Make this definition of pi work

  • Thread starter Gerenuk
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  • #1
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Main Question or Discussion Point

I quite like the following definition of pi and I wonder which minimal algebraic rules are needed to make this definition work?

[tex]\lim_{n\to\infty}\left(1+\frac{a}{n}\right)^n=1[/tex]
[tex]\therefore |a|=2\pi k[/tex]

(For example, are there algebras more general than complex numbers, where this works?)
 

Answers and Replies

  • #2
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Well, in the quaternions [itex]e^{2\pi u} = 1[/itex] where [itex] u [/itex] is any quaternion with [itex] u^2 = -1[/itex]
 
  • #3
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So quaternions probably also work.

Any more abstract algebra? Hmm, what do I need? I need addition, multiplication, scalar division, limiting process and modulus, right? What has to be fulfilled as to yield pi then?
 

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