CBSE Class 9 Answered

I couldn't understand the Contradiction method of under roots ? Can you please explain with some examples ?

Asked by amywani45 | 22 Apr, 2015, 04:25: PM

Expert Answer

$C o n t r a d i c t i o n space m e t h o d space i s space u s e d space t o space p r o v e space t h a t space n u m b e r s space l i k e space square root of 2 comma space square root of 3 space a r e space i r r a t i o n a l. F o r space e x a m p l e comma space t o space p r o v e space t h a t space square root of 2 space i s space a n space i r r a t i o n a l space n u m b e r comma space w e space s t a r t space b y space a s s u m i n g space t h a t space i t apostrophe s space r a t i o n a l. S u p p o s e space square root of 2 space i s space a space r a t i o n a l space n u m b e r space. space T h e n space w e space c a n space w r i t e square root of 2 equals p over q comma space w h e r e space p space a n d space q space a r e space t w o space c o minus p r i m e space i n t e g e r s comma space a n d space q not equal to 0. S o comma space a n y space o n e space o f space p space o r space q space i s space a n space e v e n space n u m b e r. S q u a r i n g space b o t h space s i d e s comma space w e space g e t 2 equals p squared over q squared rightwards double arrow p squared equals 2 q squared F r o m space t h e space a b o v e space e q u a t i o n comma space p squared space i s space a n space e v e n space n u m b e r. space H e n c e comma space p space i s space a l s o space a n space e v e n space n u m b e r. space left parenthesis o n l y space s q u a r e space o f space a n space e v e n space n u m b e r space c a n space b e space e v e n right parenthesis H e n c e comma space w e space c a n space w r i t e space p space a s space 2 k comma space w h e r e space k space i s space s o m e space i n t e g e r. S u b s t i t u t i n g space t h i s space i n space t h e space e q u a t i o n space p squared equals 2 q squared comma space w e space g e t 4 k squared equals 2 q squared rightwards double arrow q squared equals 2 k squared H e n c e comma space q squared space i s space a l s o space a n space e v e n space n u m b e r comma space w h i c h space i m p l i e s space apostrophe q apostrophe space i s space a l s o space a n space e v e n space n u m b e r. space W e space g e t space b o t h space p space a n space q space a r e space e v e n space n u m b e r s comma space w h i c h space C O N T R A D I C T S space o u r space i n i t i a l space a s s u m p t i o n space t h a t space p space a n d space q space a r e space c o minus p r i m e. H e n c e comma space square root of 2 space c a n space n o t space b e space w r i t t e n space i n space space f o r m. space H e n c e comma space i t apostrophe s space n o t space r a t i o n a l. I t apostrophe s space a n space i r r a t i o n a l space n u m b e r. T h a t space c o m p l e t e s space t h e space p r o o f space t h a t space square root of 2 space i s space a n space i r r a t i o n a l space n u m b e r space b y space m e t h o d space o f space c o n t r a d i c t i o n.$