Proving the easy

Proofs loom large in elementary number theory for two reasons.  First, many of the theorems (including those contained in some of the problems in this book) are so simple to state and so easy to understand that one is deceived into thinking that they must also be easy to prove, and this is not always the case.  Also, there is much more variety in the kinds of reasoning invoked than is the case in more elementary mathematics, the latter, more or less by definition, being restricted to material that is both useful and relatively easy to master.  It is not merely because number theory is not useful in commerce and engineering that it is not customarily learned before calculus, as it logically could be.  The variety of proof techniques sometimes seems so large that students regard number theory as a “bag of tricks,” but of course this is a matter of familiarity.  What is a trick the first time one meets it is a device the second time and a method the third time.

William J. LeVeque, Fundamentals of Number Theory

Now, extrapolate to other fields of endeavor.

Published in: on October 18, 2009 at 8:54 am  Leave a Comment  

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