If you google "Extended Reals", you will find lots of material. The extended reals are the union of the real numbers and positive and negative infinity. But if you google "Extended Integers" then literally nothing comes up. Why is this?

  • $\begingroup$ Extended reals are useful in measure theory, but I didn't get when the extended integers are needed. It might be the reason why no one talks about extended integers. $\endgroup$ – Hanul Jeon Sep 20 '15 at 2:38
  • $\begingroup$ Thanks for the response. I believe I have a good use for extended integers. $\endgroup$ – user73063 Sep 20 '15 at 2:43
  • 2
    $\begingroup$ There are many good uses for extended integers. (Search, for example, for $p$-adic valuations, where one commonly sets $v_p\left(0\right) = +\infty$; also search for tropical semirings and other objects.) It's just that noone uses such a generic name; after all, rational numbers, real numbers, complex numbers all have equal rights to the name of "extended integers". $\endgroup$ – darij grinberg Sep 20 '15 at 2:53
  • $\begingroup$ Hi Darij. Could you please explain how "Rational numbers" has a right to the name "extended integers"? I apologize if this is a dumb question. $\endgroup$ – user73063 Sep 20 '15 at 3:04
  • $\begingroup$ @user73063 They're a superset, so they're another way of "extending" the set. $\endgroup$ – Andrew Aug 12 '16 at 19:48

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