Can I rationalize the denominator of $\frac{1}{\pi}$?
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2So are you asking whether $\pi$ is rational? It's not. – Matti P. Mar 18 '20 at 08:01
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No. Pi is transcendental and no polynomial or rational function of Pi with rational coefficients can yield a rational. – Mar 18 '20 at 08:02
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@MattiP.: I think the OP wants to mimic $1/\sqrt2\to\sqrt2/2$. – Mar 18 '20 at 08:02
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$1/\pi$ is transcendental number. – Anton Vrdoljak Mar 18 '20 at 08:03
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6Well, there's $\frac{1/\pi}1$ ... – Hagen von Eitzen Mar 18 '20 at 08:47
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No, you cannot.
This is a consequence of the fact that $\pi$ is transcendental. This implies that there are no algebraic manipulations with rational numbers we can do (besides multiplying by $0$, which is obviously not allowed) which will yield a rational number in the denominator.
The reason we can rationalize the denominator in e.g. $\frac{1}{\sqrt{2}}$ is that $\sqrt{2}$ is algebraic (as opposed to $\pi$, which is transcendental).
Alvin L-B
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