In mathematics, as in science, there are two general methods by which we can arrive at new results. One, deduction, involves the assumption of a set of axioms from which we deduce other statements, called theorems, according to prescribed rules of logic. This method is essentially that used in standard courses in Euclidean geometry.

The second method, induction, involves the guessing or discovery of general patterns from observed data. While in most branches of science and mathematics the guesses based on induction may remain merely conjectures, with varying degrees of probability of correctness, certain conjectures in mathematics which involve the integers frequently can be proved by a technique of Pascal called mathematical induction. Actually, this technique in not induction, but is rather an aid in proving conjectures arrived at by induction.

THE PRINCIPLE OF MATHEMATICAL INDUCTION: A statement concerning positive integers is true for all positive integers if (a) it is true for 1, and (b) its being true for any integer k implies that it is true for the next integer k + 1.

If one replaces (a) by (a'),"it is true for some integer s," then (a') and (b) prove the statement true for all integers greater than or equal to s. Part (a) gives only a starting point; this starting point may be any integer - positive, negative, or zero.

Let us see if mathematical induction is a reasonable method of proof of a statement involving integers n. Part (a) tells us that the statement is true for n = 1. Using (b) and the fact that the statement is true for 1, we obtain the fact that it is true for the next integer 2. Then (b) implies that it is true for 2 + 1 = 3. Continuing in this way, we would ultimately reach any fixed positive integer.
 

Monday, April 27, 1998