# Quantities without unit of measure

haruspex
Homework Helper
Gold Member
See the first paragraph of this pdf and notice that it is said that to be a scalar magnitude is necessary to have unit of measurement.
Is it correct?

http://zeus.plmsc.psu.edu/~manias/MatSE447/03_Tensors.pdf
I would not take that web page as an authoritative source on such fine distinctions. It seems fairly low level.
A good example of a naturally occurring dimensionless scalar is the Reynolds number. Another is π.

I would not take that web page as an authoritative source on such fine distinctions. It seems fairly low level.
A good example of a naturally occurring dimensionless scalar is the Reynolds number. Another is π.
Do you know of any reliable sources on the WEB that cite the examples you mentioned, explicitly called them scalar quantities?

O.k. But as in the example of a refractive index, a quantity can also be a number without a unit. Here you measure a quotient and the units cancel out. This means, unit or not isn't the question to distinguish between scalars and vectors. Nor is it a matter of dimension. Mathematically a single number (scalar) can also be a one-dimensional vector. Physically a dimensionless scalar, is a number as the refraction index is. The hydrostatic pressure has a physical dimension (pressure) and temperature is also a physical dimension.

What is meant in the paper is the following:
Say we have the hydrostatic pressure ##H(\vec{x}_0)## and the temperature ##T(\vec{x}_0)## at a point ##\vec{x}_0## on earth.
• ##H_0=H(\vec{x}_0)## as well as ##T_0=T(\vec{x}_0)## are scalars, because they don't have a direction, only a scale - a number. They are not dimensionless and both have a unit, although they are scalars.##\\##
• ##(H_0,T_0) = (H(\vec{x}_0),T(\vec{x}_0))## is a vector of scalars with two different dimensions.##\\##
• ##\{\,(\,H(\vec{x})\,,\, \vec{x}\,)\,\vert \, \vec{x} \textrm{ a point on earth }\}## is a scalar field: each point ##\vec{x}## has an associated scalar ##H(\vec{x})## which gives us a field of scalars. Usually the entire scalar field is also briefly noted as ##H## or ##H(\vec{x})##.##\\##
• ##\{\,(\,(H(\vec{x}),T(\vec{x}))\,,\, \vec{x}\,)\,\vert \, \vec{x} \textrm{ a point on earth }\}## is a vector field: each point ##\vec{x}## has an associated vector ##(H(\vec{x}),T(\vec{x}))## which gives us a field of vectors; each vector being a pair of scalars. Usually the entire vector field is either written as ##(H,T)=(H(\vec{x}),T(\vec{x}))## or abbreviated with another letter ##V##.##\\##
• If we define ##\vec{w}(\vec{x}_0)## as the vector of wind at a point ##\vec{x_0}## on earth, then ##\vec{w}## is the classical vector with a direction, where the wind blows, and a magnitude, how strong the wind blows.##\\##
• If we define ##\{\,(\,\vec{w}(\vec{x}),\vec{x}\,)\,\vert \,\vec{x} \textrm{ a point on earth }\}## then we get a classical vector field, that is to each point ##\vec{x}## an associated vector of wind ##\vec{w}(\vec{x})##. Usually this construction is only denoted by, e.g. ##W## or ##W(\vec{x})\,.## ##\\##
So to close, in case you were to mark my initial question, you would mark the letter A. Is that it?

Quantities without unit of measure are:
A) Necessarily scalar
B) Necessarily vector
C) Can be scalar or vector
D) They are neither scalar nor vector

fresh_42
Mentor
Do you know of any reliable sources on the WEB that cite the examples you mentioned, explicitly called them scalar quantities?
Why do you insist on this wording? It's important to understand what is meant by scalar, vector, quantity, unit and dimension. Among these, quantity is certainly the least rigorously defined term.

fresh_42
Mentor
Quantities can be a scalar or a vector (C). And it doesn't matter whether they have a unit or not, only that they can be measured somehow. This makes them different from a quality.

Why do you insist on this wording? It's important to understand what is meant by scalar, vector, quantity, unit and dimension. Among these, quantity is certainly the least rigorously defined term.
I am very curious to know how to resolve this problem in detail, as I have never seen this discussed in any book. I am a physics teacher and I am writing a book on vectors in which I wish to insert such a discussion.

Quantities can be a scalar or a vector (C). And it doesn't matter whether they have a unit or not, only that they can be measured somehow. This makes them different from a quality.
I apologize for the persistence and thank you for your attention

fresh_42
Mentor
I apologize for the persistence and thank you for your attention

Caio Graco
haruspex
Homework Helper
Gold Member
So to close, in case you were to mark my initial question, you would mark the letter A. Is that it?

Quantities without unit of measure are:
A) Necessarily scalar
B) Necessarily vector
C) Can be scalar or vector
D) They are neither scalar nor vector
I would consider this a poor question because it is not clear what is allowed, though maybe there is an interesting question hiding within it.
As has been pointed out, it is always possible to construct a vector from a list of arbitrary scalars. If that is permitted then clearly the answer is C.
Also, many dimensionless scalars are so by virtue of being a ratio of two scalar entities of the same dimension. In the same way, you might argue that a unit vector derived from a given (dimensional) vector is a dimensionless vector. The direction of a velocity is given by ##\vec v/|\vec v|##. But again, this feels artificial.
If we exclude those then it gets more interesting. As I wrote, there are reasons why dimensionless constants in the real world tend to be either scalars or tensors.

berkeman
Mentor
I am a physics teacher and I am writing a book on vectors in which I wish to insert such a discussion.

haruspex