# CLASS-9PROBLEM & SOLUTION ON IRRATIONAL NUMBER

PROBLEM & SOLUTION ON IRRATIONAL NUMBER -

Example.1)  Prove that (√3 + √5) is irrational.

Solution: If possible, let (√3 + √5) be rational. Then,

(√3 + √5) is rational =>  (√3 + √5)² is rational

=>  (8 + 2√15) is rational

But, (8 + 2√15) being the sum of a rational and an irrational is irrational. Thus we arrive at a contradiction. This contradiction arises by assuming that (√3 + √5) is rational.

So, (√3 + √5) is irrational

Example.2) Show with the help of examples that : -

a) The sum of two irrationals need not be an irrational

Ans.)  Let, x = (2 + √3), and  y = (2 - √3)

Then a being the sum of a rational and an irrational is irrational. And y being the sum of a rational and an irrational is irrational.

And,  x + y =  (2 + √3) + (2 - √3)

=  4, which is rational

Thus, the sum of two irrationals need not be an irrational.

b) The difference of two irrational need not be an irrational

Ans.)  let, x = (2 + √3), and  y = (-5 + √3)

Then x being the sum of a rational and an irrational is irrational. And y being the sum of a rational and an irrational is irrational.

And, x – y = (2 + √3) - (-5 + √3)

= 2 + √3 + 5 - √3 = 7, which is rational

Thus, the difference of two irrationals need not be an irrational.

c) The product of two irrationals need not be an irrational

Ans.)  Let, x = (3 + √2), and  y = (3 - √2)

Then x being the sum of a rational and an irrational is irrational. And y being the sum of a rational and an irrational is irrational.

And, (xy) = (3 + √2) X (3 - √2)

=  9 – 2  = 7, which is rational.

Thus, the product of two irrationals need not be an irrational.

Example.3) Examine whether the following numbers are rational or irrational:-

a)  (3 + √5)²

Ans.)  (3 + √5)²

=  3² + 2.3.√5 + (√5 X √5)  [as per the formula (a + b)² = a² + 2ab + b²]

=  9 + 6√5 + 5

=  (14 + 6√5), which is irrational

b)   (5 + √3) (5 - √3)

Ans.)  (5 + √3) (5 - √3)

=  5² - (√3)²          [as per the formula a² - b² =  (a + b) (a – b)]

=  25 – 3 =  22, which is rational.

c)  5/√6

5          5          √6           5√6

Ans.)  ------ = ------- X ------- = --------,  which is irrational

√6         √6          √6            6

Irrational Numbers Between Two Rational

If ‘x’ and ‘y’ be two distinct positive rational numbers such that xy is not a perfect square, then √xy is an irrational number lying between ‘x’ & ‘y’.

Example.4) Find two irrational numbers between 3 and 3.5

Ans.) let a = 2 and b = 3.5

Then, √ab = √(2 X 3.5) = √7, which is an irrational number such that 2 < √7 < 3.5

Now, irrational between 2 and √7 =  √2 X (√7)

=  √2 X (7)1/2 = (2)1/2 X (7)1/4

So,  2 < [(2)1/2 X (7)1/4] < √7 < 3.5

So,  2 < (√2 X √7) < 3.5        (Ans.)