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CBSE Class 10 Answered

prove ?5 is irrational number
Asked by appusingh0840 | 29 Oct, 2023, 13:14: PM
answered-by-expert Expert Answer
Let √5 is a rational number.
 
Then we can write , √5 = p/q
 
where p and q are integers, q ≠ 0 and p and q are co-prime
 
( i..e. p and q do not have any common factor othr than 1 )
 
begin mathsize 14px style square root of 5 space equals space p over q end style
By squaring both sides we get ,  begin mathsize 14px style p squared over 5 equals space q squared end style  .......................(1)

From above expression , we get 5 is factor of p .
 
hence we write , p = 5 c   and    p2 = 25 c2
 
Hence eqn.(1) becomes
 
begin mathsize 14px style fraction numerator 25 space c squared over denominator 5 end fraction space equals space q squared end style
begin mathsize 14px style q squared over 5 space equals space c squared end style
Hence q is divisible by 5 , or 5 is a factor of q .
 
Hence we get 5 is  afctor of p and aloso factor of q .
 
But we have started with the assumption p and q are co-prime .
 
Hence √5 ≠ p/q if p and q are integers
 
Hence √5 is not a rational number and it is irrational number .
Answered by Thiyagarajan K | 29 Oct, 2023, 16:18: PM

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