Difference between revisions of "Pi"

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[[Pi is irrational]]<br />
<strong>Theorem:</strong> The real number $\pi$ is [[irrational]].
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[[Sum of values of sinc]]<br />
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<strong>Proof:</strong>
 
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=References=
 
=References=

Revision as of 20:11, 20 June 2016

A circle in Euclidean plane geometry is defined to be the set of points equidistant from a center point. The length around a circle is called its circumference and the length a line from the circle through the center is called a diameter of the circle. All diameters have the same length by definition of the circle. Let $A$ be a circle. The number $\pi$ is defined to be the ratio $\dfrac{C}{D}$ where $C$ is the circumference of $A$ and $D$ the diameter of $A$. It requires proof to show that the value obtained from the circle $A$, call this $\pi_A$, is the same number one obtains from another circle $B$, the value $\pi_B$.

Properties

Theorem: The value of $\pi$ is independent of which circle it is defined for.

Proof:

Pi is irrational
Sum of values of sinc

References

Proof that $\pi$ is constant for all circles without using limits

Proof that $\pi$ exists (video)

Proof that $\pi$ exists

The story of $\pi$ by Tom Apostol (video)

A simple proof that $\pi$ is irrational by Ivan Niven

100 mpmath one-liners for pi