:

${\displaystyle E(v)-E(0)={\frac {1}{2}}E(0)v^{2}/c^{2}\ }$

okay, i get it. E(v) is the total energy, E(0) is the rest energy, and the difference is the kinetic energy. i've done this before (in fact i tried to put this in the correspondence principle article, but they took it right out say that the c.p. applied only to quantum mechanics).

In the limit c --> infinity, the kinetic energy goes to zero unless you define an M such that ${\displaystyle E(0)=Mc^{2}\ }$ . So to a classical physicist a new quantity M seems to exist such that kinetic energy is ${\displaystyle {\frac {1}{2}}Mv^{2}}$. In the classical limit the relation between M and E(0) is lost, because c goes to infinity. In that limit you need to consider M as a new quantity that is incompatible with energy and you need to to give it a separate dimension. But because in the real world c is not infinite, M and E can be given the same dimensions without any problems.

Count Iblis 01:05, 20 August 2005 (UTC)

we know in the real world that c is finite. we know, in the real world, that sometimes mass and energy are interchangable, or at least have been converted from one to another. but we use this conversion factor, ${\displaystyle c^{2}\ }$, to make the exchange. it's sorta like money where the price of a commodity is the conversion factor in the exchage.
the problem is, philosophically, i can't quite bring myself to say that 299792458 meters are exactly the same thing as 1 second, or that electrical charge is the same thing as length x force^(1/2) which is what the electrostatic CGS people say it is. (that's how they say that ${\displaystyle \alpha ={\frac {e^{2}}{\hbar c}}}$ instead of ${\displaystyle \alpha ={\frac {e^{2}}{\hbar c4\pi \epsilon _{0}}}}$.
only if c=1 (or at least dimensionless, but i'd hate to carry around the dimensionless value for c everywhere it was needed, if c was not 1) can M and E be given the same dimensions without any problems. but saying that is like saying length and time is the same damn thing but they clearly have something different. you can move back and forth in the x, y, and z directions, but moving in the t direction is unidirectional ( arrow of time ) that is a qualitative difference.
that's my spin from the POV of an electrical engineer. r b-j 03:56, 20 August 2005 (UTC)
Dimensional analysis is a reduction of the algebra of a theory. It only works with quantities when the theory is constant. SI and CGS use different valid theories: SI is not coherent to CGS theory and CGS is not coherent to SI theory. You can't use dimensional analysis between cgs and si unless you use the full six dimensional analysis (three electrical). What is written as c in cgs is a great variety of constants in SI, because cgs maps things like the Lorentz-constant, various forms of the electric and magnetic constant, and even the ratio of units onto c, whereas in SI these might disappear all together. On the other hand, SI puts in things that cgs does not have, and there is the shuffle of 4 pi between the two. All of this is handled by three electrical dimensions. Wendy.krieger (talk) 07:24, 12 October 2011 (UTC)

## edit 8/24/06

Question: does the n-th root of a dimension have any meaning? For example,

${\displaystyle {\sqrt {4m}}=2m^{1/2}}$

Are these dimensions meaningful, or is this forbidden by an assumption somewhere?

units like that have meaning in some special contexts. for instance the input noise voltage of an op-amp is measured in ${\displaystyle {\frac {\mbox{V}}{\sqrt {\mbox{Hz}}}}m^{1/2}}$, because it's really about power per hertz. the way they measure electric charge in the cgs system is in fractional power units. r b-j 05:37, 25 August 2006 (UTC)

The hildebrnd value has the dimension of sqrt(pressure) --Wendy.krieger (talk) 12:24, 12 October 2011 (UTC)

## Why Charge instead of Current as fourth base dimension?

The International System of Units defines 7 base units (see http://en.wikipedia.org/wiki/SI_base_unit), corresponding to the following quantities:

[Length], [Mass], [Time], [Current], [Temperature], [Amount of substance], [Luminous intensity]

However, this article uses the following base quantities for explaining dimensional analysis:

[Length], [Mass], [Time], [Charge], [Temperature]

I do realise that you can drop the latter ones, depending on how you look at it (as also mentioned in the Definition section). However, for those that you do include, wouldn't it make sense to use the same ones as defined in the SI system?

Why, then, is charge chosen rather than current? Technically, it doesn't matter which of the two is used (they are directly related through Q = I · t), but I as a reader find this confusing, especially as the article just mentions those six quantities without further explanation:

"Every physical quantity is some combination of mass, length, time, electric charge, and temperature, (denoted M, L, T, Q, and Θ, respectively)."

Is there some kind of historical/practical/technical reason for mentioning exactly these five ones, instead of strictly using SI quantities? If so, this should be stated in the article. If not, this should be changed to match the SI system. —Preceding unsigned comment added by 85.180.79.29 (talk) 17:14, 7 December 2009 (UTC)

Charge is a more basic concept than its flow. current comes from a later refit of the magnetostatics system, which in its original form, fails to explain the potential in a hollow magnet (like the solenoid). emu are then electrics measured magnetically, is what the practical units and hence SI is defined in terms of.Wendy.krieger (talk) 07:10, 3 October 2011 (UTC)
SI does not inclued all possible quantities (eg money), and some of the base units were formerly derived (like mole, which had the dimensions of M: everyone had M/dalton for N, but SI had to use M/kilodalton, so N can not be derived from M. Correspondingly, it's sperious. C (candela) was adopted in the fps, but no one ever repests it there. Charge does not reflect the full electromagnetic relation, since simply using charge will still not give correct conversions between cgs and SI. CGS and SI have different electrical dimensions (ie QCGS is not equal to QSI, and that another dimension is needed to handle the 4pi (eg S = 1 in SI and 4pi in CGS). Wendy.krieger (talk) 07:33, 12 October 2011 (UTC)

## Pythagorean theorem

I have added a dimensional proof of the Pythagorean theorem. I believe it, at least for me (a physicist), much more convincing than all those complicated triangle squashing which are used in most other proofs. Unfortunately, though I have given an academic reference, I don't know who originally proposed it. If someone knows, please add the reference.--GianniG46 (talk) 10:32, 13 October 2010 (UTC)

In the figure, the function f(α) appearing three times is missing its second argument. Also, what's missing in the presentation is that we need the property f(θ, π/2) ≡ f(π/2-θ,π/2) in order to get to the point where the f(α, π/2) can be eliminated. I intend to change the proof to use the base angles, in which case the required property of f is f(α, β) ≡ f(β, α). That seems an easier presentation to me, and is in line with the choice of angles in the quote source (unfortunately I don't have a better source at hand but it will come eventually). If somebody would then kindly edit the figure to show f(α, β) instead of f(α) in the three instances, it would complete this edit. Sweet proof, I enjoy it too. -- Saveur (talk) 01:51, 10 November 2011 (UTC)

