Question

4. Prove that the following are not rational numbers:
(1) 2+
$\sqrt{2}$

​

Answer 1

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.

Answer 2

Answer:

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