How to suggest textbook improvement to author?
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.