how can i prove that  root 7 is an irrational no with method

 

Asked by Prachi Jha | 10th May, 2014, 07:43: PM

Expert Answer:

I t space c a n space b e space p r o v e d space w i t h space h e l p space o f space c o n t r a d i c t i o n space t e c h n i q u e  L e t s space a s s u m e space square root of 7 space i s space a space r a t i o n a l space n u m b e r space w h i c h space c a n space b e space e x p r e s s e d space a s space r a t i o space o f space t w o space i n t e g e r s space p space a n d space q comma space w h e r e space G C F left parenthesis p comma space q right parenthesis space equals space 1 comma space q not equal to 1  square root of 7 equals p over q space space space space space space  rightwards double arrow 7 equals p squared over q squared space space space space space space space space space space space space space space space open square brackets S i n c e space G C F left parenthesis p comma space q right parenthesis equals 1 comma space t h e n space G C F left parenthesis p squared comma space q squared right parenthesis equals 1 space close square brackets  rightwards double arrow p squared equals 7 q squared  rightwards double arrow p squared over q squared equals fraction numerator 7 q squared over denominator q squared end fraction  w e space k n o w space t h a t space q not equal to 1 comma space t h e r e f o r e comma space p squared over q squared space i s space n o t space i n space i t s space s i m p l e s t space f o r m rightwards double arrow square root of 7 not equal to r a t i o n a l comma space i t s space i r r a t i o n a l space n u m b e rI t space c a n space b e space p r o v e d space w i t h space h e l p space o f space c o n t r a d i c t i o n space t e c h n i q u e  L e t s space a s s u m e space square root of 7 space i s space a space r a t i o n a l space n u m b e r space w h i c h space c a n space b e space e x p r e s s e d space a s space r a t i o n space o f space t w o space i n t e g e r s space p space a n d space q comma space w h e r e space G C F left parenthesis p comma space q right parenthesis space equals space 1 comma space q not equal to 1  square root of 7 equals p over q space space space space space space  rightwards double arrow 7 equals p squared over q squared space space space space space space space space space space space space space space space open square brackets S i n c e space G C F left parenthesis p comma space q right parenthesis equals 1 comma space t h e n space G C F left parenthesis p squared comma space q squared right parenthesis equals 1 space close square brackets  rightwards double arrow p squared equals 7 q squared  rightwards double arrow p squared over q squared equals fraction numerator 7 q squared over denominator q squared end fraction  w e space k n o w space t h a t space q not equal to 1 comma space t h e r e f o r e comma space p squared over q squared space i s space n o t space i n space i t s space s i m p l e s t space f o r m rightwards double arrow square root of 7 not equal to r a t i o n a l comma space i t s space i i r a t i o n a l space n u m b e r

Answered by Avinash Soni | 11th May, 2014, 11:24: AM