Abusing the “Proportional To” symbol

Question

As far as I know, the statement “$a \propto b$” is equivalent to “$a = kb$ for an arbitrary value $k$”. Is there anything wrong with the following?

$$ \begin{align} f(x) &\propto 2\,f(x) \\ 6 &\propto \pi \\ \begin{bmatrix} -5 \\ 10 \end{bmatrix} &\propto \begin{bmatrix} 1 \\ -2 \end{bmatrix} \\ A\mathbf{x} &\propto \mathbf{x} \end{align} $$

I want to use this in an informal setting and am wondering if it complete nonsense. Has this notation been (ab)used like this anywhere? Should I first clarify that “$a \propto b \Longleftrightarrow a = kb$ for an arbitrary $k$” at the top of the page?

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(Other notations like “$a$ is a multiple of $b$ $\Longleftrightarrow$ $a|b$”, are too restrictive for my intended use because they sort-of imply integer multiplicity and can’t be chained together nicely, like $2(a, -a) = (2a, -2a) \propto (1, -1)$ for example.)

Thanks for your input.

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Edit: I was asked to give some context: I’m just a student hand-writing his linear algebra homework. I come across myself wanting to emphasise that a vector is a scalar multiple of another in a string of equalities without having to write “where $a, a’ ∈ ℝ$” everywhere or writing words. Snippet: $$ \mathbf{q}_2 = \mathbf{v}_2 - \text{Proj}_\mathbf{q1}\mathbf{v}_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix} + \frac{1}{5}\begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} \frac{6}{5} \\ -\frac{3}{5} \end{bmatrix} \propto \begin{bmatrix} \frac{2}{\sqrt{5}} \\ -\frac{1}{\sqrt{5}} \end{bmatrix} = \mathbf{u_2} $$ I realise it’s laziness, but I got curious. I thought it was pretty.

Answer 1

This usage of proportionality is unusual. Two "proportional" vectors are also parallel, hence the parallel symbol is more suitable for vectors $$ \begin{bmatrix} -5 \\ 10 \end{bmatrix} \parallel \begin{bmatrix} 1 \\ -2 \end{bmatrix} $$ $$ A\mathbf{x} \parallel \mathbf{x} $$ Functions are essentially vectors so $f \parallel g$ is also suitable.

Answer 2

To give a different viewpoint, there is actually a standard set which is the set of vectors up to scaling, and it is called $\mathbb{R}\mathrm{P}^n$ or $\mathbb{C}\mathrm{P}^n$, depending on the underlying scalar field. They are pronounced "real projective space" and "complex projective space," respectively.

Since $\mathbb{R}\mathrm{P}^n$ is defined as a set of equivalence classes, where two nonzero vectors are in the same equivalence class if and only if they are scale multiples of each other, it would be fine to use the default equivalence symbol $\sim$, so long as you define what you mean. For example, \begin{equation} \begin{pmatrix}1\\2\end{pmatrix} \sim \begin{pmatrix}2\\4\end{pmatrix}. \end{equation} (The thing I don't particularly like about $\propto$ is that it seems like a non-symmetric relation, though it is a clever use of the symbol, and if you define what you mean it is perfectly OK. In fact, you might say $v\propto w$ iff $v=cw$ for some $c\in\mathbb{R}$, and then you are allowed to say $\mathbf{0}\propto \mathbf{x}$. This is in contrast to $\mathbb{R}\mathrm{P}^n$ where we do not allow comparison with the zero vector.)

Answer 3

There is nothing wrong with using the notation like this since it is an actual generalization of the common usage of $\propto$. So if you find it very useful, why not? You are just giving a name to a relation (the proper definition was given by Monstrous Moonshine in the comments).

But, yes, you should most likely introduce the notation first before using it since it is not well known in this generalized version.

Answer 4

Proportionality as described by Wikipedia is described as the relationship between two variables, such that “$a∝b$ is equivalent to $a=kb$" as you have described.

Therefore, none of the examples are correct examples of proportionality, since (1) compares two functions, (2) compares two numbers, (3) compares two vectors, and (4) uses only one variable.

The correct way to describes these relationships would be:

(1): $g(x) = 2f(x)$

(2): $6x = \pi$

(3):
$$ A\begin{bmatrix} -5 \\ 10 \\ \end{bmatrix} $$

$$ =\begin{bmatrix} 1 \\ -2 \\ \end{bmatrix} $$

(4): $Bx = y$

In summary, proportionality describes the relationships between two variables, so that one can be "scaled up/down" to another. In many situations it is often best to define one function in terms of another by using a constant multiplier and setting that equal to the function.

EDIT: In the context the OP has put, my opinion would still be to not use the proportionality symbol. I think that sacrificing context and quality for an abbreviation hides the true meaning of proportion, and by writing $a,a′∈ℝ$ the context for what numbers are used becomes clearer.

I think that outside of its definition of describing the relationship between two variables, an equality symbol with a constant multiplier is much more useful (it can be widely applied) and is clear-cut and well-defined.