Prove that √5 is irrational. | Class 10 Mathematics Chapter Real Numbers, Real Numbers NCERT Solutions

Question:

Prove that √5 is irrational.

Answer:

Let us consider √5 is a rational number.
Therefore, √5 = p
q, where p and q are integers and q ≠ 0
If pa and q have any common factor ,then dividing by that common factor,
We have, √5 = a
b, where a and b are co primes.
a = √5b
On squaring, a² = 5b²                ...........   (1)
Due to the presence of 5 on RHS, we say that 5 is a factor of a².

5 divides a²
Since, a is prime , therefore 5 divides a                                   [ theorem ]
Therefore , a = 5k , where k is an integer.
Putting a = 5k in (1) , we have
25k² = 5b² 

b² = 5k²
This shows that 5divides b². But b is a prime no and so 5 divides b also.
Thus 5 is a common factor of a and b . This is contradiction to the fact that
a and b have no common factor other than one.
Thus our consideration is wrong and so √ 5 is not a rational number.
Hence,  proved.

 


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Welcome to the NCERT Solutions for Class 10 Mathematics - Chapter . This page offers a step-by-step solution to the specific question from Excercise 3 , Question 1: Prove that √5 is irrational.....