How to find number between two …

Suppose we have two rational numbers a and b, then the irrational numbers between those two will be, √ab. Now let us find two irrational numbers between two given rational numbers.

1. Find an irrational number between two rational numbers 2 – √3 and 5 – √3

Let x be the irrational number between two rational numbers  2 – √3 and 5 – √3. Then we get,
2 – √3 < x < 5 – √3
⇒ 2 < x + < √3 < 5
We see that  x + √3 is an irrational number between 2 – √3 and 5 – √3 where 2 – √3 < x < 5 – √3.

2. Find two irrational numbers between two given rational numbers.

Now let us take any two numbers, say a and b. Let x be any number between a and b. Then,
We have a < x < b….. let this be equation (1)
Now, subtract √2 from both the sides of equation (1)
So, a – √2 < x < b – √2……equation (2)
= a  < x + √2 < b
Addition of irrational number with any number results into an irrational number. So, x + √2 is an irrational number which exists between two rational numbers a and b.