# What is the Meaning of Dimension

1. Jul 26, 2007

### JOPearcy

What is the meaning of dimension?

It is my understanding that dimension related to the measuring out of something. If there is some observable phenomena that we can measure by defining units of measure and counting the quantity of these units that there is an associated dimension which is not unit based but the units reside within or are composed of the dimension being measured.

Thus examples of dimensions include the SI base dimensions length, mass, time-duration, electric current, thermodynamic temperature, amount of substance and luminous intensity.

It is my understanding that while dimensions are typically referred to in the sense of 3 spatial dimensions and sometimes including the 4th dimension of time, that the number of dimensions are not limited to the dimensions of space and time but include all manner of observable phenomena which can be quantified and measured.

Am I correct in this or not?

2. Jul 26, 2007

### mgb_phys

The word is used for two slightly different things ( both come from the latin word for measured )

1, The base units of some quantity, eg. you would say energy has dimensions of force*distance.

2, A direction you can measure something in independant of other changes, eg the normal X,Y,Z dimensions plus time.

Last edited: Jul 27, 2007
3. Jul 26, 2007

### JOPearcy

Is there a good detailed definition of what dimension means in science available on the net, something that explains it in depth, like a proof?

When I work through my understanding of the definition of dimension, I see a single definition for dimension from which a common subset of dimensions are typically referred to when the term dimension is used.

So, for example, length, mass, time, electric current, thermodynamic temperature, quantity of substance, luminous intensity and numerous derived dimensions from the first seven are all dimensions. But frequently when the term dimension is used, it is in specific reference to the dimension length, space defined in 3 dimensions of length or space-time in 3 dimensions of length and 1 dimension of time.

However, these dimensions of length and time are defined as dimensions through the same process as the other dimensions have been defined and the dimensions of length and time are a subset of a greater body of know and defined dimensions.

The following are statements which I believe are correct and I use understand what dimensions means.

Quantity is a property of a phenomenon, body, or substance, to which a magnitude can be assigned.

Quantities of the same kind are quantities that can be placed in order of magnitude relative to one another.

Quantities of the same kind within a given system of quantities have the same dimension.

The subdivision of quantities into quantities of the same kind is to some extent arbitrary. For example, moment of force and energy are, by convention, not regarded as being of the same kind, although they have the same dimension, nor are heat capacity and entropy.

System of quantities means a set of quantities together with a set of non-contradictory equations relating those quantities.

Base quantity means a quantity, chosen by convention, used in a system of quantities to define other quantities.

Derived quantity means a quantity, in a system of quantities, defined as a function of base quantities.

The unit of a physical quantity and its dimension are related, but not precisely identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g. length may have units of meters, feet, inches, miles or micrometres; but any length always has a dimension of L, independent of what units are arbitrarily chosen to measure it. Two different units of the same physical quantity have conversion factors between them. For example: 1 in = 2.54 cm; then (2.54 cm/in) is called a conversion factor (between two representations expressed in different units of a common length quantity) and is itself dimensionless and equal to one. There are no conversion factors between dimensional symbols.

“Quantity dimension” also phrased “dimension of a quantity” or simply phrased “dimension” is the dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities.

Quantities having the same dimension are not necessarily quantities of the same kind.

In deriving the dimension of a quantity, no account is taken of any numerical factor, nor of its scalar, vector or tensor character.

The dimension of a base quantity is generally referred to as ‘base dimension’, and similarly for a ‘derived dimension’.

4. Jul 27, 2007

### Renge Ishyo

I prefer to attempt a more broad description.

1 Dimension - A straight line.

2 Dimensions - Take another line and arrange it across the first one at some angle (90 degrees in cartesian coordinates). The two together define a flat plane of some shape.

3 Dimensions - Take another line and place it across the previous two lines but not in the plane defined by the previous two. This will create depth.

