find two irrational number between root 2 & root 3
find two irrational between 9/11 and 5/7
find two rational number between root 3 & root 6

Asked by Sanjana Patro | 24th Jun, 2014, 09:33: PM

Expert Answer:

1. space F i n d space t w o space i r r a t i o n a l space n u m b e r space b e t w e e n space square root of 2 space a n d space square root of 3 C o n s i d e r space t h e space n u m b e r s space fraction numerator square root of 2 plus square root of 3 over denominator 2 end fraction space a n d space fraction numerator square root of 2 plus square root of 3 over denominator 4 end fraction S i n c e space square root of 2 space a n d space square root of 3 space a r e space i r r a t i o n a l comma space square root of 2 plus square root of 3 space i s space a l s o space i r r a t i o n a l. A n d space h e n c e space fraction numerator square root of 2 plus square root of 3 over denominator 2 end fraction space space i s space a l s o space i r r a t i o n a l. S i m i l a r l y comma space space i s space a l s o space i r r a t i o n a l. C l e a r l y comma space space space square root of 2 less than fraction numerator square root of 2 plus square root of 3 over denominator 4 end fraction less than fraction numerator square root of 2 plus square root of 3 over denominator 2 end fraction less than square root of 3 T h u s comma space t h e space t w o space i r r a t i o n a l space n u m b e r s space b e t w e e n comma space square root of 2 space a n d space square root of 3 space a r e space fraction numerator square root of 2 plus square root of 3 over denominator 2 end fraction space a n d space fraction numerator square root of 2 plus square root of 3 over denominator 4 end fraction. 2. space F i n d space t w o space i r r a t i o n a l space b e t w e e n space 9 over 11 space a n d space 5 over 7 F i r s t space l e t space u s space t h e space o r d e r space o f space r e l a t i o n space b e t w e e n space 9 over 11 space a n d space 5 over 7 9 over 11 equals 63 over 77 5 over 7 equals 55 over 77 T h u s comma space 5 over 7 less than 9 over 11 N o w space l e t space u s space f i n d space t h e space d i f f e r e n c e space b e t w e e n space 9 over 11 space a n d space 5 over 7 9 over 11 space minus 5 over 7 equals 8 over 77 N o w space t o space m a k e space 8 over 77 space i r r a t i o n a l comma space space l e t space u s space d i v i d e space t h i s space n u m b e r space b y space square root of 2 T h u s comma space w e space h a v e space fraction numerator 8 over denominator 77 square root of 2 end fraction. A d d space t h i s space d i f f e r e n c e space t o space 5 over 7 T h u s comma space w e space h a v e 5 over 7 plus fraction numerator 8 over denominator 77 square root of 2 end fraction S i m i l a r l y comma space w e space c a n space d i v i d e space t h e space n u m b e r space space b y space square root of 3 T h u s comma space w e space h a v e space fraction numerator 8 over denominator 77 square root of 3 end fraction A d d space t h i s space d i f f e r e n c e space t o space 5 over 7 T h u s comma space w e space h a v e 5 over 7 plus fraction numerator 8 over denominator 77 square root of 3 end fraction T h e r e f o r e comma space t h e space i r r a t i o n a l space n u m b e r s space b e t w e e n space 9 over 11 space a n d space 5 over 7 space a r e : 5 over 7 plus fraction numerator 8 over denominator 77 square root of 2 end fraction comma space 5 over 7 plus fraction numerator 8 over denominator 77 square root of 3 end fraction 3. space F i n d space t w o space r a t i o n a l space n u m b e r space b e t w e e n space square root of 3 space end root space a n d space square root of 6 : square root of 3 almost equal to 1.732050.... square root of 6 equals 2.449489.... T h u s space l e t space u s space t a k e space t h e space n u m b e r s space 1.8 space a n d space 1.9 1.8 equals 18 over 10 space a n d space 1.9 equals 19 over 10 T h u s comma space 18 over 10 space a n d space 19 over 10 space a r e space t w o space r a t i o n a l space n u m b e r s space b e t w e e n space square root of 3 space end root space a n d space square root of 6 square root of 3 space end root less than 18 over 10 less than 19 over 10 less than space square root of 6 I n space f a c t space t h e r e space a r e space i n f i n i t e l y space m a n y space r a t i o n a l space n u m b e r s space b e t w e e n space square root of 3 space end root space a n d space square root of 6.

Answered by Vimala Ramamurthy | 25th Jun, 2014, 11:00: AM