How to suggest textbook improvement to author? There's a popular math textbook I enjoyed a lot when I was a grad student. Recently I noticed an extension to one of the many interesting exercises provided by the textbook. I'd like to communicate this idea with the textbook author (single author).

The exercise is to calculate and show an amazing coincidence, and the way it is phrased in the textbook suggests that the author didn't think there could be an explanation. I think I found an interpretation, and I think it might make this exercise richer if the author adds a couple of subquestions guiding readers to discover this viewpoint for on this coincidence.

My Question Is This:


  Is it considered inappropriate to email the author and suggest this as a possible improvement?
  The interpretation to that exercise is somewhat technically intensive and hard to explain in a few words. Should I try to make it brief or should I make it solid that undoubtedly my interpretation is correct? 


I'm worried that my suggestion might appear either rude (as if the author doesn't know this already) or annoying (just too lengthy), or it might seem like a clumsy attempt to advertise myself (like a lot of layman claiming to have made a breakthrough).

I think it's 50/50 whether or not the author might have known this interpretation all along but simply didn't include it in the exercise for various reasons. Even under this circumstances, this coincidence is so intriguing that I still think a few words from the author encouraging readers to explore it would completely change the tone of this exercise.

Any thoughts would be appreciated, thank you. This question might be mostly opinion based thus not suited for this site, and I'll be fine with that. 

P.S.

I'm on a 6-year-long hiatus from the academia, going on to the 7th year, I currently don't affiliate to any institution. I've been keeping a certain level of capability to do this subfield in math, and I'm about 95% confident that my result is correct. Currently I have no plans to return to the academia within the next 2 to 3 years, but things might change.
Is it considered inappropriate to email the author and suggest this as a possible improvement? 


It is very appropriate and would probably be very welcome. As the author of a textbook myself, I am always happy to receive any sort of feedback about my book (if nothing else, it proves to me that someone is reading it ;-)), and have received several emails of the exact kind you describe, and other suggestions for improvement, several of which I ended up using. Just be mindful that the author may not think your contribution is as important or worthwhile as you find it, or may be too busy to think about it, so set your expectations low and be respectful of the author's ultimate authority about what to do with your suggestion, including possibly ignoring it.

More generally, I would say it is appropriate to contact anyone who makes their email publicly available, regarding any issue that has a reasonable chance of being of interest to the recipient and is not driven by spammy or nefarious intent. Your example very clearly falls within this range.


  The interpretation to that exercise is somewhat technically intensive and hard to explain in a few words. Should I try to make it brief or should I make it solid that undoubtedly my interpretation is correct?


A detailed and well-exposed technical description of your idea would be most helpful, but I would attach it as a PDF write-up and make the email itself short with only a general description of the idea and referring to the attachment for more details. This minimizes the burden and possibility of annoyance to the author. It would also be a good idea to offer to send the author the LaTeX source in case that might be helpful (but better not include it in the initial email), and may increase the chances that the author will incorporate your idea into the book.