The concept of the dimension can be explained without any physics in it. Once you define a certain unit as a base number for a measurement and decide that this unit of measurement will be plotted along one line and another measurement on another line, etc. then the units take on the form of a mathematical dimension when they are arranged in the way described above. But the units themselves are not necessary in order to explain why they are considered a dimension (in fact, numbers are not even necessary to describe the idea of a dimension if my description above can be trusted).

5. Jul 27, 2007

### cristo

Staff Emeritus
The definition of a dimension is exactly that-- a definition. You cannot prove a definition!

The mathematical definition of dimensions would be something like parameters used to describe position or other characteristics of an object within a space. The dimension of this space is the total number of parameters.

For example, in 3D space we need three parameters to define the position of an object in the space. Therefore, the space has three dimensions.

6. Jul 27, 2007

### rbj

7. Jul 27, 2007

### JOPearcy

I've read through all those definitions before. If you can locate International vocabulary of basic and general terms in metrology (VIM) you might find it interesting.

By a definition that explains dimension in depth, like a proof, I did not mean a proof of the definition of dimension. If you are posting on this forum I am hoping that it means you have had experience with such things as writing a proof. To write a proof you have to be very detailed and cover all your bases.

To give such a rigorous definition of dimension and its use in science would answer my question because it would go into enough depth to show whether I am right or wrong in my view of dimension.

It would give a thorough enough of a definition to handle how it is used in science and then gives examples for both common uses and some uncommon uses. This might in fact include several actual proofs.

One such proof might be why and how length is found to be a dimension using the definition of dimension.

Another such proof might be why luminous intensity is found to be a dimension using the definition of dimension.

Yet another such proof might be why amount of substance is found to be a dimension using the definition of dimension.

Another such proof I’d like to see is how an effectively dimensionless dimension can be defined, the dimension of 1 such that it is derived as a ratio of dimensions of the same type, such as in deriving angle. When I have found such reference to the dimension of 1 I have found it particularly mind twisting and I’d really like to see an in-depth explanation for this.

8. Jul 27, 2007

### rbj

at this point i just throw up my hands. i guess there there is a dimensionless dimension and such are just numbers.

9. Jul 27, 2007

### JOPearcy

RBJ,

The place where I saw such a reference did not explain itself very well. I'm not sure the reference was correct or if it was worded poorly and I misunderstood it.

It goes like this.

The “quantity of dimension one” also worded as “dimensionless quantity” is a quantity for which all the exponents of the factors corresponding to the base quantities in the representation of its dimension are zero.

The values of quantities of dimension one are simply numbers.

The term ‘dimensionless quantity’ is for historical reasons commonly used. It stems from the fact that all exponents are zero in the symbolic representation of the dimension for such quantities. However, the term ‘quantity of dimension one’ reflects the convention in which the symbolic representation of the dimension for such quantities is the symbol 1 (see ISO 31-0 :1992, subclause 2.2.6).

EXAMPLES
Plane angle, solid angle, linear strain, friction factor, refractive index, mass fraction, amount-of-substance fraction, Mach number, Reynolds number, degeneracy in quantum mechanics, number of turns in a coil, number of molecules.

I’d like a better explanation about the ‘quantity of dimension one’ and how it is determined to be a dimension from the definition of dimension. I think one of the implications is that like other dimensions, just because something has the same dimension does not always mean they can be added.

For example, two lengths may not simply be added if they are perpendicular to each other, even though the both have the dimension of length.

Plane angle, solid angle, linear strain, friction factor and refractive index all have the dimension 1. Can you simply add them together like 5 scalar values?

I don’t clearly understand the how, why and associated implications of the dimension of 1.

10. Jul 27, 2007

### matt grime

Dimension in physics refers to units. A dimensionless quantity is one with no units, such as the coefficient of restitution. This is nothing to do with mathematics.

Dimension in mathematics: if V is a vector space, then its dimension is the cardinality of a minimal spanning set or maximal linearly independent set of vectors. (What this is for infinite dimensional vector spaces depends on whether you want a Hamel basis, i.e. do you allow or disallow infinite direct sums).

