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Question Number 166910 by rexford last updated on 02/Mar/22

g i v e n t h a t i s p r i m e, p r o o f t h a t \sqrt{p} i s

i r r a t i o n a l

Commented by rexford last updated on 02/Mar/22

t h a n k y o u

Answered by mr W last updated on 02/Mar/22

a s s u m e \sqrt{p} = r a t i o n a l = \frac{a}{b} w i t h g c d (a, b) = 1

\Rightarrow p = \frac{a^{2}}{b^{2}} \Rightarrow p i s a p e r f e c t s q u a r e

\Rightarrow c o n t r a d i c t i o n

a c t u a l l y \sqrt{p} i s a l w a y s i r r a t i o n a l, i f

p i s n o t a p e r f e c t s q u a r e . t h e r e f o r e

\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7}, \sqrt{8}, \sqrt{10}, \sqrt{11}, \sqrt{12},

\sqrt{13}, \sqrt{14}, \sqrt{15}, \sqrt{17}, \sqrt{18}, \sqrt{19}, \dots a r e a l l

i r r a t i o n a l n u m b e r s .

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