Question

Is `pi^2` rational or irrational?

Answer 1

`pi^2` is irrational

Explanation:

`pi` is transcendental, meaning that it is not the root of any polynomial equation with integer coefficients.

Hence `pi^2` is transcendental and irrational too.

If `pi^2` were rational, then it would be the root of an equation of the form:

`ax+b = 0`

for some integers `a` and `b`

Then `pi` would be a root of the equation:

`ax^2+b = 0`

Since `pi` is not the root of any polynomial with integer coefficients, let alone a quadratic, this is not possible.

Further, if `pi^2` was the root of any polynomial equation with integer coefficients then `pi` would be the root of the same equation with each `x` replaced by `x^2`. So since `pi` is transcendental, so is `pi^2`.