Template:Proof: Difference between revisions

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2=Assume <math>\sqrt{2}</math> is ''p''/''q'', where ''p'', ''q'' ∈ ℤ>0 and gcd(''p'', ''q'') = 1. Then ''p''2/''q''2 = 2, and ''p''2 = 2''q''2. Since ''p''2 is even, and gcd(''p'', ''q'') = 1, gcd(''p''2, ''q''2) = 1, ''q''2 must be odd. Hence ''p''2 ≡ 2 (mod 4), which is impossible because the square of an integer must always be congruent to 0 or 1 (mod 4). {{qed}}}}

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 =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). {{qed}}}}
-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).
+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> or <nowiki><math></nowiki> tag).
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