Question

1.
Prove that the following are irrational.
(i)1/√2
(ii) √3+√5
(iii) 6 +√2
(iv)√5
(v) 3+2√5​

Answer 1

Answer:

See it all below

Step-by-step explanation:

(i)  

2

​

 

1

​

 

Let us assume  

2

​

 

1

​

 is rational.

So we can write this number as

2

​

 

1

​

=  

b

a

​

 ---- (1)

Here, a and b are two co-prime numbers and b is not equal to zero.

Simplify the equation (1) multiply by  

2

​

 both sides, we get

1=  

b

a  

2

​

 

​

 

Now, divide by b, we get

b=a  

2

​

 or  

a

b

​

=  

2

​

 

Here, a and b are integers so,  

a

b

​

 is a rational number,  

so  

2

​

 should be a rational number.

But  

2

​

 is a irrational number, so it is contradictory.

Therefore,  

2

​

 

1

​

 is irrational number.

(ii) 7  

5

​

 

Let us assume 7  

5

​

 is rational.

So, we can write this number as

7  

5

​

=  

b

a

​

 ---- (1)

Here, a and b are two co-prime numbers and b is not equal to zero.

Simplify the equation (1) divide by 7 both sides, we get

5

​

=  

7b

a

​

 

Here, a and b are integers, so  

7b

a

​

 is a rational  

number, so  

5

​

 should be a rational number.

But  

5

​

 is a irrational number, so it is contradictory.

Therefore, 7  

5

​

 is irrational number.

(iii) 6+  

2

​

 

Let us assume 6+  

2

​

 is rational.

So we can write this number as

6+  

2

​

=  

b

a

​

 ---- (1)

Here, a and b are two co-prime number and b is not equal to zero.

Simplify the equation (1) subtract 6 on both sides, we get

2

​

=  

b

a

​

−6

2

​

=  

b

a−6b

​

 

Here, a and b are integers so,  

b

a−6b

​

 is a rational  

number, so  

2

​

 should be a rational number.

But  

2

​

 is a irrational number, so it is contradictory.

Therefore, 6+  

2

​

 is irrational number.