Prove that the following are irrationals | Class 10 Mathematics Chapter Real Numbers, Real Numbers NCERT Solutions

Question:

Prove that the following are irrationals:

Answer:

(i) Let us assume 1/2 is a rational number.

1/2 = p/q , where q ≠ 0 and p and q are co primes.

On reciprocal,

√2 = qp                                 ................(1)

Since, q and p are integers and q/p is also a rational number
As we know √2 is an irrational number.
From (1) 

√2 ≠ q/p

Thus our assumption is wrong 1/2 is not a rational number.
Hence, proved 

 

(ii) Let us suppose 7√5 is a rational number.
7√5 = p/q, where p and q are co primes and q ≠ 0
On solving , √5 = (p/
q)7                .....................(1)

Since p, q and 7 integers and (p/q)is also a rational number.
And we know √5 is an irrational number.
From (1) 

√5 ≠ (p/q) / 7

So our supposition Is wrong 7√5 is not a rational number.
Hence, proved.

 

(iii) Let us suppose 6 + √2 is a rational number.

6 + √2 = a/b, where a, b are co primes and b ≠ 0.
On solving,

√2 = a/- 6                                  .....................(1)

Since a, b and 6 are integers and a/- 6 is also a rational number.
And we know that √2 is an irrational number.
From (1)

√2 ≠ a/- 6

Thus our Superposition is wrong 6√2 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 3: Prove that the following are irrationals:....