Is $π^{2}$ $pi^2$ rational or irrational?

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Is $π^{2}$ $pi^2$ rational or irrational?

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Answer 1 · 2016-10-29T10:07:12

$π^{2}$ $pi^2$ is irrational

Explanation:

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

Hence $π^{2}$ $pi^2$ is transcendental and irrational too.

If $π^{2}$ $pi^2$ were rational, then it would be the root of an equation of the form:

$a x + b = 0$ $ax+b = 0$

for some integers $a$ $a$ and $b$ $b$

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

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

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

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

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