The diagram is not missing a ${\displaystyle \beta }$. I believe the original proof was using ${\displaystyle f(\alpha )}$ where ${\displaystyle \alpha }$ is the angle opposite the shortest side. PAR (talk) 04:40, 18 November 2011 (UTC)
I agree that the diagram is not missing a ${\displaystyle \beta }$. It is not required in the equation because f(α, β) =f (α, f(pi/2-α)= f'(α), where f' is a different function from f, so it is the equation needs to be modified. The proof given here actually is something of a mess. It does not say or show why the angles have the relationship given in the second and third parts of the diagram. This should be explicitly described. It also does not convincingly provide a reason that one should assume the first equation either. The problem as set up has three lengths and three angles. Elementary trigonometry would give relationships among these six quantities, allowing some of them to be eliminated, but these relationships are no more complicated than the pythagorean theorem itself. Assuming the equation as written is tantamount to assuming the pythagorean theorem, actually. A more natural choice, given the methods of dimensional analysis would be to assume area =a^m b^n f(α), or even a^m b^n c^o, where m,n and o are unknown and prove it from there. I think this example is not a good one to illustrate the methods of dimensional analysis and contributes little to the article. There are already enough examples. The whole paragraph on the pythagorean theorem should be eliminated.DonSiano (talk) 14:55, 18 November 2011 (UTC)
The equations in the quoted source [2] do not have the function f as a function of a single argument, such as the f(alpha) now defined here. -- Saveur (talk) 06:39, 19 November 2011 (UTC)
I checked the source, and that's true, which makes it rely even less on dimensionless quantities. It's nice, but really not a good illustration of the technique of dimensional analysis. PAR (talk) 08:09, 19 November 2011 (UTC)
I think I agree that it's not a good illustration of dimensional analysis. The whole thing relies on the principle "area proportional to squared length", which is a geometrical principle. The proof probably belongs to a geometrical discussion or in a list of proofs of the theorem, with a more authoritative source anyway. -- Saveur (talk) 18:54, 19 November 2011 (UTC)
Given the above discussion I will remove the Pythagorean theorem example from this article. Acting in good faith, but if this is premature or objectionable then feel free to revert my edit. -- Saveur (talk) 18:01, 20 November 2011 (UTC)
Yes, go ahead and remove it. It is not a good example.DonSiano (talk) 20:55, 21 November 2011 (UTC)

## Sources

Comment for those who are working on this article: I just happened to notice that not a single assertion in the intro (the first two paragraphs) is sourced. -- Saveur (talk) 09:08, 20 November 2011 (UTC)

It's sometimes the case that the intro isn't sourced, and it appears as if this is proper in this case. --Izno (talk) 09:46, 20 November 2011 (UTC)
I can't find a source for "Dimensional analysis is routinely used to check the plausibility of derived equations and computations" either in the intro or the text. Maybe I should have made a comment about the article in general - although it has a lot of content that is well developed, I don't see many sources in sections such as Mechanics, Other fields of physics and chemistry, Commensurability, Polynomials and transcendental functions, Incorporating units. -- Saveur (talk) 17:55, 20 November 2011 (UTC)
Indeed, a different and more important issue. --Izno (talk) 20:28, 20 November 2011 (UTC)

## New sections on dimensional equivalence

I intend to add sections for dimensional equivalences between quantities, in SI and natural units (for c = ħ = 1), for the common quantities throughout physics. Just thought I’d mention this upfront.-- 21:32, 20 January 2012 (UTC)

It would necessarily need to account for that most of the natural units are unrationalised, and for being so, the dimensional analysis would be incomplete, since by itself, it can not be used for converting units. You need something like an extra dimension to handle rationalisation. In any case, for the most part, you could just set L, M, T and Q to algebraic values, in the manner of eg T=L, (c = 1 does this). I never got around to setting h or e to 1. The electrodynamic dimensions (which set, eg c = ε = 1), reduces stuff down to a practical number of spaces (15 or so), where one has, eg T = L , Q = AL², M = A²L³. h' is A²L4, is clearly e², since e is in AL². Wendy.krieger (talk) 10:58, 22 January 2012 (UTC)
I'll come back to this later - right now i'm caught up in other QM articles.-- 11:38, 22 January 2012 (UTC)
I should have been clearer: the third unit is a fixed energy unit (say the electron-volt). That forms a complete system for quantities using dimensions of mass, length and time becuase they can be written in terms of speed, action and energy:
${\displaystyle M=E/c^{2},\quad L=\hbar c/E,\quad T=\hbar /E\,\!}$
where
mass = energy/unit speed2, length = action.velocity/energy, time = action/energy,
but speed and action are de-dimensionalized into dimensionless variables if c = ħ = 1, so the remaining dimension is energy. This is all that needs to be said.-- 00:07, 23 January 2012 (UTC)

## Transcendental functions

The JCE article "Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions" (doi:10.1021/ed1000476) claims that part of this article is wrong. I therefore removed the section for the time being. --Taweetham (talk) 04:27, 11 March 2012 (UTC)

