Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is a question of notation. I have seen in many articles that people often denote $+\infty$ when talking about 'positive infinity' of the real numbers. Is that a convention, or it can be written as anyone pleases? I never liked the notation $+\infty$ because it seemed that the $+$ sign is redundant. In my opinion there is no confusion if someone writes $\infty$ for the positive infinity and $-\infty$ when talking about the negative infinity.

Still, the fact that I've seen the $+\infty$ notation in almost every article I've read in a while made me ask this question.

Is the $+$ in the notation $+\infty$ necessary? Do $\infty$ and $+\infty$ mean the same thing? (of course I'm talking about the real line here)

share|cite|improve this question
The "+" is optional and used for emphasis, in my experience. I liken it to how come composers put a "natural sign" in a measure to emphasize that there is no longer an accidental (sharp or flat) being applied to a note. – The Chaz 2.0 Apr 3 '12 at 14:44
Confusion can arise with the unsigned infinity of the projective line, which is also denoted $\infty$. For example, $\lim_{x \to 0} 1/x = \infty$ in the projective line, but $\lim_{x \to 0} 1/x \neq +\infty$ in the extended reals. – Chris Eagle Apr 3 '12 at 15:04
I can't resist putting in my $+2$ cents' worth. – André Nicolas Apr 3 '12 at 15:17
@AndréNicolas: It took me $+$ a minute to catch it... – The Chaz 2.0 Apr 3 '12 at 15:43
I once had a reviewer complain about writing $h: (0, \infty) \to (\frac{1}{2}, \infty)$ instead of $h: (0, +\infty) \to (\frac{1}{2}, +\infty)$. It seemed to me that in the context of the story, the $+$ was unnecessary, so I did not change it. It's a matter of taste, I guess. – TMM Apr 3 '12 at 17:33
up vote 5 down vote accepted

The answer to your question depends on individual opinion/definition. So here is my opinion.

I take $\infty$ to mean $+\infty$. Why? Because if you insist that one has to write plus in front $\infty$ every time one means positive infinity, then it is like saying that the symbol $\infty$ isn't well defined. So why not just adopt the convention from the real numbers where $+x$ means $x$. We don't write $+1$, we just write $1$.

Now that said, if you are writing a paper where it is essential that the reader catches whether something is $\infty$ or $-\infty$, then you might want to add the plus-sign in front when you mean (positive) infinity.

Or, if a limit is equal to either positive or negative infinity you might write $\pm \infty$ (thereby indirectly writing a $+$.

That is my opinion.

Note for example that in Stewart's calculus book the interval from negative infinity to (positive) infinity is written $(-\infty , \infty)$, so different from what Thomas Andrews has come across in his answer.

share|cite|improve this answer

As with all notation, it depends on the context. For example, when dealing with a sequence $a_1,...,a_n,...$ we write $\lim_{n\to\infty} a_n$. On the other hand, the value of this limit might be $+\infty$ or $-\infty$. So you can sometimes write:

$$\lim_{n\to\infty} a_n = -\infty$$

On the other hand, when dealing with a function on the real line, say, $f(x)=\frac{e^x}{1+e^x}$, the behavior exists and is different for large negative and large positive numbers. So we dinstinguish:

$$\lim_{x\to +\infty} f(x)=1$$


$$\lim_{x\to -\infty} f(x)=0$$

In this case, it doesn't make sense to talk about $\lim_{x\to\infty} f(x)$.

Other places you'll see infinite values are in intervals, like:

$$[a,+\infty)$$ $$(-\infty,b]$$ $$(-\infty,+\infty)$$

The key is to realize that $\infty$ in all these instances are shorthands for definitions. So $[a,+\infty)$ is the set of all real numbers at least as big as $a$, for example.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.