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Make this definition of pi work

  1. Mar 13, 2010 #1
    I quite like the following definition of pi and I wonder which minimal algebraic rules are needed to make this definition work?

    [tex]\therefore |a|=2\pi k[/tex]

    (For example, are there algebras more general than complex numbers, where this works?)
  2. jcsd
  3. Mar 13, 2010 #2
    Well, in the quaternions [itex]e^{2\pi u} = 1[/itex] where [itex] u [/itex] is any quaternion with [itex] u^2 = -1[/itex]
  4. Mar 13, 2010 #3
    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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