How to cite equations in a research paper?

Today, I will talk about how to cite equations in research papers, and give a few advices that may be useful to researchers. The idea of this blog post comes from a discussion that I had with some colleagues about the proper way of citing equations. As you know, some papers will contains equations like this:

   u= q × p     (1)

And then authors will also refer to this equation in the text such as “Equation (1) explains how to calculate the utility of a pattern“. This way of citing equations is generally viewed as acceptable and very common. But is this the best way of refering to equations?

There are a few things that need to be discussed.

First, there are researchers that will say that “equation” is a word with a broad meaning and that it is better to use a more specific term when possible. Following this principe, one could say “Formula (1) explains how to calculate the utility of a pattern” in the above example. There are several terms that can be used depending on the context such as “formula“, “reccurrence“, “inequality” and “identify“. So what is the difference?

  • A mathematical expression is an mathematical phrase that contains some numbers or variables connected by some operators. For example, 1+1 is an expression, and 1+1=2 is also an expression.
  • An equation is a mathematical expression that contains the = symbol. For example 1 = 1 is an equation, and a = b + c is also an equation. Such statements can also be called an equality.
  • An identify is an equality that is true no matter what values are given to the variables that it contains. For example, x + x = x × 2 is an identify.
  • An inequality is a matematical expression where we compare two expression that are not equal by using the symbols such as < > = ≥ ≤ ≠. For example, x+c < d is an inequality. It is not an equation!
  • The definition of a formula is quite broad and some will disagree on the exact meaning. But generally, a formula expresses how to calculate some variable based on one or more variables. For example, equation (1) in the example is a formula for calculating the utility of a pattern. It tells how to find the value of a variable “u” from the values of two variables “q” and “p”. Thus, it can be called a formula. Another example is the Pythagorean theorem which is a formula that can be written as an equation: a^2 + b^2 = c^2.
  • and there are others…

Second, some researchers will suggest to avoid using abbreviations to refer to equations such as “eq. (1)” and “eqn. (1) but instead to write in full “equation (1)” or “formula (1)”.

Third, it is recommended to not just refer to equations in the text but also to add some words to help the reader to remember what the equation is about. This is explained clearly with some examples in the Handbook of writing for the mathematical sciences of Higam:

“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.” (color and bold formatting added by me)

That is all for today! Hope that this has been interesting and will be useful in your papers.

Philippe Fournier-Viger is a professor of Computer Science and also the founder of the open-source data mining software SPMF, offering more than 120 data mining algorithms.

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