Acronyms and words as variables and in mathematical notation

Question

I am unsure if this question is warranted.

Often mathematical symbols and objects are represented by a single character, e.g. variables are most often single characters like $x$, and often to describe a variable further we use an additional character in a subscript or a superscript, like $x_t$.

In physics people often choose intuitive letters to represent quantities as variables, for example, using $t$ for time or $v$ for velocity (starting character is used to make it more recognizable).

Of course this always isn't the case and I acknowledge that there are the cases of function names such as $\sin(x)$ or even objects like $\sup A$ and $\inf A$ for a set $A$.

However my question is what is the general opinion (concerning formal formatting) on using words, abbreviations, and acronyms as variable quantities in mathematical sentences.

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An example, we have that the revenue of a simple trade of a good is the quantity multiplied by the price. A few possible formats include:

1. $R = q\times p$
2. $\text{revenue} = \text{quantity} \times \text{price}$
3. $r = q_{\text{good}} \times p_{\text{good}}$

Perhaps #3 isn't so pretty with this simple formula, but if we take the mark-to-market formula of quantity times the difference in market price and trade price, we can get some other formats:

4. $\text{MTM} = q\times(p_{\text{market}} - p_{\text{trade}})$
5. $\text{MTM} = q\times (\text{MP} - \text{TP})$

and so forth.

Of course I see words and acronyms often in the equations from the softer sciences like economics where my example comes from, but I am sure this question applies to pure mathematics in some cases.

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I do not know which notation style we should tend towards these cases, especially if it was in the context of an academic paper.

Thanks for any opinions!

Answer 1

First you can/should define the variables, which you want to use in a formula. In general you can define a variable as you like. Here are my suggestions:

$p$=price of the good

$q$=quantity of the good

$r$=revenue

Now you can simply write down the formula for the revenue.

$r=p\cdot q$

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For the price I would superscript the abbreviations for the market and the trade.

$p^M$=market price

$p^T$=trade price

In both cases you have prices as variables. Here it is a good idea to make the distinction at the superscripts.

$q$= quantity of the good

$\Pi^{mm}$profit due to a market to market operation. It can be also dentoted as $\Pi^{m}$ or something similar. The main thing is, that you define the variables.

$\Pi^{mm}=q \cdot (p^M-p^T)$

Answer 2

I think it depends a lot on the context. For a one-off formula, you're often better off spelling it all out. You don't have to explain what all of the terms mean.

But in a situation where various values, especially in different formulas, are related to each other (as, say, $p_\text{market}$ is to $p_\text{trade}$), the use of subscripts on a common variable can highlight, visually, that relationship. It becomes much more readily apparent, more so than in something like $\text{market price}-\text{trade price}$. It also facilitates generalization, if there are other $p$'s floating around.

I don't think there's a reliable hard-and-fast rule to rely on, if your overall objective is clarity in presentation. You just sort of have to know your audience.

Answer 3

The general rule for mathematical notation is that a variable should be denoted by a single italic letter, possibly with embellishments. Letters used for purposes other than denoting variables, such as descriptive subscripts (as in your example 3) or the names of standard mathematical functions, should be in roman type. The convention is that juxtaposition denotes multiplication in the case of italic letters and spelling in the case of roman letters. Superscripts are best avoided, particularly when powers are used in the same setting.

It is hard for mathematicians to read papers in economics where these conventions are not followed. Arguably, there is a case for other conventions to be used in an applied field when multiplication is always denoted explicitly and so many variables are used that their meaning is easily forgotten without a prompting spelt-out or acronymic name. Even so, I think it is still best to put these names into subscript, as per your example 3.