# Talk:Dimensional analysis/Archive 1

 Archive 1 Archive 2 →

## 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
$\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.
$\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)

$\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

$R=C v_x^a g^b \theta^c$

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 $\hbar \$, $c \$, and $G \$ (along with $\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

$T = m v^2 \$

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

$T = m v \$

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 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:

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

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

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

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 $\sin(\theta)=\theta+\theta^3/3!+...$ is an oriented quantity, while $\cos(\theta)=1+\theta^2/2!+...$ is without orientation. If we take θ to be dimensionless, then the expression $q=\sin(\theta)+\cos(\theta)$ 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: $\sin(\theta+\pi/2)=\cos(\theta)$ appears inconsistent, but should be interpreted as an instance of $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ 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:

$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
$T = m_0 c^2 \left( \frac{1} {\sqrt{1-v^2/c^2}} - 1 \right)$  ?

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

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

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

$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 $E(0) = M c^2 \$ . So to a classical physicist a new quantity M seems to exist such that kinetic energy is $\frac{1}{2} M v^2$. 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, $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 $\alpha = \frac{e^2}{\hbar c}$ instead of $\alpha = \frac{e^2}{\hbar c 4 \pi \epsilon_0}$.
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,

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

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 $\frac{\mbox{V}}{\sqrt{\mbox{Hz}}} m^{1/2}$, 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 $\beta$. I believe the original proof was using $f(\alpha)$ where $\alpha$ is the angle opposite the shortest side. PAR (talk) 04:40, 18 November 2011 (UTC)
I agree that the diagram is not missing a $\beta$. 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)