Is applying to 20 graduate schools too much? I am applying to math graduate programs, have a list of schools of different range, but I am so worried about being accepted to a grad school, that am considering to apply to 20 schools. 

My undergraduate institution is one of the top schools in math, #1 or #2 in several math fields. 

My worry is that I messed up grades of some important classes, also totally messed up my gre sub, since I didn't prepare for it at all. But I do have lots of research experience in fields related to applied math (I want to do applied math, now I am pure math major).

Should I apply to 20 or there is no point in applying to so many, maybe just 15? I also don't want to disturb my rec letter writers so much.
I am not thinking about the money I have to spend on the application process

Thank you :)
Just my opinion: Do you really need to apply to 20 schools? I think it is very hard to be focused when applying to many schools. You also have to spend lots of money, and more importantly, you have to find referees willing to write 20 recommendation letters! 

I have read many articles and most of them suggest 6-8 school: 3 reach schools, 2 match schools, and 2 safe schools and that seems reasonable.

20 seems to be too much and a waste of your time and money.