Difference between units and dimensions

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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].

Answer 4

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 :) )

Answer 5

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.

Answer 6

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.

(This is a physicist's answer, I don't know what a mathematician would say about this formulation.)