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how can you say that there are more irrational numbers between two consecutive numbers than rational numbers?prove.
Asked by vaishnavi | 25 Apr, 2016, 02:23: PM
answered-by-expert Expert Answer
R a t i o n a l space n u m b e r s space a r e space n u m b e r s space t h a t space c a n space b e space w r i t t e n space i n space t h e space f o r m space p over q comma space w h e r e space q not equal to 0. I r r a t i o n a l space n u m b e r s space a r e space n u m b e r s space t h a t space c a n n o t space b e space e x p r e s s e d space i n space t h e space f o r m space p over q comma space q not equal to 0 R e a l space n u m b e r s space i s space t h e space s e t space w h i c h space c o m p r i s e s space o f space b o t h space r a t i o n a l space a n d space i r r a t i o n a l space n u m b e r s. A n d space t h e r e space i s space n o space s u c h space n u m b e r space s u c h space t h a t space i t space c a n space b e space b o t h space r a t i o n a l space a n d space i r r a t i o n a l. T h a t comma space i s space i n t e r s e c t i o n space o f space r a t i o n a l space n u m b e r s space a n d space i r r a t i o n a l space n u m b e r s space i s space a n space e m p t y space s e t. S o space a space n u m b e r space c a n space b e space e i t h e r space r a t i o n a l space o r space i r r a t i o n a l. S i n c e space t h e r e space i s space a space c o n s t r a i n t space f o r space a space n u m b e r space t o space b e space r a t i o n a l comma space t h e r e space a r e space o n l y space c o u n t a b l e space n u m b e r space o f space r a t i o n a l s. A space s e t space i s space c o u n t a b l e space e i t h e r space i f space i t space i s space f i n i t e comma space o r space i t space i s space i n f i n i t e space a n d space t h e r e space i s space a space o n e minus o n e space c o r r e s p o n d e n c e space b e t w e e n space t h e e l e m e n t s space o f space t h e space s e t space a n d space t h e space s e t space o f space n a t u r a l space n u m b e r s. I t space i s space l i k e space t h i s comma space b e t w e e n space t h e space n u m b e r space t h e r e space a r e space s o space m a n y space r a t i o n a l space n u m b e r s W e space c a n space c o n s t r u c t space s o m e space r a t i o n a l space n u m b e r s space l i k e space t h i s colon 1 over 1 comma space 1 half comma space 1 third comma space 1 fourth comma space 1 fifth comma space 1 over 6 comma space 1 over 7 comma space 1 over 8 comma.. B e t w e e n space 2 space a n d space 3 comma space w e space h a v e comma 2 over 1 comma space up diagonal strike 2 over 2 end strike comma space 2 over 3 comma up diagonal strike space 2 over 4 end strike comma space 2 over 5 comma space up diagonal strike 2 over 6 end strike comma space 2 over 7 comma space up diagonal strike 2 over 8 end strike comma.. S o m e space o f space t h e space f r a c t i o n s space a r e space s t r i k e d space o u t space w h i c h space a r e space h a v i n g space e q u i v a l e n t space f r a c t i o n s. S o space i n space t h i s space w a y comma space w e space c a n space s a y space t h a t space t h e r e space a r e space u n i q u e space r a t i o n a l space n u m b e r s space w h i c h space a r e space c o u n t a b l y space i n f i n t e. B u t space b e t w e e n space e v e r y space t w o space r a t i o n a l space n u m b e r s comma space t h e r e space a r e space i n f i n i t e space n u m b e r space o f space i r r a t i o n a l space n u m b e r s space a n d h e n c e space t h e r e space a r e space i n f i n i t e space n u m b e r space o f space i r r a t i o n a l space n u m b e r s space i n space t o t a l.
Answered by Vimala | 26 Apr, 2016, 09:41: AM

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