I am not concerned with your poor grammar. What is that supposed to mean–the plural pronoun?
Unless you want me to report this post I expect you to explain this phrasing immediately!
Er, I think you’re joking, but in case you’re serious: though your username is quite possibly your real name and therefore perhaps an indication of your gender, I don’t actually know it and didn’t want to presume, and thus used singular “they”, as I frequently do without any great thought put into it.
Sure there is. If I use the numbers -3 or 0 or √5 or 6i, they mean exactly the same thing no matter how I use them. But “infinity” doesn’t have a single value. The number of integers and the number of real numbers are both infinite, but those are different values.
Now, depending on how you define “number”, that might be a distinction without a difference. But to say that claiming infinity is not a number is the same as claiming 0 is not a number isn’t fair. It’s difficult to find definitions of number that don’t include 0; there are plenty of definitions of number that don’t include infinity.
I explained in a thread about Usernames, several years ago, that mine was sort of a shortened version of the name I most often go by, Doug Montgomery. It would take quite a stretch of the imagination to construe ‘my gender’ as other than male.
Do we understand each other?
There are plenty of notions of number that don’t include 6i or sqrt(5) or -3 or 1/2 or all the rest of them, as well. Hence people calling “6i” imaginary in the first place. If numbers are meant to count sizes of sets, as they were originally intended to do, well, all of these are right out. None of these have anything to do with that, nor make any sense on such an account of what numbers are. But all this means is that there are different kinds of different numbers, for different purposes.
Speaking of sizes of sets, the number of integers and the number of real numbers are only different if you are choosing to count them as different; if you’d like to work in a system of cardinalities in which all infinite cardinalities are identified (a perfectly coherent and reasonable thing to do, for many purposes), they are the same.
But, again, counting sizes of sets is not the only thing numbers do, given such numbers as sqrt(5), etc.
In fact, you say sqrt(5) means exactly the same thing, no matter how you use it. I disagree (or rather, I say this depends on what “means exactly the same thing” means, and only amounts to tautology; yes, sqrt(5) is always sqrt(5), and in the same sense, ∞ is always ∞). Are there not multiple square roots of 5? Not only 2.2360679775 and -2.2360679775, but also 4 and 7 in mod 11 land, and the matrix [[1, 2], [2, -1]] among many others, and so on and so on.
Shall we say then that “sqrt(5)” then is not a number but a process, and only 2.2360679775 is a number proper? You can contort yourself like that if you want, but this is madness. sqrt(5) refers to a number, or rather, a multiplicity of numbers in different numerical systems.
And the same is true of all the rest of them. The cardinality 6 (existing within a discrete context in which there is no such thing as -1 or 1/2), the ordinal 6, 6 as a measure of ratios of unsigned lengths, 6 as a signed integer, 6 as an entity within the “complex numbers”… all of these are in some sense different concepts under the umbrella of 3, and none of this is to deny that 6 is a number. (And at times, we may wish to draw finer distinctions that are coarsely lumped together under 6: when we say that the limiting value of (x - sin(x))/x^3 at x = 0 is 6, we are lumping together behaviors like “oscillates around”, “approaches always staying under”, “approaches and actually reaches”, etc., all into the single entity 6, which at other times we may wish to distinguish between. Again, none of this means 6 isn’t a number (or if you like, many different numbers in different contexts)).
I’m sorry for causing offense by refraining from assuming things about a stranger I did not know.
Rather, “not a number but a concept”
The numerator and denominator here should be switched (but anyone who cares already realized that).
Sorry, ignore this
You’re approaching the concept of a limit, but you’re doing it improperly. (Which is OK: most people before Newton, even very smart ones, didn’t really know how to think about limits). Again, look up L’Hopital’s rule to understand how to properly think about ratios involving zero or infinite quantities. The issue here is that different ratios may both involve infinite quantities (and thus look, in some sense, like infinity / infinity), but they might converge to very different limits, and that’s why infinity * zero is undefined.
Fine. 
I am not a mathematician so whatever I type is going to be deeply flawed, but I am having a hard time figuring out how zero times infinity could be anything other than zero. The zero should negate any sum no matter how big or what it is. Nothing multiplied by something is nothing.
Let’s put it this way: zero times infinity definitely encompasses zero (and for the very reasons you think it does), but (in suitable contexts) also other possible values as well.
So many typos… In addition to the previously noted two,
Scare quotes belong on “imaginary”, not “6i”.
Should have a trailing ellipsis in each instance.
“3” here should of course have been “6”
Not sure about zero, but I am certain that infinity times eight is cloverleaf.
Well, tilted cloverleaf. To get a conventional cloverleaf, you also have to multiply it by sqrt(i).
I think the problem in this thread and many like it is that you are defining “number” in mathematics and other posters are defining “number” in arithmetic. You simply aren’t speaking the same language.
r = |sin 2Ω| is the equation for a cloverleaf in polar …
I don’t see that arithmetic vs. wider math makes much difference here.
In even grade school arithmetic, we already have a multitude of number concepts: numbers that count “How many?” (0, 1, 2, 3, ), numbers that count differences (here come negatives), numbers that count ratios (here come fractions). We see already that the basic concept of number expands and expands to cover new abstractions.
And for many of these applications, incidentally, it would not be unreasonable to consider the particular abstraction of infinity as a number:
If numbers mean only “How many?”, then -1 and 1/2 aren’t numbers. And ∞ is a number. (“How many primes are there?”)
“No, 1/2 is a number because you can have something 1/2 a foot long”. Oh, ok. You can have one length that’s 1/2 as long as another, so 1/2 is a number. Fine.
If numbers mean only ratios of lengths, then, again, -1 isn’t a number, and ∞ is. (“How many times as long as a point is a stick?”)
“No, -1 is a number because I can owe you one dollar.” Well, yeah… money’s a game whose rules we’ve made up. If we wanted, you could owe me red banana dollars, and have rules for how to handle that situation.
“Well, ok, -1 is a number because it’s what you get when you subtract 6 from 5”.
One might say “You can’t subtract 6 from 5. Sure, it looks like you can, but properly speaking, we must say there’s no such thing as the number -1, just the concept of the difference, between something and something one higher. -1 isn’t a number; it’s a difference. Just like ∞ isn’t a number; it’s a limit”.
But this is a foolish way of hamstringing our perspective and our language.
In arithmetic, we already see that there are multiple kinds of numbers, useful for different purposes, and thus should already be comfortable with the idea that we could make new numerical systems encapsulating ∞ cleanly. And so we might well consider ∞ a number, in just the same way as all these others.
No, -1 is a number, from a very useful group of numbers. You can, for instance, have a negative velocity because velocity is the vector quantity of speed. Without negatives, vector calculations (like composing billiards shots or plotting space junk trajectories), would be a pain in the ass.
But, for example, a rectangle of, say, 5 x -3, does not have an area of -15 because negative area does not make sense. Even if it is being used in some sort of math where it offsets with rectangles of positive dimension, its area is still not negative. But if you had a rectangle of 0 x ∞, you would not actually have a rectangle, so you could not express its area in any meaningful way (you would have a line).
Zero times anything is zero. But also, infinity times anything is very big. Something’s got to give way.
Cite.