Difference between units and dimensions

Though this question may seem related to Physics, I think that at the very root this is a mathematical question and so I have posted this on math.stackexchange.

Background: Initially I thought that the terms-unit and dimension, refer to the same thing. Physical quantities are categorised into fundamental/basic physical quantities and derived physical quantities. A fundamental physical quantity cannot be broken down into simpler physical quantitities, cannot be obtained from other fundamental quantities and all the known physical quantities can be obtained using fundamental quantities. There was a line in my book which stated

Mass, length, time, thermodynamic temperature, electric current, amount of substance, luminous intensity are the seven fundamental quantities and are often called the seven dimensions of the world. Thus the dimension of mass is [M], that of length is [L] and so on. The dimensions of a derived physical quantity are the powers to which the units of the fundamental physical quantities have to be raised in order to represent that derived physical quantity completely.

This is very confusing and I am finding it difficult to understand the difference between the two terms-dimension and unit.

Question: What exactly is the difference between the meaning of the terms unit and dimension?

• Feet, inches, and meters are different units. But they all measure the same thing and each can be converted to the others, so they are the same dimension: length. – MJD May 22 '17 at 12:02

The difference is quite subtle and of little practical importance if done accordingly.

The difference is that a unit incorporates a property of scale while dimension doesn't. For example in the case of length it could be measured in meters, decimeters or kilometers, but nevertheless the dimension is length.

If you use unit-analysis instead of dimensional analysis you would have to take into account that different units of length only differs by a (dimensionless) constant.

• Redundancy: "a constant constant". – Ruslan May 22 '17 at 18:15
• Re "...of little practical importance if done accordingly." I'm not sure what "if done accordingly" means. I would also disagree that the distinction can be quite useful, e.g., in talking about radians and degrees as dimensionless units. I've also found the distinction important when implementing units or dimensions in a programming language. – Chris Chudzicki May 22 '17 at 21:17
• @Ruslan Thanks for pointing that out, corrected! – skyking May 23 '17 at 5:11
• @ChrisChudzicki When you use unit analysis you may end up with something that differs only by a dimensionless constant (ie real number) which is to be regarded as acceptable. As for dimensionless units they are as units regarded just a dimensionless constant (ie real number). The point is that you can get the same work done using unit analysis instead - there are details in the procedure that may differ, but basically the approach is equivalent. – skyking May 23 '17 at 5:18
• Can you please clarify what you mean by "if done accordingly"? – MrAP Jun 25 '17 at 3:53

You would say speed has dimensions of distance / time. Checking that a formula has the right dimensions is always a good check on a calculation. Units are the choice which has been made to quantify a dimension, e.g. distance in cubits or mega-parsecs.

I think there is no difference between dimension and unit in your case. They can be used interchangeably. However, the same word "dimension" is also used in another context, namely, describing the amount of numbers needed to describe a point in your space uniquely. These two use cases should not be confused. They are very different.

For example, the space we live in is 3-dimensional. This means we describe a point inside by giving three values of the dimension (= unit) [M].

It's a subtle difference, indeed. I can't really find a good definition of it, other than what @skyking said ("a unit incorporates a property of scale while dimension doesn't"), but here's something that could help you understand better. What looks right to you?

• A) My height is a length;
• B) My height is meters.

Which one?

• A) Speed is distance over time;
• B) Speed is inches over seconds.

A talks about dimensions, B talks about units. It is correct to say that speed is expressed in (or measured in) inches per seconds, but not that it is inches over seconds or inches per seconds.

(Plus, inches are not SI :) )

One thing I haven't seen brought up yet is that units can refer to more specific ideas or quantities than dimensions do. Example:

The Becquerel is a unit of radioactivity, referring to an average of 1 decay per second. It has dimensions of $\mathrm{time}^{-1}$, and is measured in $\mathrm{seconds}^{-1}$.

The Hertz is a unit of frequency, referring to exactly one cycle per second. It also has dimensions of $\mathrm{time}^{-1}$ and is measured in $\mathrm{seconds}^{-1}$.

Does this mean that 1 Becquerel is the same thing as 1 Hertz? No! The former is used to talk about the average frequency of random events, while the latter is for regular events.

• Sorry to resurrect this post, but I'd like to discuss this. To be picky, the Becquerel is not 1/seconds, but it is decays/seconds, and the Hertz is cycles/seconds. I think this distinction matters even more in physics, as Randall Munroe pointed out in an excellent and fun way here: what-if.xkcd.com/11 – Simone Mar 2 '18 at 10:27

A dimension represents the definition of an inherent physical property which remains independent of the particular scheme used to denote its measure. For example, the quantity of matter present in a lump of metal has the dimension of mass and the physical size of the edge of a book has the dimension of length.

A unit represents the particular, arbitrary scheme used to denote the magnitude of a physical property. Thus, the mass of matter in the lump of metal may be expressed in kilograms or slugs and the length of the book expressed in meters or feet depending on the system of units selected. Usually the quantity to be measured influences the choice of units to be employed, that is, meters or feet to measure the length of the book rather than kilometers or miles.

I have copied and pasted it from somewhere but with the hope that it helps. Thank you

Using numbers in physics

Units are artificial constructs that allow people to express a dimension as a number. This allows physicians to use arithmetic calculations to predict an experimental result. Very powerful invention. And a dimension is an arbitrarily chosen physical property.

In English, a number can be also called a "quantity"; hence, a "physical quantity" is a number of physical units.

For example: "this ball will fall for a quite short span of time" can be more precisely expressed as "this ball will fall 3 seconds". Note that in both cases dimension is time, but only in the latter case it could be computed.

The "dimensions are powers" confusion

Alas your book (maybe Elements of Physics judging from a cursory Google Books search) uses different definitions. It assumes a base of fundamental quantities. Say [mass, length, time]. In such base a unit of an area can be described as [0, 2, 0] and unit of acceleration as [0, 1, -2]. They don't call the fundamental quantities "dimensions". They don't call these vectors of exponents "dimensions". Dimension is defined as a single exponent that comprises that vector. Just one small number. Confusing, isn't it? As you see, with such definition, "dimensions are powers". To obtain a unit of area, you need to assign a dimension of 2 to a unit of length. So one of dimensions is literally 2 and other dimensions are zero. Personally, I think such definition of "dimension" is utter crap, especially when talking to mathematicians.