Sounds like the right thing to do. I would be dubious of the entry in the first place. I can't get a free copy of the article, but this reference to it is free: [[3]]
I do not know how the argument to the sine function can ever have dimension (with units attached), but I can conceive of the log of a dimensionful quantity. Quoting the article:
The value of a dimensional physical quantity Z is written as the product of a unit [Z] within the dimension and a dimensionless numerical factor, n.
${\displaystyle Z=n\times [Z]=n[Z]\ }$
So the log of that is:
${\displaystyle \log(Z)=\log(n[Z])=\log(n)+\log([Z])\ }$
So we know what ${\displaystyle \log(n)\ }$ is, but the ${\displaystyle \log([Z])\ }$ term is a sorta turd. Now, just like if there is a ratio of like-dimensioned quantities, this turd will get sorta canceled by another turd. Consider this
${\displaystyle 1\ {\mbox{ft}}=0.3048\ {\mbox{m}}\ }$ is identical to ${\displaystyle 1={\frac {0.3048\ {\mbox{m}}}{1\ {\mbox{ft}}}}.\ }$
That means
${\displaystyle \log(1)=0=\log \left({\frac {0.3048\ {\mbox{m}}}{1\ {\mbox{ft}}}}\right)=\log(0.3048\ {\mbox{m}})-\log(1\ {\mbox{ft}})\ }$
and
${\displaystyle 0=-1.18809945516965+\log({\mbox{m}})-\log({\mbox{ft}})\ }$
${\displaystyle \log({\mbox{m}})=\log({\mbox{ft}})+1.18809945516965.\ }$
So an equation with these turds left all by themselves is nonsensical. But if it's a sum of the logs of like-dimensioned units and with opposite signs, you can make sense of it.
Nonetheless, there is only one good lesson in this: Make sure that the argument to any transcendental is dimensionless. If it is not, you're either doing something wrong or, like with the log function, you could do something to make that argument dimensionless. 71.169.191.83 (talk) 18:38, 11 March 2012 (UTC)
I would change that could to should. Regarding the argument of ${\displaystyle \sin(x)=\sum _{n=0}^{\infty }(-1)^{n}x^{2n+1}/n!}$, this will make sense if odd powers of x have the same dimension or, in other words, even powers are dimensionless. Then ${\displaystyle \cos(x)=\sum _{n=0}^{\infty }(-1)^{n}x^{2n}/n!}$ makes sense too. In Siano's development, radians are considered to be a unit, and radians squared are dimensionless. This allows one to reject an expression of the form ${\displaystyle \sin(x)+\cos(x)}$ as dimensionally inconsistent. Expressions like ${\displaystyle \cos(x+\pi /2)=-\sin(x)}$ appear inconsistent, but this is a special case of ${\displaystyle \cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)}$ which is dimensionally consistent. In other words, properly stated, ${\displaystyle \cos(x+\pi /2)=-\sin(x)/\mathrm {rad} }$ PAR (talk) 20:37, 11 March 2012 (UTC)
Can't vouch for Siano, but I've been taught from square 1 that radians are fundamentally dimensionless. It's distance (arc length) divided by distance (radius). Or, if you like how hyperbolic trig functions are related to circular trig, it's θ/2 as the area of the sector swept out in a unit circle. But I really see neither rad nor rad2 as having dimension. Just like the percent mark % is really just a specific factor of 1/100, the degree symbol ° for angles really is just a factor of π/180.
I guess I don't agree with Siano. There are different notions of units, some are dimensionful (m is length, kg is mass, s is time, C is electric charge) and some are not, (percent, degrees, dB). In fact, I think except for a quantitative difference in scale, dB is to Np as ° is to rad. The former (in each pair) is for human convenience and the latter is a pure number and mathematically natural. And all are dimensionless. And the symbols Np and rad have scaling factor of 1, so are essentially superfluous. I view the symbol dB as equivalent to the dimensionless factor of log(10)/20 . 71.169.184.131 (talk) 04:21, 12 March 2012 (UTC)
I look at whether it works, not whether I was taught it. If you have 5 variables and 3 units, mass, length, time for example, then you can describe the system in terms of 5-3=2 dimensionless parameters. If you notice that some of the lengths in your 5 parameters are orthogonal, x-length and y-length for example, then you have 4 units and one dimensionless parameter - you have cut the complexity of the problem in half and doubled your insight. I've done this a number of times - rather than go through a detailed analysis, do dimensional analysis to get an idea of whats going on, and many times, I've cut the problem by half or third using Siano's method. And it works - a detailed analysis bears out the results every time. The idea of radians being dimensioned follows from this x-length, y-length, etc. idea. I've used that before too, and it works. Siano's method is not flaky, it works. If you don't reject it simply because it wasn't what you were taught, but rather investigate it, I'd be interested in a case where it fails. And that's what dimensional analysis is all about - the more knowledge you bring to a problem, the more insight you get. And if you can come up with new units in a proper way, you are bringing more knowledge to the problem, and the result is - less dimensionless parameters needed to describe your system.
dB and Np are dimensionless numbers, because they are logs of ratios of like quantities, no way out of that. But radians and degrees need not be, although they are related by a scale factor. PAR (talk) 06:17, 12 March 2012 (UTC)

I see nothing wrong with the passage that was commented out (except perhaps that it was a bit long-winded). Could someone state explicitly why, wherefore, or inasmuch as which it is incorrect?  --Lambiam 14:12, 2 June 2012 (UTC)

## Commensurability

"Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque and energy share the dimension ML2/T2, they are fundamentally different physical quantities."

Is there a way to reconcile this fact with commensurability? Can the consistency of dimensional analysis be maintained by say changing the dimension of torque (to perhaps include an orientation, or a direction) or something else, or is this a fundamental problem somehow? Robleroble (talk) 22:31, 13 March 2013 (UTC)

Yes, you can do that by assigning a dimension to angles. Torque times a rotation angle is energy, but because angles are conventionally dimensionless, a torque has the same dimension as energy. But this same problem will then crop up elsewhere, so it is still a fundamental problem. The issue is that physics is fundamentally dimensionless, the units we use are ultimately arbitrary human constructs.
The reason why dimensional analysis works is because it is a scaling argument in disguise. Given a number of physical quantities, you can always make them dimensionless by multiplying them with combinations of c, hbar and G, then you can apply any mathematical function to those variables, and then you can give that any dimensions you like by multiplying this by an appropriate combination of c, G and hbar. So, a priori, any physical quantity Y could be related to some given physical quantitities x1, x2, x3, ... via any given function, if you are allowed to use the constants c, G and hbar. So, dimensional analysis only yields non-trivial results if you make the additional demand that you are not allowed to use one or more of the constants hbar, c, and G. But this means that you are looking at e.g. the classical limit hbar to zero, or the nonrelativistic limit c to infinity, which can be interpreted as looking at the physics at certain scaling limits.
It is then only in these scaling limits that certain variables are incompatible. E.g. you can compare lenghts to time intervals, the conversion factor is c. But you lose this conversion factor precisely in the scaling limit that corresponds to the classical limit. Count Iblis (talk) 23:55, 13 March 2013 (UTC)
Check out the "extensions" section. In Siano's extension, the energy is without direction (${\displaystyle \mathbf {1} _{0}}$) while, for example, the x-component of torque is directed, having dimension proportional to ${\displaystyle \mathbf {1} _{x}}$. This is essentially what Count Iblis is saying. In Siano's extension, angles have a direction. For example, the direction of an angle formed by 2 lines in the xy plane has a dimension proportional to ${\displaystyle \mathbf {1} _{z}}$ PAR (talk) 00:05, 14 March 2013 (UTC)
I've often wondered why we don't simply treat angles dimensionally. I've even come across a couple peer reviewed academic papers that are in favor of this [1] [2]. The most obnoxious part about treating angles dimensionally seems to be that we would have to divide by 1 radian constantly. For example, sin(θ) becomes sin(θ/rad) and exp(iθ) becomes exp(iθ/rad). I suppose we could just define a physical constant c=1 rad=180/π deg=1/(2π) turn, but then this would pop up everywhere. Are there some other unintended effects that would prevent me from simply treating angles like a dimension?
[1] K. R. Brownstein. Angles---Let's treat them squarely. American Journal of Physics. July 1997. Volume 65, Issue 7, pp. 605. [DOI].
[2] Jean-Marc Lévy-Leblond. Dimensional angles and universal constants. American Journal of Physics. September 1998. Volume 66, Issue 9, pp. 814. [DOI].
--Mk29 (talk) 14:20, 16 August 2013 (UTC)

## Compound units / ampere

"For example, an ampere is a measure of electrical current, which is fundamentally electrical charge per unit time and is measured in coulombs (a unit of electrical charge) per second, so 1A = 1C/s."

