How to Find Irrational Number Between Two Rational Numbers?

To find an irrational number between two rational numbers, consider the average of the given rational numbers and add the square root of 2 or any other known irrational number to obtain a number that lies between them.

To find an irrational number between two rational numbers a and b, follow these steps:

Calculate the Average

• Find the average of the two rational numbers by adding them and dividing by 2: ​(a+b)/2.

Add an Irrational Number

• To introduce an irrational number, add the square root of 2 (or any other known irrational number) to the average obtained in step 1.

Example

If the average is (a+b)/2, then an irrational number between a and b could be √2 + (a+b)/2.

This process ensures that the resulting number lies between the two given rational numbers and is irrational because it includes the irrational component (square root of 2). The concept can be extended to other irrational numbers based on the desired level of complexity.

Second Method:

Taking the square root of the product of two rational numbers, √ab​, can yield an irrational number between them, but it’s not guaranteed in all cases.