1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Which number set does log(0) belong to?

Tags:
  1. Feb 28, 2016 #1

    Garlic

    User Avatar
    Gold Member

    Hello everyone,
    Which number set does log(0) belong to? Or does it belong to any number sets?
     
  2. jcsd
  3. Feb 28, 2016 #2

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    ##\log(0)## is not defined.
    The most you can say is that ##\displaystyle \lim_{x\rightarrow 0+} \log(x) = -\infty##.

    You could say it belongs to the Extended Real Numbers, I guess.
     
    Last edited: Feb 28, 2016
  4. Feb 28, 2016 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Which then is part of the "extended real numbers".
     
  5. Feb 28, 2016 #4

    Garlic

    User Avatar
    Gold Member

    Is this statement true: "Any number z is an element of the complex numbers set"
    Are there any sets for numbers which aren't defined in the complex number set?
    What happens when a theoretical mathematics researcher or a philosopher thinks outside the "normal numbers" box? Do they define an arbitrary set, or is there an outermost number set?
     
  6. Feb 28, 2016 #5

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    One example:

    https://en.wikipedia.org/wiki/Quaternion
     
  7. Feb 28, 2016 #6

    Garlic

    User Avatar
    Gold Member

  8. Feb 28, 2016 #7

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    You certainly can define your set of "numbers" as you like (if a definition is useful, or maybe more important from a mathematical point of view, interesting, is another matter).
    For example, numbers not related to the quaternions are the p-adic numbers.
     
  9. Feb 28, 2016 #8

    Garlic

    User Avatar
    Gold Member

    Thank you for your explanation :smile:
     
  10. Feb 28, 2016 #9

    bcrowell

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Some of the examples you've been discussing are not ordered fields. For example, the complex numbers are not an ordered field, because there isn't an ordering defined on them. The extended reals aren't an ordered field because they aren't a field.

    But if you restrict yourself to ordered fields, then the surreal numbers are in some sense the "outermost." ("In some sense" means that it depends on what set-theoretical foundations you take.) The surreals are a proper class, not a set.

    Re your original question, I would guess that it's possible to prove that in an ordered field, log(0) is undefined.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Which number set does log(0) belong to?
  1. Does 0^0 = 1 (Replies: 106)

  2. Anti-log of a number (Replies: 3)

Loading...