Talk:Dimensional analysis

From Wikipedia, the free encyclopedia

Mathematics Portal

This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.

Mathematics rating:

B Class

Mid Priority

Field: Applied mathematics

Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments.
Please add to or update the comments to suggest improvements to the article.
Geometry guy 15:23, 10 June 2007 (UTC)

1 Old talk
2 Dimensionless logs?
3 An equation with torque on one side and energy on the other would be dimensionally correct, but cannot be physically correct!
4 Huntley's refinement
5 Restriction on functions
6 removal of dimensional analysis
7 Units vs. dimensions calculations
8 There are no conversion factors between dimensional symbols???
9 notation
10 edits of 07/28/06
11 edit 8/24/06
12 Lay readers and notation
13 Dimension of interest on money
14 Concern about "a more complex example" -- Non-expert comment

[edit] Old talk

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

[edit] 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)

[edit] 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)

[edit] 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)

[edit] Restriction on functions

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.

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] Lay readers and notation

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)

[edit] 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)

[edit] 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)