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Real numbers are imaginary, and imaginary numbers are real.


[I]maginary numbers describe a physical state of something, so as much as a number can exist, these do. But … real numbers, [being ideal], are imaginary.

David Manheim

(I changed some parts that I don’t agree with but the phrasing and initiative are his.)

The “rational” numbers are ratios and the “counting” numbers are, um, what you get when you count. But “real” and “imaginary” numbers have nothing to do with reality or imagination (each is both real and ideal in the same sense).

 

How about we start referring to them this way?

  • ℝ = the complete numbers. ℝ is the Cauchy-completion of the integers, meaning that ℝ has completely fills in enough options so that any sequential pattern will be able to dance wherever it wants and never need to step its shoe on another element outside the system in order to fulfill its pattern.
  • Any field adjoined to the √−1 becomes “twisting numbers”. This derives from the “twisting” feeling one gets when multiplying numbers from ℂ. For example 3exp{i 10°} • 5exp{i 20°} = 15exp{i 30°}, they spiral as they multiply outwards. Keep multiplying numbers off the zero line and they keep twisting. Tristan Needham coined the word “amplitwist” for use in ℂ.
  • ℂ = the complete, twisting numbers. Since ℂ=ℝ adjoin √−1.
  • “Complete spiral numbers” sounds nice as well.

Just to give a few examples of other acceptable numbers systems:

  • ℚ adjoin √2
  • the algebraics
  • ℚ adjoin √[a+√[b+c]]
  • sets
  • DAGs
  • square matrices … with many kinds of stuff inside
    Magma to group2.svg
  • special matrix families
  • certain polynomials (sequences) … taking many kinds of things (not just “regular numbers”) as the inputs
  • clock numbers (modulo numbers)
  • Archimedean fields and non-Archimedean fields
  • functions themselves … and the number of things that functions can represent boggles the mind. Especially when the range can be different than the domain. (Declarative sentences can have a codomain of truth value. Time series have a domain of an interval. Rotations of an object map the object to itself in a space. And more….)
  • And many, many more! Imagination is the limiting reagent here.

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