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.)