sqrt(2) is irrational.

3 is rational.

The product of an irrational and a non-zero rational is irrational.

A more fundamental proof would follow the lines of the proof that sqrt(2) is irrational.

There cannot be a proof since your assertion is not necessarily true.

sqrt(2)*sqrt(3) = sqrt(6). All three are irrational numbers.

See http://mathforum.org/library/drmath/view/52619.html for a proof of the irrationality of sqrt(3). The proof that sqrt(5) is irrational is identical (substituting 5 for 3 in the proof).

There were beliefs for pi being irrational since the 9th century. However, the first proof was given in 1768 by Johann Heinrich Lambert.

pi cannot be expressed as a terminating or non-repeating decimal. Consequently, it is irrational. The pure mathematical proof is not simple.