Could this be expressed in another way? 1A is the defining quantity in the SI system, so by definition 1C = 1As. This gives a circular argument: 1A = 1C/s = 1As/s = 1A. I agree that the physics are correctly described, but it's not how the SI system works. Jens 130.243.105.40 (talk) 09:44, 8 August 2014 (UTC)

## Did Newton refer to dimensional analysis as the "great principle of similitude"?

The article asserts that Newton (in 1686) would have called the method of dimensional analysis "the great principle of similitude". Reference is given as "Walter R. Stahl, Dimensional analysis in mathematical biology, Bulletin of mathematical biophysics, Vol. 23, 1961, p. 355". - The reference to Stahl is correct, the reference to Newton is not. Nowhere did Newton ever say anything like that, or similar to that assertion. The assertion is simply false. Never cite secondary sources! 91.37.165.191 (talk) 19:51, 8 September 2014 (UTC)

As a wikipedia contributor, I'm probably not qualified (and haven't time) to determine from primary sources (couched in 17th century conceptual-framework and terminology) whether your assertion is true (regarding everything Newton ever wrote). I also can't verify your qualifications (and compare them to mine or Stahl's). What I can do is verify secondary sources. So now we do know for a fact that a scientific journal (math. biophys.) published an expert's assertion that Newton (in Principia Mathematica, 2 §7 (1686)) indeed referred to dimensional analysis as the Great Principle of Similitude. Other sources (including [4] and [5]) give further discussion of Newton's and other's historical contributions. Cesiumfrog (talk) 09:09, 9 September 2014 (UTC)

Thank you. In this case, instead of attributing Stahl's assertion simply to Newton as a fact, wikipedia should perhaps tell the reader the truth that "Stahl asserted that Newton would have called ..." etc. etc. 91.37.155.29 (talk) 12:28, 9 September 2014 (UTC).

I add that I have checked the "sources" you have given me, to no effect. To say it once again: The assertion that Newton in Principia, Book II, Prop. 32 (probably the "§ 7" you refer to, but Newton has no "paragraphs" but "sections") would deal with "dimensional analysis" explicitly or implicitly, speaking of a "great principle of similitude", is simply not true. It is also not true what your source "Brennan" asserts on p. 44 of his book, when he refers to Newton's Principia, Book II, prop.32. No trace of "distinct entities as length, inertia, and mass" to be found there. Nowhere speaks Newton of "concepts" as Brennan asserts. I propose to correct the page by cancelling the mistaken reference to Newton.91.37.155.29 (talk) 14:52, 9 September 2014 (UTC)

Since your interpretation of principia appears to disagree with those of all published experts, why not first submit your argument and explanation to a peer-reviewed history-of-science journal? Or in cases such as this (whenever one wikipedian disputes some mainstream consensus) do you think it would be a better practice if we immediately rewrote the article to say "Stahl, Brennan, West, and others, are all wrong according to an anonymous editor and the concepts underlying dimensional analysis were completely unimagined and unprecedented before the 19th century"? Cesiumfrog (talk) 12:11, 10 September 2014 (UTC)

Please will you note that I'm not "interpreting the principia". Also, I do not assert what you impute to me, that "the concepts underlying dimensional analysis were completely unimagined and unprecedented before the 19th century". Rather, I'm telling you that something which you quote (!) from Principia, Book II, Sect. 7, cannot be found there, no matter how many "published experts" assert the same nonsense. You can most easily verify that I'm right. So I'm pointing to an evident error in the Wikipedia which should be corrected immediately, and you should perhaps be grateful to me for wasting my time in correcting your mistake. Or am I to learn that Wikipedia prefers the false opinion of "experts" over the simple truth? 91.37.163.73 (talk) 05:26, 11 September 2014 (UTC)

91, I checked this out at this reference. I'm pretty sure you're correct. And I removed the sentence. 71.169.182.51 (talk) 12:55, 11 September 2014 (UTC)
I've also checked that link, and although that translation from Latin does not use our modern terminology, it is broadly consistent with the expert summaries (for example, a principle of similarity from geometry is applied to a problem in mechanics and used to conclude on the proportionality of particular physical measurements). More discussion can be found in chapter 4 of J.C. Gibbings' Dimensional Analysis book, with further detail in the references listed at the end of that chapter (some of which include quotes from Principia). Another quote from Newton appears in the history chapter of Dimensional Analysis by Jonathan Worstell. Shouldn't our history section (like that of all these sources) begin prior to Maxwell? Cesiumfrog (talk) 08:09, 18 September 2014 (UTC)
The let's get a reference that is verifiable, unambiguous, in English, and with widely-accepted repute. I don't really have a dog in this fight, but it's just that when I checked this out from what i could find on the internet, 91's case seemed to be supported. There is no evidence, so far, that Newton said or wrote such a thing. It would be fine by me to credit Newton with the basic concept of dimensional analysis, but let's get a solid reference that no one can argue with. 64.17.96.139 (talk) 21:36, 18 September 2014 (UTC)
It's a little bit tendentious for you to ask for 'a reference' after a dozen sources (including multiple textbooks, as well as scholarly articles devoted to the history of dimensional analysis) have already been listed above. If this doesn't count as evidence for you, what does? An anonymous forum post? Cesiumfrog (talk) 04:05, 20 September 2014 (UTC)

## Can we have a consistent format in the article for depicting dimensions?

If this is going to be the "truthiness":

The dimension of a physical quantity can be expressed as a product of the basic physical dimensions mass, length, time, electric charge, and absolute temperature, represented by sans-serif symbols M, L, T, Q, and Θ ...

Then let's stick to that convention throughout the entire article. I see the same dimensions expressed in italics: M, L, T, Q... and I see something that looks like dimensions expressed as: [M], [L], [T], ...

All in the same article. Can we make this article self-consistent? And then, what convention shall we use? I don't care what convention, but it should be consistent from the beginning of the article to the end. 65.183.156.110 (talk) 21:38, 10 April 2015 (UTC)

Done – I have cleaned this up, using roman sans-serif symbols throughout, without brackets. This is the convention adopted by the BIPM (see {{dimanalysis}} and its reference). —Quondum 14:03, 31 May 2015 (UTC)

## Old talk 2001

In the most primitive form, dimensional analysis is used to check the correctness of algebraic derivations: in every physically meaningful equation, the dimensions of the two sides must be identical. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers, which is typically achieved by multiplying a certain physical quantity by a suitable constant of the inverse dimension.
I do not agree. Dimensional analysis is used to solve PDEs. The statement just describes e.g., stochiometry.

I admit that I don't know how to use dimensional analysis to solve PDE's (do you have any references?), but this paragraph was really just the beginning, showing the most primitive "dimensional analysis" as taught in college chemistry classes: make sure that the dimensions are right. I agree there's much more to Dimensional Analysis than that, and the rest of the article shows it, so I think the criticism is not justified. --AxelBoldt

What I was trying to say is the "monorail" algorithm for using units to solve stochiometry problems is not really dimensional analysis, but to be fair I will start cracking some books on this.

The above mentioned reduction of variables uses the Buckingham Pi theorem as its central tool. This theorem describes how an equation involving several variables can be equivalently rewritten as an equation of fewer dimensionless parameters, and it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown. Two systems for which these parameters coincide are then equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one.
This is not quite correct either. The resulting dimensionless parameters generally need to be determined experimentally, or there must be some sort of experimentally verified constitutive relationship. No one as yet can predict a Froude or Mach number, we can only measure them.

