NCERT Class 10 Solutions: Real Numbers (Chapter 1) Exercise 1.3 –Part 1

Q-1 Prove that 5 is irrational.


GIven the Real number table and prove the number are irrational

Given the Real Number Tables

GIven the Real number table and prove the number are irrational

Let us assume, to the contrary, that 5 is rational.

That is, we can find integers a and b (0) such that 5=ab

Suppose a and b have a common factor other than 1, then we can divide by the common factor and assume that a and b are co-prime.

Therefore b5=a

Squaring on both sides 5b2=a2 (Equation 1)

The above implies that a2 is divisible by 5 and also a is divisible by 5.

Therefore we can write that a=5f for some integer f.

Substituting in equation 5b2=(5f)2

5b2=25f2 Or b2=5f2

b2 Is divisible by 5 which mean b is also divisible by 5.

Therefore a and b have 5 as a common factor.

This contradicts the fact that a and b are co-prime.

We arrived at the contradictory statement as above since our assumption 5 is not correct.

Hence, we can conclude that 5 is irrational.

Q-2 Prove that 3+25 is irrational.


If possible let a=3+25 be a rational number.

Squaring both side



5=a22912 (Equation 1)

Since 'a' is a rational number that expression a22912 is also rational number.

5 Is a rational number

This is contradiction. Hence, 3+25 is irrational.

Hence proved

Q-3 Prove that the following are irrationals:

  1. 12

  2. 75

  3. 6+2


  1. 12


    Let a=(12)2 be a rational number.


    2a is rational number since product of two rational numbers is a rational number.

    Which will imply that 2 is a rational number. But it is a contradiction since 2 is an irrational number.

    Therefore our assumption is wrong

    Therefore 12 is irrational

  2. 75

    Let a=75 be a rational number.


    Here, a7 is a rational number

    Since product of two rational number is a rational number.

    The above will imply that 5 is a rational number

    But 5 Is an irrational number

    This contradicts our assumption.

    Therefore we can conclude that 75 is an irrational number

  3. 6+2

    If possible let a=6+2 be a rational number

    Squaring a2=(6+2)2


    2=a23812 (Equation 1)

    Since a is a rational number the expression a23812 is also rational number

    2 is a rational number

    This is a contradiction

    Hence 6+2 is irrational

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