Proof by Contradiction

In this section, we will learn about using contradiction for mathematical proofs. Proof by contradiction is a method of proof where you first assume the claim you wish to prove is false, and then prove through a series of logical deductions that this assumption results in a contradictory claim. If this happens, and we have made no errors, this assumption that the claim was false must have been incorrect. Thus, the claim must be true.

While this idea may make sense abstractly and we see the proof method is confirmed by formal logic, the authors believe the method is best demonstrated by examples if you hope to build some intuitive understanding of the approach, learn when it is likely to be effective, and construct your own mathematical proofs.

First, let's review some ideas we all probably learned in primary school. Recall a real number x is called rational if it can be written as a ratio:

Here, a and b ≠ 0 are relatively...