In geometry, one can take the term dimension to mean this, roughly: if I have a geometric shape, with some associated quantity, and if I double all the lengths of the shape (scale by two), then if the associated quantity scales like 2^d, then d is the dimension. Example: in 2-d take an plane polygon. If you double its sidelengths you mulitply its area by 2^2, or for a polyhedron, doubling the sides gives a factor of 2^3 change in volume.

There are many other different definitions of dimension.

I hope that helps.

11. Jul 27, 2007

### matt grime

if you felt like it - they're just numbers. The resulting answer may be physically meaningless. Don't forget you can multiply any physical quantity by 1 (unit x)/(unit y) to convert the associated 'units' into anything you choose.

Remembering what the units of a measurement are is just a good way of making sure you're doing something physically sound. It has no real mathematical content, as far as I'm concerned. (Although dimensional analysis is useful in applied mathematics, apparently).

12. Jul 28, 2007

### JOPearcy

matt grime,

I agree that there are many definitions of dimensions, I believe there is one definition for dimension as it is used in science, math, physics, engineering and so on which is used for the process of defining any particular dimension or system of dimensions.

What does the dimension of length, mass, luminous intensity or an element of the coordinate space F^n?

In all cases you have some quantity you are measuring. In order to measure that quantity you need to define some base quantity you are measuring in relation to and thus you define a unit of the quantity you are measuring. The dimension of the quantity you are measuring is what the unit of quantity consists of or exists in.

So I can be measuring something in the units of inches, meters or yards. The units for making measurement are not the dimension but they lie in the dimension or consist of the dimension, the dimension in this instance being length.

The same holds true for thermal temperature. We begin with something we can measure. We then define some unit of quantity as a basis or comparison point for our measurement. The dimension of that measurement is then what our unit of quantity exists in and consist of. So you end up with some quantity of units of the dimension you are measuring.

13. Jul 28, 2007

### belliott4488

JOPearcy - I think you are still confusing the issue by mixing the two uses of the word "dimension" in Physics. You could simply substitute the word "units" for one of those, the way people do when the state, for example, "energy has units of mass times length over time squared", and most people would understand exactly what you mean. (Strictly, "units" refers to a specific system, such as SI, but as I said, you will still be understood if you use that word in place of "dimension" when doing dimensional analysis.)

The other use of "dimension" is mathematical, and as far as I know, is unrelated to the first use. Mathematicians speak of, for example, a "5-dimensional Euclidean space", and they aren't thinking about units of anything. Such geometrical spaces have plenty of applications in Physics, for modeling space and time as well as other more abstract spaces, such as the Hilbert Space of Quantum Mechanics.

I'm not familiar with the use of "dimension one" to describe what I would have called a dimensionless number, but it clearly has nothing to do with a one-dimensional geometric space. It looks to me like a mere convention of terminology, stemming from the fact that if you express such a quantity in powers of basic units of length, mass, time, and electric charge, then all the exponents would be zero, thus giving you "one". Sort of. I'd just call it "dimensionless" and be done with it.

One more comment - you stated that you can't necessarily add quantities with the same dimensions, for example the lengths of two perpendicular lines. Of course you can. It just depends on the question you're asking and answering. If I were to ask the total length of the sides of a rectangle, you'd add up the lengths of the sides, even thought they're perpendicular. What you can't ever do is add a length to a time. Three meters plus two seconds makes no sense.

14. Jul 28, 2007

### JOPearcy

Belliot4488,

From the tone of your reply, are you following me from another forum just to continue making statements of how wrong I am so I can continue to make statements of how wrong your are?

Your comment “I think you are still confusing the issue by mixing the two uses of the word "dimension" in Physics” implies to me you have knowledge of this argument from this other forum.

Are you only following me here to continue arguing without listening? If so, what is the point? To try and follow my discussion where ever I might go so you can shout me down as being wrong without listening or understanding what I am saying?

If you are not interested in understanding, why spend the effort to follow me?

