Step 1 : First of all, we check our rational numbers and assume ‘x’ as a value which exists between given rational numbers means if we have two rational numbers like ‘a’ and ‘b’ exist, then any number between them is-

a < x < b …........equation (1)

Here ‘x’ is any assumed number.

Step 2: After assuming any value, now we perform arithmetic operations on the given equation (1) like we subtract √2 on both side, then -

a - √2< x < b - √2 …........equation (2)

Step 3: After above two steps, now we solve equation (2) -

a - √2< x < b - √2,

=> a < x + √2 < b,

As we all know that any addition between irrational number and any number produces an irrational number. So, here ‘x’ is an irrational number.

Now we take an example to understand an irrational number that exists between two rational numbers

Example: Prove that an irrational number exists between following two rational numbers

2 - √3 and 5 - √3?

Solution: We use following steps for evaluation of irrational number between two rational numbers -

Step 1: First of all, we arrange given rational numbers in relationship order-

2 - √3 < x < 5 - √3 …..............equation (1),

Here ‘x’ is a assumed variable.

Step 2: Now we apply arithmetic operations on given equation (1),

2 - √3 < x < 5 - √3,

=> 2 < x + √3 < 5,

As we all know that any addition between irrational number and any number produces an irrational number. So, here ‘x’ is an irrational number.

Now we can say that an irrational number exists between 2 - √3 and 5 - √3.