That's what I was trying to say: the Pi theorem tells you how to turn the measured variables into dimensionless parameters, and then you have to empirically find the relationship between those dimensionless parameters. No one can predict a Mach number, but people can predict the proper formula for Mach numbers. How can we clarify the above paragraph? --AxelBoldt

I think you mean "proper units" for Mach numbers...
That's the 64 dollar question. People that know how to do this (e.g., Barenblatt) just smile enigmatically when asked "how you do dat?" The best that I have been able to determine is that the process is like that cartoon of the physicist at the blackboard, where in a long chain of formulas, the one in middle is labeled "magic here".

I removed the "typed family of fields" comment, since there is no such thing in mathematics.

Such a thing can be well-defined. See G. W. Hart "Multidimensional Analysis".

Also,

Note also that the dimensionless numbers are not really dimensionless. The actual

structure of a dimensionless number is unity in the type. For example, consider the so-called dimensionless unit of strain: L/L. The L/L units are usually dropped, either implicitly or explicitly, but it is a mistake to regard strain as a physically meaningful quantity without some notion of the L in the denominator, which acts as a gauge length. For another example, consider the physical meaning (none)of adding strain (dimensionless) to Mach (dimensionless).

I don't understand this. Are you arguing that even dimensionless numbers should keep their dimensions? I can't make mathematical sense of that. Is L/L a different unit in your system than M/M? --AxelBoldt

It's not "my system", it's physics. L/L is a different unit than M/M. Yes, I am saying that dimensionless numbers should keep their dimensions. Think about it carefully. The real numbers used for computing physical quantities are meaningless without units. Velocities must be expressed in terms of L/T, whether it be meters/sec or furlongs/fortnight. The problem is that while the real numbers obey the axioms for a field, units obey group axioms. We can do math on the reals alone (analysis), the units alone (group theory) or real numbers with units attached to each quantity (typed family of fields). In the scalar world all of this is pedantic frippery. However, linear systems constructed to solved differential equations describing matter will have units attached. As it turns out, with care, units may be mixed within the system, and a solution determined using LU decomposition (say) will remain dimensionally correct. You can integrate this stuff too... (heh heh) If strain didn't keep its L/L dimensions, then strain energy per unit volume (FL/L^3) would just be F/L^2 which is units of stress.

No further comments or analysis tolerated on dimensional analysis

Can't you see how many people wrote about it?

• Barenblatt, G. I., "Scaling, Self-Similarity, and Intermediate Asymptotics", Cambridge University Press, 1996
• Bridgman, P. W., "Dimensional Analysis", Yale University Press, 1937
• Langhaar, H. L., "Dimensional Analysis and Theory of Models", Wiley, 1951
• Murphy, N. F., Dimensional Analysis, Bull. V.P.I., 1949, 42(6)
• Porter, "The Method of Dimensions", Methuen, 1933
• Boucher and Alves, Dimensionless Numbers, Chem. Eng. Progress, 1960, 55, pp.55-64
• Buckingham, E., On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis, Phys. Rev, 1914, 4, p.345
• Klinkenberg A. Chem. Eng. Science, 1955, 4, pp. 130-140, 167-177
• Rayleigh, Lord, The Principle of Similitude, Nature 1915, 95, pp. 66-68
• Silberberg, I. H. and McKetta J. J., Jr., Learning How to Use Dimensional Analysis, Petrol. Refiner, 1953, 32(4), p179; (5), p.147; (6), p.101; (7), p. 129
• Van Driest, E. R., On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems, J. App. Mech, 1946, 68, A-34, March
• Perry, J. H. et al., "Standard System of Nomenclature for Chemical Engineering Unit Operations", Trans. Am. Inst. Chem. Engrs., 1944, 40, 251
• Moody, L. F., "Friction Factors for Pipe Flow", Trans. Am. Soc. Mech. Engrs., 1944, 66, 671

Who the heck do you think you are?

little guru

## Restriction on functions 2006

This whole section (except the matrix part) is treated in two other places in the article, and is a bit of overkill. The bit about certain matrices, while true, is very much of a sideshow curiosity, and is not usually used in dimensional analysis. The section also destroys the flow of the article, and seems to be at the very least, out of place. I am deleting it for now, but would be willing to reconsider if there is a strong objection.

## No merge 2013

"Dimensional analysis" is a discipline, "units of measurement" is a standard. NO MERGE VOTE.

The old alert was correct, a "need to sync": Wikipedia:Summary style

{{sync|Units_of_measurement#Calculations_with_units_of_measurements}}

## L/T-only dimensions

The additions where Maxwell's attempt at natural units is used to analyse several quantities does not belong here; it merits only a mention in the History section. It is contrary to the spirit of the article (dimensional analysis gives useful information, and this simply discards most of it), it is misguided (this approach leads naturally to all quantities including L and T being regarded as dimensionless since there are enough physical constants to normalize to do this, as in Planck units), if only some dimensions are going to be made equivalent, the very first such equivalence would be L = T, and finally, it makes no point whatsoever. I'm sorry for the hard work (which is also original research), but I feel that it must be removed. —Quondum 14:15, 31 May 2015 (UTC)

