Definition
Proof by contradiction
Proof by contradiction proves a claim by showing that assuming it is false leads to an impossibility.
What it means
Start by pretending the claim is wrong. Follow that pretend idea using facts you already trust. If it forces two statements that cannot both be true, the pretend idea has failed. That leaves the original claim as the only possible conclusion.
Remember it
Try the opposite; if it crashes into an impossibility, the original claim survives.
Example
Claim: there is no greatest whole number. Suppose there were a greatest whole number, G. Then G + 1 would also be a whole number, and it would be greater than G. That contradicts the choice of G as greatest, so no greatest whole number exists.
Watch for
A result that merely seems strange is not a contradiction; the assumption must force facts that cannot both hold.
Common mix-up
A counterexample disproves a claim by finding one failure. Proof by contradiction proves a claim by making its opposite impossible.
Exact meaning
Proof by contradiction is a proof method in which the negation of a claim is assumed and valid deductions from that assumption lead to a contradiction; therefore the assumption is false and the original claim is true.