Template:Proof

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This is a box for proofs. This calls Template:Databox but it automatically appends Template:Qed in the end of the contents and adds the page to Category:Pages with proofs.

Usage

You type:

{{Proof
| title=Proof that the root of any integer is either an integer or irrational
| contents=Assume <math>\sqrt[n]{m}</math> is {{nowrap|''p'' / ''q''}}, where {{nowrap|''p'', ''q'' ∈ ℤ<sup>+</sup>}} and {{nowrap|gcd(''p'', ''q'') {{=}} 1}}. Then 
{{nowrap|''p<sup>n</sup>'' / ''q<sup>n</sup>'' {{=}} ''m''}}, and {{nowrap|''p<sup>n''</sup> {{=}} ''mq<sup>n''</sup>}}. This means ''p'' is divisible by ''m''. Therefore, there exists some integer ''r'' such that {{nowrap|''p'' {{=}} ''mr''}}, so we now have {{nowrap|''m<sup>n</sup>r<sup>n</sup>'' {{=}} ''mq<sup>n</sup>''}}. Dividing both sides by ''m'' gives {{nowrap|''m''<sup>{{nowrap|''n'' − 1}}</sup>''r<sup>n''</sup> {{=}} ''q<sup>n</sup>''}}. This means that ''q'' must also be divisible by ''m'', which is a contradiction, since ''p'' and ''q'' were assumed to be relatively prime.
}}

You get:

Proof that the root of any integer is either an integer or irrational

Assume [math]\displaystyle{ \sqrt[n]{m} }[/math] is p / q, where p, q ∈ ℤ⁺ and gcd(p, q) = 1. Then pⁿ / qⁿ = m, and pⁿ = mqⁿ. This means p is divisible by m. Therefore, there exists some integer r such that p = mr, so we now have mⁿrⁿ = mqⁿ. Dividing both sides by m gives m^{n − 1}rⁿ = qⁿ. This means that q must also be divisible by m, which is a contradiction, since p and q were assumed to be relatively prime. [math]\displaystyle{ \square }[/math]

Note: Pipe characters (|) must be escaped as {{!}} or {{pipe}}, equals signs must be escaped as {{=}} (this may apply to tags as well, e.g. <div style{{=}}"…">), and other special characters and character sequences may need to be escaped accordingly, except for ones inside <nowiki>, <pre>, <math>, and <syntaxhighlight> tags.

See also

• Template:Theorem