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Prove that the following are irrationals : (i) (ii)  (iii) 

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CBSE/Class 10/Mathematics/UNIT I: Number systems/Real numbers/NCERT Solutions/Exercise 1.3

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(i) Assume that is Rational. Hence, is rational as a and b are integers. Hence is rational which contradicts the fact that is irrational. Therefore, our assumption was incorrect and is irrational. (ii) Assume that 7 is rational. where a and b are some integers. therefore, a and 7b are ratonal... read more

(i) Assume that  is Rational.

Hence, 

  is rational as a and b are integers.

Hence  is rational which contradicts the fact that  is irrational. 

Therefore, our assumption was incorrect and  is irrational. 

(ii) Assume that 7 is rational.

 where a and b are some integers.

therefore, 

a and 7b are ratonal and so is 

But this contradicts the fact that  is irrational.

Hence our assumption was incorrect and  is rational.

(iii)  Assume  as rational

Since a and b are integers,  is also rational and  should be rational as well.

This contradicts the fact that  is irrational and hence our assumption was incorrect.

 is irrational. 

 

 
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Whenever you find some prime numbers in root you can conclude that they are irrational.
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1) 1⁄√2 Rationalise this number 1 *√2⁄√2 *√2 = √2/2 √2 is irrational number and 2 is rational number division of rational number by irrational number is irrational . Hence 1/√2 is an irrational number. 2)7√5 7 is rational number √5... read more

1) 1⁄√2 

Rationalise this number 1 *√2⁄√2 *√2 

= √2/2 

√2 is irrational number and 2 is rational number division of rational number by irrational number is irrational . Hence  1/√2 is an irrational number.

2)7√5   

7 is rational number √5 is an irrational number product of rarional and irrational number is irrational number.

3)6+√5 

6 is a rational number √5 is irrational number sum of rational and irrational number is an irrational number.

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(i)Let is a rational number=. Squaring both sides ½=, So b^2=2a^2.So b^2 is divisible by 2 and b is divisible by 2. Let b=2c where c is an integer. Then b^2=4c^2, Hence 2a^2=4c^2. and a^2=2c^2.so a^2 is divisible by 2 and a is divisible by 2.This shows a and b have a common factor 2. But for a... read more

(i)Let is a rational number=. Squaring both sides ½=, So b^2=2a^2.So b^2 is divisible by 2 and b is divisible by 2. Let b=2c where c is an integer. Then b^2=4c^2, Hence 2a^2=4c^2. and a^2=2c^2.so a^2 is divisible by 2 and a is divisible by 2.This shows a and b have a common factor 2. But for a rational number a/b there cannot be a common factor other than 1.So this contradicts 1/√2 is rational. So it is an irrational number.

(ii) Let 7√5 is a rational number=a/b.  

then√5=a/7b is a rational number. But we know√5 is irrational. So 7√5 is irrational.

(iii)Let 6+√2 is a rational number=a/b

then√2=(a/b)-6=(a-6b)/6 is a rational number.

This contradicts √2 is irrational.

So 6+√2 is irrational.

 

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