What is a counterexample in math?

Answer 1

A counterexample is an example (usually of a number) that disproves a statement. When seeking to prove or disprove something, if a counter example is found then the statement is not true over all cases. Here's a basic and rather trivial example. I could say "There is no number greater than one million". Then you could say, "No! Take 1000001 for example". Because that one number is greater than one million my statement is false, and in that case 1000001 serves as a counterexample. In any situation, an example of why something fails is called a counterexample.