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Most scientific papers refer to almost any formula that contains the "=" sign by the word "equation". Consider this case:

Let value = function(input) (1) where input is a known input value and value is the result of the function computation. In this situation, there are no effective unknowns (we are guaranteed that no unknowns are hidden in the function expression either).

Most papers (if not all) often refer to expression (1) in sentences like "equation (1) is equivalent to", "referring to equation (1), we see that..".

Some pedantic referees, however, suggest the use of a more proper word (e.g. formula (1) instead of equation (1)).

Is there a grammar reference that solves this seemingly insignificant issue?

Further clarification

An example of an expression not considered an equation by a referee: (a+b)^2 = a^2 + 2ab + b^2 (1).

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  • Can you make a real example of a formula that you do not consider to be an equation? Jul 29, 2016 at 12:55
  • @FedericoPoloni I have to mention that a referee made this observation. To me, from my actual experience, this is close to nitpicking or even outright incorrect since almost all journals use the term equation with a broader semantic spectrum (which encompasses both formulae and strict equations). To me it seems unnatural to use formulations such as "We deduce from formula (1)..".
    – teodron
    Jul 29, 2016 at 14:04
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    Strictly speaking the referee is right: the example expression is an equality, rather than an equation.
    – Massimo Ortolano
    Jul 29, 2016 at 16:19
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    @MassimoOrtolano Maybe you mean an identity? All these subtly different words are quickly getting out of hand... Jul 29, 2016 at 17:38
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    Recent question on Math Educators about the meaning of the word "equation" (albeit one with an accepted answer with which I disagree): matheducators.stackexchange.com/questions/11203/… Jul 29, 2016 at 19:41

3 Answers 3

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From Higham, Handbook of writing for the mathematical sciences (emphasis mine):

Referencing Equations

When you reference an earlier equation it helps the reader if you add a word or phrase describing the nature of that equation. The aim is to save the reader the trouble of turning back to look at the earlier equation. For example, "From the definition (6.2) of dual norm" is more helpful than "From (6.2)"; and "Combining the recurrence (3.14) with inequality (2.9)" is more helpful than "Combining (3.14) and (2.9)". Mermin [200] calls this advice the "Good Samaritan Rule". As in these examples, the word added should be something more informative than just "equation" (or the ugly abbreviation "Eq."), and inequalities, implications and lone expressions should not be referred to as equations.

Pro-tip: arguing with a referee on such trivialities is rarely worth your time. Just do what they suggest.

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    While I agree with everything there, it doesn't answer the poster's question, which is in regards to the word "formula". Jul 29, 2016 at 19:39
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Mathematically, one could say that a formula is an equation that defines one side (typically the left-hand side) in terms of the other side. An equation on the other hand states that two expressions in terms of previously defined quantities are equal.

For example, Pythagoras' theorem a^2 + b^2 = c^2 I would call a typical example of an equation: a,b,c are predefined lengths of a rectangular triangle, and it makes a non-trivial statement on how they are related.

On the other hand, if you simply assign a function value, y = f(x), then it makes sense to call that a formula, since you define y in terms of the right-hand side.

Edit: Federico makes a fair point.

Let me just add that "Mathematics into Type" by Ellen Swanson, published by the AMS recommends, somewhat similar to Higham referenced by Federico:

6.4.2 Equations

Do not capitalize. An author is apt to refer to the same display as equation (3), property (3), or definition (3); it can become rather confusing if the word is treated as a proper noun when references is made to it in so many ways.

which implies that at least alternative names are possible.

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I'd say your referee is right.

When you have variables on both sides of an equality, i.e. when you show some relationship between variables, we tend to say this is a formula.
E.g. x + 2y = 3z is a formula.

If, however, one of the sides of your equality contains no variables, just value(s), then we say it's an equation.
E.g. x + 2y = 3 is an equation.

Edit: The "equation" environment in LaTeX can be quite confusing this way, as it is often used for formulas, rather than equations...

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    Sorry, but from a mathematical viewpoint I think this is nonsense: if in x+2y=3z you move 3z to the other side, then at once x+2y-3z=0 becomes an equation?? Jul 29, 2016 at 15:55

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