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Hypercomplex numbers (a+bi with a,b hyperreal)

  1. Jul 9, 2009 #1
    Does the field of "hypercomplex numbers" (*C; a+bi with a,b hyperreal numbers) satisfy the transfer principle? Are all of the arithmetical facts with complex numbers true with the hypercomplex numbers?

    One might be able to construct a nonstandard complex analysis with the hypercomplex numbers, with identical results as as standard complex analysis, with the standard part function st(z) = st(Re(z)) + i st(Im(z)) used to define complex derivatives and integrals.
    Last edited: Jul 9, 2009
  2. jcsd
  3. Jul 9, 2009 #2
    Yes, there is no problem doing what you want. I learned nonstandard analysis from Robinson's book, which is still a classic. From that approach, you see how it is not limited to real numbers, but encompasses all of analysis.

    P.S. The term "hypercomplex numbers" is used more commonly with a completely different meaning. For what you want, maybe say "complex numbers in nonstandard analysis".
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