some subjects are concerned with techniques or methods, others are concerned with specific topics or examples. number theory is the study of a specific example, the integers, and generalizations.
analysis is a technique, the method of approximation via limits, derivatives, integrals, series,...
analysis as a tool can be used to study number theory. e.g. in complex analysis, the zeroes and poles of holomorphic functions are discrete sets. so one can ask whether certain discrete sets can or cannot be the zeroes of a given holomorphic function.
such considerations led to the solution of fermats last theorem when frey asked whether there could be an elliptic curve defined by a function with zeroes constructed from the putative solutions of fermats equation.
riemann also constructed his zeta function from the discrete sequence of prime numbers and asked where the resulting zeroes of that function would lie?
dirichlet used similar considerations to analyze the behavior of a complex function constructed from sequences of primes and prove there must be infinitely many primes in certain arithmetic progressions.
i think the lesson is that not every one needs to know number theory, but a number theorist, along with everyone else, needs to know analysis. i.e. you may be missing out on a beautiful subject, but without number theory you can still do most other subjects, whereas without analysis you have trouble doing anything deep.
