LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

PROBLEM & 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.)**