# Is copying equations plagiarism?

I defended my master thesis successfully yesterday. Everything in my master thesis was written by me and there is no problem. However I'm a bit skeptical about one thing. So in my thesis I have a part that explains a very well known model. To explain the model you basically need to calculate the gradient of some function, and calculating the gradient is basically lengthy and spans many subsections. So there are many papers and I believe even books that show how to calculate the gradient step by step. So I found one paper and I read it and understood it and understood how the gradient of the function is derived. Next I compared the calculations in the paper to other books and I found that they are very similar.

Since I'm introducing the model in my thesis I had to explain how to calculate the gradient. I explain everything in my own words. However when I wrote the equations they look almost like a copy from the paper!! Since I had the same understanding, so the calculations just almost look the same. So I was really confused what to do about this. Since I want to include the derivation of gradient in my thesis and it looks almost the same from the paper. So I was wondering if this is plagiarism! Since I was skeptical, I checked that the equations also look somehow almost the same from other papers. So to make sure that I don't be held for any kind of plagiarism and credit the authors, I mention at the beginning of the section that starts to explain the model is that my following explanations are based on the explanation given in the paper, where I learned about the model. Calculating the gradient is kinda lengthy and spans couple subsections, so in the subsections I assumed that since I credited the authors at the beginning then I didn't credit them again later for the other derivations of the gradient.

Now my question, is what I did correct?! I mean they were only the derivations of the gradient, so they were equations and I can't just "rephrase" them!

I remember once asking one postdoc that I'm using a lot of equations from one paper and he said that I just need to credit the authors that I'm using the same notation, which what I did. However, when I look at the formulas I feel somehow uncomfortable, because they look like a copy from that paper. Should I be worried?!

I really tried to credit the authors and I cited the papers in many places where I was explaining the models. But only the look of those large chunks of equations seems like copying.

As I said earlier I defended my thesis today, so I just hope that I didn't do anything wrong with that because I really worked hard for my thesis and every single word is mine.

P.S what I know is that copying common knowledge is not plagiarism. So this was my assumption behind the calculation of the gradient. Because the derivation of the gradient is not a contribution and the derivation is done in many papers. (like the gradient of the log-likelihood of logistic regression).

So in short, suppose that calculation of the gradient of the log-likelihood of logistic regression is very lengthy in general. And I just copied the equations from a paper that does the derivation. However when I motivated Logistic regression in my thesis I said that my explanations in the next subsections are based on the explanation in the paper where I copied the equations from. Is that okay?

• If this is a standard calculation, why not just reference it? (and not include the equations) – Joe Manlove Mar 17 '15 at 16:03
• @JoeManlove because the thesis, as far as i was told, should be self-contained. So I had to explain the details of the models that I address in my thesis. – Jack Twain Mar 17 '15 at 16:50
• I'd ask your advisor about this. Unless they add something to the exposition, redoing pages of calculations that are in the literature is not usually necessary. It's not a plagiarism issue if cited, but is a bit tedious to write (and read). – Joe Manlove Mar 17 '15 at 16:58
• If one needs to cite Pythagoras upon every theorem use I'd say western civilization has one hell of a lawsuit on our hands. – easymoden00b Mar 17 '15 at 19:00
• If nervous, document where you got the formulas from. Which might be good practice anyway if they aren't well-known and/or obvious. – keshlam Sep 13 '15 at 21:58

Mathematical equations are not normally considered subject to plagiarism, since there are only so many ways they can be written, and because they represent ideas rather than material that can be "stolen" without proper attribution.

So long as you cite the material you are drawing from, and explain the equations in your own words, you should be fine.

