# Is pi^2 rational or irrational?

Oct 29, 2016

${\pi}^{2}$ 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:

$a x + b = 0$

for some integers $a$ and $b$

Then $\pi$ would be a root of the equation:

$a {x}^{2} + b = 0$

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 ${\pi}^{2}$.