Difference between revisions of "Pi"

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=Properties=
 
=Properties=
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[[Pi is irrational]]<br />
<strong>Theorem:</strong> The value of $\pi$ is independent of which circle it is defined for.
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[[Sum of values of sinc]]<br />
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[[Wallis product]]<br />
<strong>Proof:</strong> █
 
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=Videos=
<strong>Theorem:</strong> The real number $\pi$ is [[irrational]].
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[https://www.youtube.com/watch?v=JmnjjE0b5z0 The story of $\pi$ by Tom Apostol (1995)]<br />
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[https://www.youtube.com/watch?v=72N7yjcVFC8&feature=youtu.be&t=11s Proof that $\pi$ exists (2014)]<br />
<strong>Proof:</strong> █
 
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{{:Sum of values of sinc}}
 
  
 
=References=
 
=References=
 
[http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits Proof that $\pi$ is constant for all circles without using limits]<br />
 
[http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits Proof that $\pi$ is constant for all circles without using limits]<br />
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[http://www.oocities.org/cf/ilanpi/pi-exists.html Proof that $\pi$ exists]<br />
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[http://projecteuclid.org/download/pdf_1/euclid.bams/1183510788 A simple proof that $\pi$ is irrational by Ivan Niven]<br />
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[http://fredrikj.net/blog/2011/03/100-mpmath-one-liners-for-pi/ 100 mpmath one-liners for pi]<br />
  
[https://www.youtube.com/watch?v=72N7yjcVFC8&feature=youtu.be&t=11s Proof that $\pi$ exists (video)]
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[[Category:SpecialFunction]]
 
 
[http://www.oocities.org/cf/ilanpi/pi-exists.html Proof that $\pi$ exists]
 
 
 
[https://www.youtube.com/watch?v=2fCTnF75AL0#t=142 The story of $\pi$ by Tom Apostol (video)]
 
 
 
[http://projecteuclid.org/download/pdf_1/euclid.bams/1183510788 A simple proof that $\pi$ is irrational by Ivan Niven]
 
 
 
[http://fredrikj.net/blog/2011/03/100-mpmath-one-liners-for-pi/ 100 mpmath one-liners for pi]
 

Latest revision as of 12:12, 29 August 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

Pi is irrational
Sum of values of sinc
Wallis product

Videos

The story of $\pi$ by Tom Apostol (1995)
Proof that $\pi$ exists (2014)

References

Proof that $\pi$ is constant for all circles without using limits
Proof that $\pi$ exists
A simple proof that $\pi$ is irrational by Ivan Niven
100 mpmath one-liners for pi