Pi is Irrational/Proof 3
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Theorem
Pi ($\pi$) is irrational.
Proof
From Rational Points on Graph of Sine Function, the only rational point on the graph of the sine function in the real Cartesian plane $\R^2$:
- $f := \left\{ {\left({x, y}\right) \in \R^2: y = \sin x}\right\}$
is the point $\left({0, 0}\right)$.
But $\left({\pi, 0}\right)$ is also on $f$.
Hence $\pi$ cannot be rational.
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.17$: More About Irrational Numbers. $\pi$ is Irrational