Mathematics distinguishes itself from the other sciences in that it is built upon a set of axioms and definitions, on which all subsequent theorems rely. All theorems can be derived, or proved, using the axioms and definitions, or using previously established theorems. By contrast, the theories in most other sciences, such as the Newtonian laws of motion in physics, are often built upon experimental evidence and can never be proved to be true.
It is therefore insufficient to argue that a mathematical statement is true simply by experiments and observations. For instance, Fermat (1601–1665) conjectured that when n is an integer greater than 2, the equation x n + y n = z n admits no solutions in positive integers. Many attempts by mathematicians in finding a counter-example (i.e. a set of positive integer solution) ended up in failure. Despite that, we cannot conclude that Fermat’s conjecture was true without a rigorous proof. In fact, it took mathematicians more than three centuries to find the proof, which was finally completed by the English mathematician Andrew Wiles in 1994.
To conclude or even to conjecture that a statement is true merely by experimental evidence can be dangerous. For instance, one might conjecture that n 2 − n + 41 is prime for all natural numbers n. One can easily verify this: when n = 1, n 2 − n + 41 = 41 is prime; when n = 2, n 2 − n + 41 = 43 is prime, and so on. Even if one continues the experiment until n = 10, or even n = 20, one would not be able to find a counter-example. However, it is easy to see that the statement is wrong, for when n = 41 the expression is equal to 412 which definitely is not prime.
While experimental evidence is insufficient to guarantee the truthfulness of a statement, it is often not possible to verify the statement for all possible cases either. For instance, one might conjecture that 1 + 3 + 5 + + (2n − 1) = n 2 for all natural numbers n. Of course one easily verifies that the statement is true for the first few (even the first few hundreds or even thousands of cases if one bothers to do so) values of n. Yet we cannot conclude that the statement is true. Maybe it will fail at some unattempted values, who knows? It is not possible to verify the statement for all possible values of n since there are infinitely many of them. So how can we verify the statement? A powerful tool is mathematical induction.
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2. The Basic Principle
An analogy of the principle of mathematical induction is the game of dominoes. Suppose the dominoes are lined up properly, so that when one falls, the successive one will also fall. Now by pushing the first domino, the second will fall; when the second falls, the third will fall; and so on. We can see that all dominoes will ultimately fall.
So the basic principle of mathematical induction is as follows. To prove that a statement holds for all positive integers n, we first verify that it holds for n = 1, and then we prove that if it holds for a certain natural number k, it also holds for k + 1 . This is given in the following.
Theorem 2.1. (Principle of Mathematical Induction)
Let S (n) denote a statement involving a variable n. Suppose (1)
S (1) is true;
(2) if S (k ) is true for some positive integer k, then S (k + 1) is also true. Then S (n) is true for all positive integers n.
Prove that 1 + 3 + 5 +
+ (2n − 1) = n 2 for all natural numbers n.
We shall prove the statement using mathematical induction.
Clearly, the statement holds when n = 1 since 1 = 12 .
Suppose the statement holds for some positive integer k. That is, 1 + 3 + 5 +
+ (2k − 1) = k 2 .
Consider the case n = k + 1 .
By the above assumption (which we shall call the induction hypothesis), we have 1+ 3 + 5 +
+ [ 2(k + 1) − 1] = [1 + 3 + 5 +
+ (2k − 1) ] + (2k + 1)
= k 2 + (2k + 1)
= (k + 1) 2
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