Difference between revisions of "Proof by contradiction"
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'''Euclid's proof of the infinitude of primes''' | '''Euclid's proof of the infinitude of primes''' | ||
| − | Assume there exists a finite number of [[primes]] <math>p_1, p_2,\ldots, p_n</math>. Let <math>N=p_1p_2p_3...p_n+1</math>. N is obviously not divisible by any of the known primes since it will leave a remainder of one upon division by any one of them. Thus, N must be divisible by some other prime not in our list, which contradicts the assumption that there is a finite number of primes. <math>\Box</math> | + | Assume there exists a finite number of [[prime|primes]] <math>p_1, p_2,\ldots, p_n</math>. Let <math>N=p_1p_2p_3...p_n+1</math>. N is obviously not divisible by any of the known primes since it will leave a remainder of one upon division by any one of them. Thus, N must be divisible by some other prime not in our list, which contradicts the assumption that there is a finite number of primes. <math>\Box</math> |
==See also== | ==See also== | ||
*[[Proof writing]] | *[[Proof writing]] | ||
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php How to Write a Solution] by [[Richard Rusczyk]] and [[Mathew Crawford]] | *[http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php How to Write a Solution] by [[Richard Rusczyk]] and [[Mathew Crawford]] | ||
Revision as of 21:28, 19 June 2006
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 utilized proof by contradiction include the irrationality of 
and the infinitude of primes. (If you know how, please make the following proofs look better). This technique usually works well on problems where not a lot of information is known, and thus we can create some using proof by contradiction.
Examples
Proof that the square root of 2 is irrational
Assume is rational, i.e. it can be expressed as a rational fraction of the form 
, where a and 
are two relatively prime integers. Now, since
, we have 
, or 
. Since 
is even, 
must be even, and since is even, so is . Let 
. We have
.
Since 
is even, 
is even, and since is even, so is a. However, two even numbers cannot be relatively prime, so cannot be expressed as a rational fraction; hence is irrational. 
Euclid's proof of the infinitude of primes
Assume there exists a finite number of primes 
. Let 
. N is obviously not divisible by any of the known primes since it will leave a remainder of one upon division by any one of them. Thus, N must be divisible by some other prime not in our list, which contradicts the assumption that there is a finite number of primes.
