section_path: Methods of proof/Proof by contradiction
section_name: Proof by contradiction
paragraph_text: Proof by contradiction
  
  In proof by contradiction, also known by the Latin phrase reductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, a logical contradiction occurs, hence the statement must be false. A famous example involves the proof that 2 {\displaystyle {\sqrt {2}}} is an irrational number:
  
  To paraphrase: if one could write 2 {\displaystyle {\sqrt {2}}} as a fraction, this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator.