Proof by contradiction

Revision as of 20:13, 17 June 2006 by Quantum leap (talk | contribs)

Proof by contradiction is an indirect type of proof that assumes the proposition (that which is to be proven) is true and shows that this assumption leads to an error, logically or mathematically. Famous results which utilised proof by contradiction include the irrationality of $\sqrt{2}$ and the infinitude of primes.


Assume $\sqrt{2}$ is rational, i.e. it can be expressed as a rational fraction of the form $\frac{b}{a}$, where a and $b$ are two relatively prime integers. Now, $\sqrt{2}=\frac{b}{a}$ $2=\frac{b^2}{a^2}$ $b^2=2a^2$ Since $2a^2$ is even, $b^2$ must be even, and since $b^2$ is even, so is $b$. Let $b=2c$. We have, $4c^2=2a^2$ $a^2=2c^2$ Since 2c^2 is even, $a^2$ is even, and since $a^2$ is even, so is a. However, two even numbers cannot be relatively prime, so $\sqrt{2}$ cannot be expressed as a rational fraction; hence $\sqrt{2}$ is irrational.

Euclid's proof of the infinitude of primes: Assume there exist a finite number of primes $p_1, p_2, ..., p_n$. Let $N=p_1p_2p_3...p_n+1$. By the original assumption, N is not in the set of primes, so it is composite and divisible by some prime p_i. If p_i|N and $p_i|p_1p_2...p_n, p_i$ must also divide $1$. However, no prime number evenly divides $1$, so our original assumption that there are only a finite number of primes is false.