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'''Reductio ad absurdum''', also called [[proof]] by contradiction, is a method of mathematical proof. It involves assuming the opposite of what one is trying to prove, and showing that this would lead to a contradiction. An example of this is [[Euclid]]'s proof of the infinitude of the [[Prime number|primes]].
'''Reductio ad absurdum''', also called [[proof]] by contradiction, is a method of mathematical proof. It involves assuming the opposite of what one is trying to prove, and showing that this would lead to a contradiction. It works by the law of the excluded middle. The proof typically follows this structure:
#Create an initial assumption
#Follow a series of axiomatically valid steps
#Reach a contradiction
#Therefore the initial assumption is incorrect
An example of this is [[Euclid]]'s proof of the infinitude of the [[Prime number|primes]]:
#Assume there are finitely many primes
#Take the product of all primes and call it ''N''. Since ''N+1'' is not in our finite set of primes, it must be composite
#By the [[fundamental theorem of arithmetic]], ''N+1'' has a prime factorization. But ''N+1'' is not divisible by any of the previous primes
#Since ''N+1'' is composite, there must be a prime missing from our set of primes. But this set contains all primes
#Therefore, our initial assumption ("there are finitely many primes") is invalid
[[Category:Mathematics]]
[[Category:Mathematics]]