IRRATIONAL NUMBERS    

All real numbers which are not rational are called irrational numbers. , , are some examples of irrational numbers.

    There are decimals which are non-terminating and non-recurring decimal.

    Example: 0.303003000300003…

    Hence, we can conclude that

    An irrational number is a non-terminating and non-recurring decimal and cannot be put in the form where p and q are both co-prime integers and q ¹ 0.

Solved Examples Based on Irrational Numbers

Question:

Prove that is not a rational number.

Solution:

Let is a rational number

        \        [p and q are co-prime and q ¹ 0]    

        Squaring both sides

        Þ    

        Þ     is even or p is even.

        Let p = 2r

        Þ    

        Þ    

        Þ     is even so q is even.

        Hence, p and q have 2 as a common factor or p and q are not co-prime.

        So, our supposition is wrong.

        \     is not a rational number.

Question:

Prove that is an irrational number.

Solution:

Suppose is a rational number and can be taken as , b ¹ 0 and a, b are co-prime.

        Þ                [Rational]             

        Squaring both sides

        Þ    

        Þ    

        LHS is which is irrational while RHS is rational.

        So, our supposition is wrong.

        Hence, is not a rational number.

Question:

Show that there is no positive integer n for which is rational and can be expressed in the form , b ¹ 0 and a & b are co-prime.

Solution:

Let there be a positive integer n for which is rational.

        Which means              … (i)

        or             

        Rationalizing LHS, we get

        Þ    

        Þ    

        Þ    

        or            …(ii)

        Adding and subtracting (i) and (ii) we get

             and

        Þ     and

        Þ    RHS of both are rational.

        \     and are also rational.

        Þ     and are perfect squares of positive integers.

        This is impossible as any two perfect squares differ at least by 3.

        Hence, there is no positive integer n for which () is rational.