An elementary definition of a scalar is that it is ''a quantity having only magnitude''.
I've been trying to reconcile this statement that I've always accepted with (1) the more exact definition of a scalar and (2) the definition of magnitude in Euclidean space. I am struggling with a particular paradox that I've tried to explain to myself in a brief note. I'm not convinced that my reasoning nor my conclusion are entirely valid. This is what I wrote:
From the first introduction to vectors in secondary school, up to fundamental courses at the collegiate level, a frequently encountered statement is:
''A vector is a quantity having both magnitude and direction. A scalar is a quantity having only magnitude.''
Though a useful introduction the the concept of vectors, the problem with the above statement is that it is misleading.
In the context of applications to mechanics, the further discussion is restricted to Euclidean vectors.
First, consider the more precise definition of a scalar; namely a quantity which is invariant under all possible rotational transformations of the coordinate system in which it is described.
In Euclidean space, by definition the magnitude (Euclidean norm) is always positive. Also, the magnitude of a Euclidean vector is invariant under all possible transformations of the coordinate system in which it is described. Therefore magnitude is a scalar.
Now consider that there are two vectors, $\mathbf{a}$ and $\mathbf{b}$. The difference of the lengths of these two vectors, $\Delta = \left\|\mathbf{b}\right\| - \left\|\mathbf{a}\right\|$ can be negative. Yet $\Delta$ is also invariant with rotation of the coordinate system, and is therefore a scalar. But by definition, if it is less than zero, it is not a magnitude ($\Delta$ does, of course, have a magnitude --its absolute value--in addition to a sign).
The above example suggests that ''all magnitudes are scalars, but not all scalars are magnitudes''.
A more accurate wording of the classic introductory definition of vectors and scalars might therefore be,
''A vector is a quantity having both magnitude and direction. A scalar is a quantity having magnitude and a sign.''
Is what I am saying correct and/or are there errors or fallacies in my reasoning?
EDIT This might be semantics, but it is possible to interpret the elementary definition as "''A vector is a quantity having both magnitude and direction. A scalar is a quantity having only magnitude (and not direction).'' In which case it would not preclude a sign, and would make sense.
The point of the elementary definition is to build physical intuition for common elementary examples, which turn out also to be useful examples of more general phenomena.
Of course the loose definition will not do when you start pinning down subtleties connected with the real numbers, and when we start defining vector spaces by their properties, rather than imagining vectors as arrows in physical three-dimensional space.
But physical space is what we might call a motivating example - it is one key reason why we care about making a precise definition, and creating a model (mathematical structure) with useful properties.