Introduction: The process which takes known results such as axioms, definitions,
and previously proven theorems, and follows the principles of logic to derive new results, is called deductive reasoning. This process has been employed in the preceding sections and in previous mathematics courses to prove various assertions. There is, however, another reasoning process used to justify an assertion by a kind of experimental method. In this process the experiment is performed numerous times, under carefully controlled conditions. If the same result is consistently found, then "inductively" the conclusion is, "under the prescribed experimental conditions, that result will always be found." Of course, such a conclusion cannot be 100% certain. This is one of the kinds of reasonings used by scientists to advance knowledge in their disciplines and is appropriate for their work. The experimental method is unsuitable, however, for proving mathematical assertions. The problem with using this type of induction in mathematics, as well as the sciences, is that the same result may have been obtained in each experiment due to an unknown or accidental influence. In fact, the example in Chapter 5 regarding the primeness of n2 — n + 41 illustrates that this type of inductive reasoning is unsatisfactory for proving mathematical assertions.