Do you know/use this notation?

[Igrek]

We had a discussion with our math teacher weather or not this notation is internationally accepted. It stands for the set of real numbers, without zero.

[[font="Courier"]R

[simplethinker]

This? ℝ₀
Does it include irrationals and negatives? I don't think that a set would be defined just to leave out 0, it seems kind of pointless.

[DarkerLine]

You might use it, for example to define 1/x as a function from ℝ-{0} to ℝ (because it's not a function if defined from ℝ to ℝ)

And no, I've never seen this notation used before.

[simplethinker]

I thought of that too DarkerLine, but it doesn't really have a lot of applications.
Plus, if 0 is excluded from the set of all reals, doesn't that remove it's identity element, thus demoting it from being a complete set?

[DarkerLine]

It stops being an additive group, but it becomes a multiplicative group (since now, every element has a multiplicative inverse). And regardless of its usefulness as a mathematical object, it is very useful as mathematical notation, for example saying x∈ℝ₀ could be a shorthand method of writing "x is a nonzero real number" which is useful all the time.

I would probably write ℝ-{0} as above or ℝ\{0}, though.

[simplethinker]

What do you mean ℝ-{0} and ℝ\{0}?

[DarkerLine]

The same thing as ℝ₀ (which I've never seen used).

In general, X-Y or X\Y (when X and Y are sets) is called the "difference" of two sets and consists of all elements of X that aren't elements of Y. So it's clear that ℝ-{0} consists of all elements of ℝ that aren't elements of {0}, that is, all nonzero real numbers.

[Igrek]

I'll tell my math teacher that she is right, that it isn't used (anymore)in other countries.
We use it a lot in situations like:
log(x) only has a real answer when x∈ℝ⁺₀.

[simplethinker]

wouldn't just x>0 work? or do you have one of those teachers that is anal about using precise notation for everything? (not that I'm saying it's a bad thing, but some teachers do go a bit overboard)

[Igrek]

We are allowed to say x>0 but when you have to tell the domain of f(x) it is only possible to use dom(f(x))=ℝ⁺₀.

[DarkerLine]

ℝ⁺₀ seems to be horrendous notation to me for positive reals which don't include zero, considering ℕ₀ is so commonly used for natural numbers which do include zero.

[Igrek]

[simplethinker]

I thought that the natural numbers never (or at least aren't supposed to) included 0. I still don't see the point in modifying the actual sets to include/exclude 0. If there isn't a set that contains what you want, I don't think you should modify the notation for ones that already do exist and hope that people know what you're talking about. My opinion is that formal notation is formal notation, so either use it or don't, not modify it to suit your purposes.

[edit] scratch that never part, I didn't read the rest of the paragraph. It is debatable whether to include it or not (but I still think that the nats don't include 0)

[DarkerLine]

[Igrek]

[thornahawk]

In all the years I have perused the mathematical literature, I haven't seen the notation you describe. Then again, I have been limited to the literature of America, Germany, and Russia. Maybe it was used somewhere else.

And yes, I'd use set difference notation to for the set you describe (ℝ-{0}).

thornahawk

[Igrek]

It is dutch notation.

An other similar question, do you write

log_a b
or
^alog b

(We use the second option)

[DarkerLine]

Wow, you dutch people use really crazy notation. I've never seen logs with superscripts like that before. In fact, I've never seen superscripts used as prefixes (although subscripts used as prefixes do occur, for example for modules, in abstract algebra).

But even http://nl.wikipedia.org/wiki/Logaritme uses the subscript notation! (log_ax) Although it does mention the other possibility.