The book The Unreasonable effectiveness of number theory mentions amongst other things that number theory crops up in stability questions in dynamical systems, e.g. the small-divisor problem in KAM-theory where diophantine approximation is used.
Another book with several applications is: Number Theory: An Introduction to Mathematics whose blurb contains "As a source for information on the 'reach' of number theory into other areas of mathematics, it is an excellent work."
