show-that-3-is-an-irrarional-number-

Question Number 158567 by MathsFan last updated on 06/Nov/21

show that \sqrt{3} is an

irrarional number

Commented by mr W last updated on 06/Nov/21

a s s u m e \sqrt{3} = \frac{p}{q} w i t h g c d (p, q) = 1

p^{2} = 3 q^{2}

\Rightarrow p = 3 k

q^{2} = 3 k^{2}

\Rightarrow q = 3 h

\Rightarrow g c d (p, q) = 3

\Rightarrow c o n t r a d i t i o n w i t h g c d (p, q) = 1

\Rightarrow \sqrt{3} \neq \frac{p}{q}

Commented by MathsFan last updated on 06/Nov/21

thank you sir