\sqrt {2} = 1.414...
and,
\sqrt {3} = 1.732...

So, we can easily find irrational numbers between this, which are:

1. 1.565665666....
2. 1676776777....

Introduction
In mathematics, an irrational number is a real number that cannot be expressed as a ratio of two integers. Irrational numbers are non-repeating, non-terminating decimals. Examples of irrational numbers include √2, π, and e.

Finding Irrational Numbers between √2 and √3
To find two irrational numbers between √2 and √3, we can use the fact that the decimal representation of an irrational number never ends or repeats. One way to generate such numbers is to take the square root of a non-square integer between 2 and 9.

Method 1: Square Root of 5
One such integer is 5. Taking the square root of 5 gives us a non-repeating, non-terminating decimal between √2 and √3. We can approximate this decimal to any desired degree of accuracy using a calculator or by long division.

√5 = 2.236067977...

To find another irrational number between √2 and √3, we can add or subtract a small amount from this number. For example, we can add 0.1 to √5 to get:

√5 + 0.1 = 2.336067977...

This is another irrational number between √2 and √3.

Method 2: Square Root of 7
Another integer we can use is 7. Taking the square root of 7 gives us another non-repeating, non-terminating decimal between √2 and √3.

√7 = 2.645751311...

Again, we can add or subtract a small amount from this decimal to get another irrational number between √2 and √3. For example, we can subtract 0.1 from √7 to get:

√7 - 0.1 = 2.545751311...

This is another irrational number between √2 and √3.

Conclusion
In conclusion, we can find two irrational numbers between √2 and √3 by taking the square root of a non-square integer between 2 and 9. We can then add or subtract a small amount from this decimal to get another irrational number between √2 and √3. These numbers are non-repeating, non-terminating decimals that cannot be expressed as a ratio of two integers.