# Which number set does log(0) belong to?

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1. Feb 28, 2016

### Garlic

Hello everyone,
Which number set does log(0) belong to? Or does it belong to any number sets?

2. Feb 28, 2016

### Samy_A

$\log(0)$ 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
3. Feb 28, 2016

### HallsofIvy

Staff Emeritus
Which then is part of the "extended real numbers".

4. Feb 28, 2016

### Garlic

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?

5. Feb 28, 2016

### Samy_A

One example:

https://en.wikipedia.org/wiki/Quaternion

6. Feb 28, 2016

### Garlic

7. Feb 28, 2016

### Samy_A

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.

8. Feb 28, 2016

### Garlic

Thank you for your explanation

9. Feb 28, 2016

### bcrowell

Staff Emeritus
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.

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