15. Jul 28, 2007

### JOPearcy

Belliot4488,

If you are not following me from this other forum them I apologize for being overtly defensive. I have had a discussion on another forum where a group of posters clearly did not understand what I was saying, incorrectly stated I was wrong over something I know I was correct about and then repeatedly insulted at which point I became insulting back.

I don’t want to get into a discussion with people where the substance of that discussion is no attempt to understand the other person backed by insults.

If you are really interested in understanding what I am saying, then I will try my best to explain it, piece by piece, so that hopefully there will not be a misunderstanding.

But I have no interest in having someone effectively shout me down saying “Wrong, Wrong, Wrong, How can you be educated” while they don’t even understand what I am saying.

I have spent 6 years of intensive study in electrical engineering, math and physics at Cal Poly Pomona. I excelled in the sciences. In other subjects, English, history and such I had to struggle. When I started my English Grammar skills were just good enough to get me by. But in areas of abstract and analytical skills of science I was top of my class.

The underlying principles of how dimensions are defined using the definition of dimension was taught to me at Cal Poly, first and foremost in the physics department but also in certain advanced math and engineering courses.

I do know what I am talking about. Whether or not I can explain it well enough for you, whether or not you are able to understand the underlying principle and/or whether or not I can find an authoritative online definition to point to explaining what I am talking about is all unknown.

16. Jul 28, 2007

### belliott4488

JOPearcy,

Nope, I never saw your posts on any other forums. I honestly just thought you were not aware of the two separate uses of the word "dimension" in Physics.

You say that you are aware of these things, so could you please explain again why you are comparing them? How is the use of the word "dimensions" in the statement "velocity has dimensions of length over time" related to its use in the statement "Special Relativity can be described in terms of Minkowsky space-time with four dimensions"? I would have thought that it is an accident of language that we even use the same word for these two concepts.

17. Jul 29, 2007

### matt grime

Belliott is entirely correct - you appear to be confusing the physical notion of 'unit' with the mathematical usages of dimension.

18. Jul 30, 2007

### JOPearcy

Belliott4488,

I want to apologize again for being so snappy. You might figure out by now that I had some posts on another forum whose tone angered me.

I want to discuss this politely and rationally.

Velocity is a dimension. It is a derived dimension, but it is still a dimension. The dimensions that the dimension of velocity is derived from are a system of dimensions.

The dimensions of space-time are a system of dimensions.

What is space?

19. Jul 30, 2007

### cristo

Staff Emeritus
Velocity is not a dimension. You appear to be confused by the definitions. I suggest you review the mathematical definition of a dimension.

20. Jul 30, 2007

### belliott4488

Um, okay ... but I still think your use of language is somewhat obscuring the point you're making. For example, I'm not sure how you define a "system of dimensions".

In any case, I've never heard any say that "velocity is a dimension," but it is not unusual to say, "the dimensions of velocity are length over time." Maybe they're the same thing to you?

What I don't see at all, however, is your connection between geometric dimensions, which are related to the coordinates needed to identify a point uniquely, and dimensions in the sense of units of measurement. For example, if we talk about a space of N dimensions, points are identified by N coordinates, in terms of which we can define distances (assuming the space has such a concept - I don't want to get side-tracked by spaces with no norm or anything like that), such that all the dimensions are measured by length units. The fact that space-time is commonly spoken about as having time as a dimension does not mean that the fourth dimension is measured in units of time. In fact, it is often introduced as being measured by ct (speed of light multiplied by t), so that it has units of length, just like the spatial dimensions. (You'll also see descriptions where the time dimension is measured by t alone, but then the metric, which defines how distances are measured, introduces c wherever t is used, so it comes out the same.)

The point is that for a geometric space to make sense, there must be an equivalence of all the dimensions, in the sense that you can rotate coordinate axes and mix the values measured along the axes. You cannot have a sensible geometric space where different geometric dimensions have different dimensions in the sense of units (e.g. one dimension of length and one of mass - that's not a geometrical space).

Does that make any sense to you? I hope so.