My apologies for not responding sooner to your comments, but I've only just been made aware of the above. I hadn't realised that one needed to change notification preferences, such that article talk-page comments were conveyed via email, like user talk-page entries are. Sorry about that, Chief. I'm something of a "newbie" at writing material for the Wiki, but trust that the community of Wikipedians will let me know (and hopefully forgive) the occasional lapse of correct form. I can assure you I wouldn't waste a good deal of the precious little time I have left, let alone Wiki reader's time, with any "misguided" or irrelevant information. Nothing useful is being discarded; I myself have utilised "classical" dimensional analysis for many years, and have a deep appreciation of its value. However, it's of limited efficacy with regards to the cutting edge of Complex Dimensional Analysis, which toolkit is proving extremely useful to theoretical/particle physicists seeking to resolve the profound impasses of complexification and the unification of GR with QM. As for my material being "original research", you'll note that every statement of bare fact has been carefully referenced to published and/or peer-reviewed work, i.e. "reliable, published sources", as required by the Wikipedia policy. Kindly let me know if anything I've written needs further citations — I'll dig them out forthwith. Wherever I have re-packaged or re-interpreted established physics, e.g. the Planck Units, I have done so (I hope) in a completely transparent manner, such that a 9th-grader could follow the mathematical logic. In fact, such students are my best proof-readers and style critics; adults tend not to tell me when they don't have a clue what I'm gabbing on about. The gifted young people whom I tutor are the next generation of scientists, and regularly use Wikipedia for clarification of classwork, lab experiments and lectures. They appreciate that I am going to the bother of writing out my somewhat "unconventional" understanding of fundamental physics; in fact, it was they who encouraged me in this direction. I'll try cutting to the chase (complex DA → 6D special relativity) a bit faster, so you'll see where all this deep dimensional reduction is heading. Thanks for the feedback. Rjowsey (talk) 05:05, 1 June 2015 (UTC)
I have just scanned through the material on the L–T arrangement of quantities again, and I'm still left with the same impression: of the arbitrariness of it. You have not responded directly to my observations.
My intention was to give warning of my intentions to strip the material from the article. There are other potential possibilities. If we get stuck, I'll invite comment from others. You might also want to review WP:SYNTH. —Quondum 06:37, 1 June 2015 (UTC)
What, specifically, is "arbitrary"? Which of your observations have I not responded to directly, other than the reductio ad absurdum of L = T? My apologies (perhaps I'm suffering from premature senility), but I utterly fail to understand why you would intend "to strip the material from the article". Sorry, I have absolutely no idea what you mean by "there are other potential possibilities". Yes, please, do invite comment from others (definitely prior to exercising an arbitrary editorial authority). The WP:SYNTH policy, viz. "Do not combine material from multiple sources to reach or imply a conclusion not explicitly stated by any of the sources" is noted: which particular "conclusion" not explicitly stated by my referenced sources requires further clarification or elucidation? I'm perfectly happy to discuss any such issues, in a collegial manner. Rjowsey (talk) 09:12, 1 June 2015 (UTC)
I've given a bit more thought to your problem(s) with the L/T material, and I think (hope) I've understood the issue(s) you might be having with the framework. I've been using this 2-dimensional space/time matrix (or fabric) for a while now, so its usefulness is profoundly obvious to me, and to the youngsters I'm tutoring. Perhaps if I explain how I'm using this math framework, its value might be more obvious to you. When teaching the fundamentals of physics, I introduce kids to the Planck units by having them substitute Maxwell's L3/T2 into the SI dimensions of all the Planck quantities. They can knock that off in about 15 minutes. Then I suggest they try arranging their "2D" units into a (log-log) space/time matrix, according to indices, which takes them about 10 minutes, and they're quite delighted at how simple it all is. Then I show them how to multiply and divide, demonstrating with a couple basic equations, e.g. F = ma, and E = mc2. Then I let 'em rip, discovering more patterns and the relationships between quantities for themselves, e.g. find all the ways quantities can multiply or divide to get units of force, energy, or power. They love it! And they learn the Planck units in about 1/10th the time compared to the "textbook" approach. Then we do the same process with the principal electromagnetic quantities. They're quickly able to see the relationship of potentials to field strength, etc. They can intuitively grasp the concept of rates of flow, flux, current, etc, and relate shifting across cells, or downwards, with differentiation, gradients and charge density. They enjoy discovering the fundamental expressions relating the quantities, e.g. V x I = P, even a few of Maxwell's equations, by themselves. Because they're curious, and interested, and they're learning by finding patterns, using simple mathematical tools they built for themselves! By generalising into a 4D volume (tessaract) based on quaternions, i.e. 1 dimension of real space, 2 dimensions of imaginary space, and 1 dimension of imaginary time (ct), they learn how to do vector projections from flat Minkowski space onto 3 imaginary planes. From there we can proceed easily into a discussion about special relativity, time dilation and Lorentz contraction, with the aid of a purpose-built simulator (animated graphics software). Then we delve into General Relativity, using the same animated 4D computer graphics, so absent any mind-numbing tensors, metrics, Kronecker deltas, Christoffel gammas and partial differential equations. From there we can introduce the Dirac equation (using a bispinor wavefunction simulator), thence to phase and quantum spin. KISS! It's a very natural, effortless progression, and they end up with a really solid grasp of foundational principles. These kids use Wikipedia constantly, so they badgered me to write something about the math framework they'd learned, since they found it so much more straight-forward than the standard textbook approach. I hope that helps allay your concerns about the relevance of this L/T spacetime fabric to the DA article. I'm definitely not pushing any kind of wingnut "theory" — this is just a simplified math framework which has proved really useful. If you feel that complex DA doesn't fit in here, or is irrelevant to DA, perhaps it would be better to migrate it into a new article. They say you learn best when trying to teach something; it's even more true when one is trying to write about it! BTW, thanks for the additions and corrections here and there, much appreciated. :D Rjowsey (talk) 02:29, 2 June 2015 (UTC)
Maybe you're right. I've been working with deep dimensional reduction and complex DA for many years, so it's very familiar territory, and has come to feel like an entirely natural evolution of traditional dimensional analysis. However, I concede that this is "new math" for the majority of readers, and may serve to distract and confuse people who're trying to understand how DA is used, in the everyday sense. Since my approach may be a bit too "bleeding edge" for some, I've copied my contributions into a new page, which I'll tidy up and flesh out, anon. As always, any help is greatly appreciated! Please feel free to remove/revert my edits to this page (or should I do that?). Rjowsey (talk) 22:17, 2 June 2015 (UTC)
One final thought, then I'll disappear. When you say "it must be removed" to a newb contributor, that sounds very much like "this stuff isn't suitable material for the Wiki, and should be tossed away". Had you said instead, something like "it's muddying the waters, it doesn't quite fit in here, perhaps it ought to be a new topic", that wouldn't have freaked me out at all. And I probably would've agreed, very quickly, that that would make much more sense. Just sayin'... Rjowsey (talk) 00:04, 3 June 2015 (UTC)
Done – Removed all L/T content on the Dimensional Analysis topic. Rjowsey (talk)
I apologize, I should have been more careful of my wording. Indeed, I meant that it did not fit as part of the topic that this article covers, though I have to confess that as it stood, my prejudices allowed me to be a bit brusque in the expression of my opinion: without enough background, it looked like merely an extension of Maxwell's presentation. Given more background and context to motivate and give insight into that direction of study, as well as some of the specific insights derived from this approach, the topic may be interesting in its own right. —Quondum 02:42, 3 June 2015 (UTC)
No need to apologize! You've actually done me a big favor, by forcing me to think through exactly what I was attempting to communicate, i.e. the bigger picture. So, thanks for that. It takes time to learn how to write "encyclopaedically", but I'm gradually getting there. Rjowsey (talk) 03:47, 3 June 2015 (UTC)

## Citation not required

Most[citation needed] physicists do not recognize temperature, Θ, as a fundamental dimension of physical quantity since it essentially expresses the energy per degree of freedom, which can be expressed in terms of energy (or mass, length, and time). I think I believe this, but suspect that there are no references. It would be a lot of work to survey enough physicists to find out, for no real gain. Seems to me that many things have no citation because they aren't worth writing a citable article about. Gah4 (talk) 07:20, 24 October 2015 (UTC)

## Plural or singular: dimensions or dimension?

It is confusing that this wikipedia article mixes the plural and singular forms of dimension, apparently at random. In physics, the plural form is often used. For example, the dimensions of area are L2, and the dimensions of speed are L/T, according to Maxwell's Treatise on Electricity and Magnetism, Giancoli's Physics for scientists and Engineers, and Haliday&Resnick's Fundamentals of Physics. This has to do with Maxwell's definition of the dimensions of a unit: "When a given unit varies as the nth power of one of the three fundamental units, it is said to be of n dimensions as regards that unit. For instance, the unit of volume is always the cube whose side is the unit of length. If the unit of length varies, the unit of volume will vary as its third power, and the unit of volume is said to be of three dimensions with respect to the unit of length."[6] Is there a reason why this wikipedia article sometimes uses the singular form? Ceinturion (talk) 12:38, 28 February 2016 (UTC)

