[tex]\displaystyle\bf\\presupunem~ca~\sqrt{13} \in Q \implies exista~a~si~b \in Z_{+}, ~ b\neq 0,~, (a,b)=1 , astfel~incat~\sqrt{13} = \frac{a}{b}.\\[/tex][tex]\displaystyle\bf\\\sqrt{13} = \frac{a}{b} ~ \HUGE{|}^2 \Leftrightarrow 13 = (\frac{a}{b})^2 \Leftrightarrow 13 = \frac{a^2}{b^2} \Leftrightarrow 13 \cdot b^2 = a^2.\\in~mod~clar~a~va~fii~divizibil~cu~13.[/tex]
[tex]\displaystyle\bf\\\ implies exista~a_0~astfel~incat~a=13a_0,~inlocuim.\\13b^2=169a_0^2 | : 13 \Leftrightarrow b^2=13a_0^2 \implies ~b~va~fi~divizibil~cu~13,~dar~si~a~este~divizibil~cu~13.[/tex][tex]\displaystyle\bf\\contradictie~cu~(a,b)=1,~deci~\sqrt{13} \in R-Q.[/tex]