• Moreover, in the mathematical sciences it is very well understood that two people who are doing a standard type of calculation or technique, and both doing it correctly, have a good chance of doing it in a very similar way, even up to using similar or identical notation. After all, we very much take notation and terminology from each other: imagine what a disaster it would be if everyone had their own "non-plagiarized" notation for an integral or cross product... – Pete L. Clark Mar 17 '15 at 16:49
• @PeteL.Clark, we shouldn't? That explains the puzzled looks when I ask my calculus students to calculate $\times \vec v\int\vec w\textup dt$! (EDIT: Oops, maybe it's less funny without MathJax.) – LSpice Mar 17 '15 at 22:37
• Between this answer and @PeteL.Clark's comment, this covers pretty much everything I would have posted myself. The only point I would add is that it's very common to say something like "In this section I follow the derivation of [source]" and then include the derivation from that source - but this is really an acknowledgement that you are copying the overall structure of the derivation, rather than individual equations. Also, a nitpick: I would note that the distinction between ideas and expression is important to copyright, not so much to plagiarism. – David Z Mar 18 '15 at 5:27
• @LSpice Even with MathJax turned on, it has an error (\textup is not recognized). – Tobias Kildetoft Mar 18 '15 at 8:54
• @aeismail I like your reply and agree with it. Many months ago, on this site, I also discussed "explaining things in your own words" and I was told it is plagiarism. Most persons' "explanations in their own words" are inevitably derived from somebody else's explanations. I don't believe it is plagiarism and I think people are REALLY too hyperactive about plagiarism these days. – Inquisitive Jul 22 '16 at 0:44

I do believe, like @aeismail and other commentators, that equations are knowledge, and not subject to accusations of plagiarism. Yet, you should be cautious about your mathematical notations and their correspondence with the copied equations.

If you have adapted the proofs, or the calculations, with your own symbols, and typed the equations yourself, you are fine.

Be cautious about some mathematical notations (imaginary unit, convolutions, ordinal variables, greek letters), that may be more or less standard depending on the scientific field. For instance, the imaginary unit is often "i" in maths, and "j" in electrical engineering. The reuse of other field's notations should not look too artificial.

However, I have seen persons who have cropped blocks of equations from some source in slides or reports, and who did not bother change their notations or cite the source. This behavior is closer to laziness or dishonesty than to a genuine personal work. But working on equations, their variables, to make them more understandable, clearer, is a plus.

Finally, one side of plagiarism is in automated plagiarism detection tools. You may be worried about texts, but so far, they do not seem very efficient with equations. Unfortunately, since it is difficult to search formulae online.

• "Your own symbols" sounds ambiguous. One should of course ensure notations in one's work are internally consistent. However, it is totally fine (and recommended) to use the same symbols as another work, because many notations are (semi)-fixed for some concept, down to the letter to use (say, a standard deviation is always a $\sigma$). – Blaisorblade Apr 30 '16 at 9:14
• @ Blaisorblade I meant that inn the context "Yet, you should be cautious about your mathematical notations and their correspondance with the copied equations". Additionnaly, symbols may vary from one domain to the other: E.g "i" vs "j" for the imaginary unit – Laurent Duval Apr 30 '16 at 9:21
• @Laurent Duval I think yours is a good answer. You shouldn't copy blocks of equations without citing the source. It gives the appearance that you wrote the equations or even developed them. – Inquisitive Jul 22 '16 at 0:58

You should remember that academic plagiarism is not plagiarism in the sense of copyright. Academic plagiarism is mainly the "stealing" of other people's ideas. As such rephrasing is normally not enough. Mathematical formulas are a bit of a gray area in that. Much of it is (very) well known and also subject to possible independent invention. Once you have a lot of them though, the probability of the same approach without awareness becomes unlikely.

In all cases of gray areas, especially if they concern aspects of professional ethics (such as plagiarism), be safe and reference. Also be aware that not all readers are familiar with all parts of maths and may not recognise that you are just describing a reasonably well-known concept, not introducing it yourself (I have reviewed a journal paper with this issue, directing the authors to add the reference was sufficient, it was clear that they actually didn't come up with the idea).