The SI unit of length is the meter. In SI, objects are measured in meters. In English, you use the singular form for one (or negative one?), plural for any other amount, including zero, and including unknown values. I can run down the street at a speed of more than one meter per second. What is the mass of one cubic meter of Oriental soup? Gah4 (talk) 22:24, 6 December 2016 (UTC)
Your answer does not contain the word dimension(s). Maxwell said the dimensions of speed are L/T whereas the wikipedia article currently says the dimension of speed is L/T. Which is better? Apparently the SI brochure [7] decided the singular is better: "In general the dimension of any quantity Q is written in the form of a dimensional product, dim Q = Lα Mβ Tγ Iδ Θε Nζ Jη, where the exponents α,β,γ,δ,ε,ζ and η, which are generally small integers which can be positive, negative or zero, are called the dimensional exponents." Ceinturion (talk) 16:27, 17 December 2016 (UTC)

## Is "physically meaningful" an empirical result?

The current article says: "Any physically meaningful equation (and likewise any inequality and inequation) will have the same dimensions on the left and right sides, a property known as "dimensional homogeneity".

The reason for this claim should be clarified. One can argue that it is an empirical fact that mathematical identities (such as F = MA) that describe general physical laws are observed to obey dimensional homogeneity. However, people often speak as if dimensionally inhomogeneous equations describe things that are somehow "impossible" or self-contradictory. If that type of abstract argument is valid, then dimensional homogeneity should be established in a non-empirical way from simpler assumptions.

For example, is it physically possible to have situation where ${\displaystyle x={\sqrt {t}}\sin(t)}$ where ${\displaystyle x}$ is in meters and ${\displaystyle t}$ is in seconds? If a result of some experiment is described by that equation, someone wishing to repeat the experiment using units of centimeters and minutes could figure how to rewrite the equation using those different units. So the equation is not an ambiguous description.

What we may be able to say about a dimensionally inhomogeneous equation is that it cannot be deduced from a set of dimensionally homogeneous equations. If that claim is provable then an equation like ${\displaystyle x={\sqrt {t}}\sin(t)}$ cannot describe a situation in a mathematical model that assumes _only_ a set of dimensionally homogeneous equations. For example, if we have a situation like a ball rolling along a track of given shape, the particular shape of the track can introduce a relation between the force on the ball and the distance along the track that is not a universal physical law about how distance is related to force. As another example, if we have car with a micro-controller in it that is programmed to enforce a certain relation between the displacement ${\displaystyle x}$ of the car in meters and the elapsed time ${\displaystyle t}$ in seconds, then we can't deduce ${\displaystyle x}$ as a function of ${\displaystyle t}$ using only the laws of Newtonian mechanics.

Tashiro~enwiki (talk) 04:27, 1 December 2016 (UTC)

OK, first I agree that it isn't true that "Any physically meaningful equation (and likewise any inequality and inequation) will have the same dimensions on the left and right sides, a property known as "dimensional homogeneity". But even so, I don't agree with your examples. For one example, special relativity mixes times and distances. If you insist that times and lengths don't mix, you will have a harder time. On the other hand, you really should use a properly dimensioned c when you do, or use c=1 units. But the ${\displaystyle x={\sqrt {t}}\sin(t)}$ example is not a good example. You really need a constant with 1/t units to multiply t, and a constant with length units on the outside. If those constants happen to have units of 1/s and m, respectively, then it will give the result you expect, but be dimensionally consistent. You will find equations like that in math books, but not in physics books. Gah4 (talk) 05:30, 1 December 2016 (UTC)
What is an example of a physically meaningful but dimensionally inconsistent equation? Special relativity does not mix times and space distances, there is always the speed of light involved to "homogenize" the equations. Demanding dimensional homogeneity is equivalent to the statement that the physics of a system does not depend on the unit of length or time or mass that you choose as standard. I cannot imagine a case where this would not be true. PAR (talk) 02:24, 3 December 2016 (UTC)
My favorite description of special relativity is the beginning of Spacetime Physics (http://www.macmillanlearning.com/catalog/newcatalog.aspx?isbn=0716723271). Yes you can put in the C, and make the dimensions consistent, or measure both times and distances in meters. But different inertial reference frames mix times and distances, or momentum and energy, or electric and magnetic fields, in different ways. See if you can find the book at a nearby library. Gah4 (talk) 06:42, 3 December 2016 (UTC)
Consider just simple geometry in 3-space. Suppose you have a point in 3-space and two people who are measuring things in centimeters using the usual (x,y) Cartesian coordinate system, but their coordinate systems are rotated with respect to each other. In other words, they "disagree" on what is "vertical" and what is "horizontal". Their descriptions of the position of that point will mix "vertical" and "horizontal" components in different ways. But the position of that point is "real", its actual position doesn't change depending on which coordinate system they choose, which directions they choose to be vertical and horizontal. The equations that describe the relations between the measurements of the observers do not "mix" vertical and horizontal components in any way relevant to the reality of the position of that point.
Special relativity is analogous - "space" and "time" are analogous to "vertical" and "horizontal". Physical processes occur in a space, irrespective of what two observers "choose" to call space and time. Two observers in two different inertial frames disagree on what is "space" and what is "time", but that distinction is not "real" any more that "vertical" and "horizontal" are real in 3D-geometry. Their differences arise because their coordinate systems are simply rotated with respect to each other in spacetime. An "event" is a point in spacetime, and different inertial frame observers will disagree on the time and position of that event. Their different "mixings" of space and time are irrelevant to the reality, the position of the event in spacetime.
In short, special relativity does not have any dimensionally inconsistent equations any more than 3-D geometry has dimensionally inconsistent equations, and for the same reasons. PAR (talk) 15:55, 3 December 2016 (UTC)
In the US, it is usual to measure vertical distances in feet, and horizontal distances in miles. That complicates the coordinate rotation, but you still know how to do it. In non-relativistic problems, you can keep distance and time nice and separate. In relativistic, and especially ultra-relativistic problems, it is harder to keep them separate. You might still manage, but it isn't so easy. Maybe think in a different direction: what is the difference between a gas and a liquid. You think you would always know the difference, having seen both for many years, but it turns out that, unless you have both phases together, you can't say. You can start with a gas, increase the pressure (no phase transition), increase the temperature (no phase transition), decrease the pressure(no phase transition) and decrease the temperature (still no phase transition) and get from gas to liquid without ever going through a phase transition. That is, if you go to temperature and pressure that are higher than the critical point temperature and pressure. Using different units for vertical and horizontal distances seems fine on the earth's surface, until you try to do a coordinate rotation. If you do a lot of coordinate rotations, you will likely give up and start using the same units. Note that it is common in some physics problems to use eV, more usually GeV, as both mass and energy unit, and for that matter, also a momentum unit. That is, to use units with c=1. It is also usual in some quantum mechanics problems to use ħ=1 units. To measure energy in time units, or time in energy units. Not that it is impossible to not use such unit systems, but it is enough more work to do it, than the advantage of keeping them separate. With ħ=1, you can use GeV for a frequency (1/time) unit, too. And sometimes things that should be different, end up with the same units. Torque and energy, using mass, length, and time units, are dimensionally the same, yet you probably won't find people using Joules for torque. Some unit systems make some problems easier to do, and also help catch some errors. They make other problems, harder, so don't use them for those problems. Gah4 (talk) 16:59, 3 December 2016 (UTC)
You said "I agree that it isn't true that 'Any physically meaningful equation (and likewise any inequality and inequation) will have the same dimensions on the left and right sides'".
Can you give an example of such an equation, so I can understand what you are saying? PAR (talk) 01:42, 4 December 2016 (UTC)
It seems to me that it's not confusing to measure or express torque as "Joules per radian" and radians are dimensionless being the ratio of arc length to radial arm. Torque is Joules of energy times the meters of radial arm per meter of arc length. But hey, Joules of energy per meter of arc length is the force (in Nt) applied to the radial arm. So it's the same. I don't think it's confusing to express torque in terms of Joules. 96.237.136.210 (talk) 07:59, 6 December 2016 (UTC)
I agree, this is a conundrum. We know that energy and torque are two different things. Rather than say that dimensional analysis obscures their difference, I prefer to say that dimensional analysis, as you have presented it, is incomplete - it fails to draw the distinction.
In the sections under "Extensions", Huntley and Siano offer a more complete theory of dimensional analysis which maintains the distinction. Siano built upon the work of Huntley, so its more complete. His approach to the above conundrum is to maintain a dimensional distinction between lengths in different directions. Energy (e.g. Joules) is a scalar, torque is a vector and Siano says the units are different. Not in their absolute value (Joules or Newton-meters) but in their directionality.
Siano says that 1x, 1y, 1z and 1o are "orientational symbols" which must be part of any physical dimension and they have their own multiplication table. If F is a vector force, dr is an infinitesimal vector displacement, then the infinitesimal increase in energy is F.dr = (Fx 1x+Fy 1y).(dx 1x+dy 1y) = (Fx dx + Fy dy)1o, where Fx, Fy, dx, dy have scalar units of length. Clearly energy is a scalar as evidenced by the 1o symbol.
For torque, =F x dr = (Fx dy-Fy dx)1z showing that torque does not have the same dimensions as energy, it is a directed quantity as evidenced by the 1z symbol.
Additionally, angles are no longer scalars, but directed quantities, in a direction perpendicular to the plane of the two lines forming the angle. That means sin(θ) is a directed quantity (the Taylor expansion is the sum of odd powers of θ and odd powers of any orientation symbol equals that orientation symbol). For the same reason, the cosine of an angle is a scalar (even powers of 1z equals 1o). That means that any equation like A sin(θ)=cos(θ) cannot be physically meaningful, while the old theory could not say that.
The bottom line is that Siano's extension duplicates the old theory and more. I have never found a case where Siano's theory fails to draw a distinction, but even if it did, I would take the same attitude - the dimensional analysis theory is wrong or incomplete, we should not accept that it obscures the distinction. PAR (talk) 19:52, 6 December 2016 (UTC)

