Is $\infty$ enough or do I need to write $+\infty$

Question

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)

Answer 1

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.

Answer 2

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:

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

and

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

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.