I happen to know a lot of math (pure) and basically non of the applied stuff. People keep on telling me that Diffrencial Equations are the most important thing to know in math. The problem is that these people/teachers do not know and have no appreciation for Pure math. I keep on saying I wish to study Pure math because Applied is not fun. Just because something works out mathematically in Applied math does not mean it is actually true, opposite of pure math. Also, applied math problems are not fun, you just memorize the steps you have to to take to slove the problem and the next time the same problem appears you do the same thing but with different numbers. They keep on telling me I cannot get a job that involves pure math. I am in 12th Grade and next year I am probably going to go college. I wish to study pure math, it is an obession, and I do not wish to spend time on diffrencial equation problems because they are not fun and I do not think Pure math EVER uses them. Am I right? What do you think?
You are DEAD wrong; in applied math, you often need a great deal of creativity to determine what mathematical techniques to use or develop to fit the task at hand...and you DO need to prove that what you are doing is mathematically sound; in fact, the job of a certain type of applied mathematician (known as a mathematical physicist) is to determine whether the sleight-of-hand that theoretical physicists are fond of using is mathematically kosher, and a few times, a mathematical physicist can point out a fatal flaw in a mathematical model used by a theoretical physicist.
There are also, of course, people who do research in applied mathematics itself, and these are most properly known as applied mathematicians.
Your belief that "none of the applied stuff" is used in pure mathematics is only partially true; some of the theorems that have come to be vital to areas like differential geometry, like the local existence and uniqueness theorems for systems of ordinary differential equations, are considered to be part of "applied mathematics" even though they themselves are not all that "applied" in nature...also, much of the study of functional analysis deals with its application to partial differential equations and integral equations. As yet another example, graph theory has numerous connections with algebra and topology and computer science.
I second the comments by lewisje. Mathematics is one subject. The same standards of rigor apply everywhere (well, mostly). You need to know pure mathematics to succeed in applied mathematics, and it helps to know applied mathematics if you want to come up with interesting problems in pure mathematics. And there are not only techniques from "applied" math that are essential to "pure" math as lewisje points out, there are also results from "pure" math that turn out to be extremely relevant for "applied" problems. Examples are elementary number theory and public key cryptography; Fourier transforms and signal processing; finite fields and experimental design in statistics; fixed point theory and macroeconomics; and so on and so forth.