|
|
| (13 intermediate revisions by 3 users not shown) |
| Line 1: |
Line 1: |
| <includeonly>{{Databox|{{{title|Proof}}}|<br/>{{{contents}}} {{qed}}}}[[Category:Articles with proofs]]</includeonly><noinclude> | | <includeonly>{{Databox|{{{title|Proof}}}|{{trim|{{{contents}}}}} {{qed}}}}[[Category:Pages with proofs]]</includeonly><noinclude> |
| Infobox for proofs.
| | {{documentation}} |
|
| |
|
| You type:
| | [[Category:Templates]] |
| | |
| <pre>
| |
| {{PAGENAME}}|title=Proof that <math>\sqrt{2}</math> is irrational|
| |
| contents=Assume <math>\sqrt{2}</math> is ''p''/''q'', where ''p'', ''q'' ∈ ℤ<sub>>0</sub> and gcd(''p'', ''q'') = 1. Then ''p''<sup>2</sup>/''q''<sup>2</sup> = 2, and ''p''<sup>2</sup> = 2''q''<sup>2</sup>. Since ''p''<sup>2</sup> is even, and gcd(''p'', ''q'') = 1, gcd(''p''<sup>2</sup>, ''q''<sup>2</sup>) = 1, ''q''<sup>2</sup> must be odd. Hence ''p''<sup>2</sup> ≡ 2 (mod 4), which is impossible because the square of an integer must always be congruent to 0 or 1 (mod 4).
| |
| }}
| |
| </pre>
| |
| | |
| You get:
| |
| | |
| {{{{PAGENAME}}|title=Proof that <math>\sqrt{2}</math> is irrational|
| |
| contents=Assume <math>\sqrt{2}</math> is ''p''/''q'', where ''p'', ''q'' ∈ ℤ<sub>>0</sub> and gcd(''p'', ''q'') = 1. Then ''p''<sup>2</sup>/''q''<sup>2</sup> = 2, and ''p''<sup>2</sup> = 2''q''<sup>2</sup>. Since ''p''<sup>2</sup> is even, and gcd(''p'', ''q'') = 1, gcd(''p''<sup>2</sup>, ''q''<sup>2</sup>) = 1, ''q''<sup>2</sup> must be odd. Hence ''p''<sup>2</sup> ≡ 2 (mod 4), which is impossible because the square of an integer must always be congruent to 0 or 1 (mod 4).
| |
| }}
| |
| | |
| Note: To include the character <code>|</code>, it must be escaped as <code><nowiki>{{!}}</nowiki></code>, unless it appears inside preformatted text (such as a <nowiki><syntaxhighlight></nowiki> tag).
| |
| | |
| [[Category:Articles with proofs|*]] | |
| </noinclude> | | </noinclude> |