• Chrobin
    link
    fedilink
    arrow-up
    2
    arrow-down
    1
    ·
    6 months ago

    Is this some joke I’m not getting? Cause yes, real numbers are the closure of irrational numbers, but imaginary numbers are just isomorphic to them.

    • CompassRed
      link
      fedilink
      arrow-up
      2
      ·
      edit-2
      6 months ago

      You’re thinking of topological closure. We’re talking about algebraic closure; however, complex numbers are often described as the algebraic closure of the reals, not the irrationals. Also, the imaginary numbers (complex numbers with a real part of zero) are in no meaningful way isomorphic to the real numbers. Perhaps you could say their addition groups are isomorphic or that they are isomorphic as topological spaces, but that’s about it. There isn’t an isomorphism that preserves the whole structure of the reals - the imaginary numbers aren’t even closed under multiplication, for example.

      • Chrobin
        link
        fedilink
        arrow-up
        3
        arrow-down
        2
        ·
        6 months ago

        You’re right, I mixed it up with the complex numbers being isomorphic to R^2. Thanks for clearing it up!

        Love btw how I get downvoted for an honest mistake.