prove-that-2-is-irrationl-number-

Question Number 45712 by malwaan last updated on 15/Oct/18

prove that :

\sqrt{2} is irrationl number

Commented by maxmathsup by imad last updated on 15/Oct/18

l e t s u p p o s e \sqrt{2} = \frac{p}{q} w i t h p a n d q i n t e g r s n n a t u r a l a n d Δ (p, q) = 1 \Rightarrow

2 = \frac{p^{2}}{q^{2}} \Rightarrow p^{2} = 2 q^{2} \Rightarrow 2 / p^{2} \Rightarrow 2 / p \Rightarrow \exists m \in N / p = 2 m \Rightarrow 4 m^{2} = 2 q^{2} \Rightarrow q^{2} = 2 m^{2} \Rightarrow

2 / q^{2} \Rightarrow 2 / q \Rightarrow 2 \in D_{p} \cap D_{q} b u t t h i s i s i m p o s s i b l e b e c a u s e Δ (p, q) = 1

f i n a l l y \sqrt{2} \notin Q .

Commented by malwaan last updated on 15/Oct/18

thank you sir

Commented by maxmathsup by imad last updated on 16/Oct/18

y o u a r e w e l c o m e s i r .