It is nice to see others joining the discussion. We use units as they are convenient for the problem at hand. Using dimensions of [mass], [length], and [time] is convenient for many mechanics problems, and as noted, we can use different units for quantities with those dimensions. But what I was saying above, is that sometimes we use the same units for quantities with different dimensionality. We can measure time in meters, or mass in GeV. In some problems, it is convenient to keep directionality in the dimensions, but not in others. Even though the MLT dimensionality is the same, we measure torque in N m, and not in J. In AC electronics, there is a unit of volt amps (VA), which is not watts. To get power in watts, you multiply by the power factor, which comes from the phase difference between the current and voltage. There is some discussion in Dimensional_analysis#Formalisms that helps explain this. SI adds a new base unit for electrostatic problems, the ampere second. (There is no current in electrostatics, so it doesn't make sense to have a unit for it.) Gaussian units defines the charge unit such that the constant is one in Coulomb's law, so no new base unit, and no added dimension. As above, both units and dimensionality are for the convenience of the user, and for the problem at hand. Gah4 (talk) 22:56, 6 December 2016 (UTC)

Even though the terminology is different, dimensionally power and volt-amp product are dimensionally the same, even if one is with respect to mean real power and the other is reactive "power". Torque and energy are dimensionally the same even if one is expressed in Nt-m and the other in Joules (if SI units). I think the article is correct to differentiate between the concepts of units and dimensionsion of "stuff" (or physical quantity). It seems pretty clear to me that, if you can accept temperature as nothing other than energy per particle (and per degree of freedom), then there are four different base dimensions of "stuff" (time, length, mass, charge) that are really different from each other and not describable solely from another (without a physical constant as a scale factor) and with that all other physical quantity can be described or defined. But people will argue about temperature or electric charge being unique forms of physical quantity. (I think the former not and the latter is.) But it's not about what I think. 96.237.136.210 (talk) 08:33, 7 December 2016 (UTC)
Yes, but the OP suggested that the idea that physically meaninful equations are dimensionally consistent (and those that are not are not) requires some justification. Some people take the point of view that its not always true, its like a guideline or something. Others, myself included, think that there is a very fundamental truth to it, and any example that falls short is simply pointing out a deficiency in our theory of dimensions, not that the whole concept is faulty. I agree with the OP that the latter point of view requires justification, and I'm not sure how to do it in a totally convincing manner. Dimensioned quantities are really ratios of what you measure to some standard measure. I think the fundamental point is that the laws of physics don't depend on your choice of standards. If you have an inconsistent equation that says length=mass, then changing your mass standard but not your length standard would predict different physics, but mass=mass would not. PAR (talk) 10:29, 7 December 2016 (UTC)
Someone wrote: Even though the terminology is different, dimensionally power and volt-amp product are dimensionally the same, even if one is with respect to mean real power and the other is reactive "power". While technically true, it isn't practically true. To me, dimensionality should relate to quantities that can be added. You shouldn't add real power and reactive power, and you shouldn't add torque and energy. Dimensionality is convenient for a large variety of problems, but as commonly used, not for all problems. It is commonly ignored in problems where it isn't so convenient. This is most obvious for time and distance, which in many cases should be considered as having the same dimension. Note that physicists often quote mass in GeV, instead of the more technically correct GeV/c2. Consider the idea of base units, which should describe quantities that are fundamentally different, and yet the definition of the meter is based on the definition of the second. The meter should be a derived unit, but for historical reasons, and convenience for most user, it isn't. Gah4 (talk) 15:49, 6 February 2017 (UTC)