A student asked me “Is infinity a number?” while I was introducing limits. My immediate response was “infinity is a concept” and I started talking about vs. . I didn’t really know where I was going with that though, so I let the students debate for a bit and finally related the whole conversation back to the problem at hand. The discussion was interesting, but other than acknowledging the difficulty of comprehending infinity, I didn’t contribute much (which is usually good, but I certainly didn’t provide clarity).

How do you define infinity?

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In Projective geometry, infinity is a point! It’s just a very interesting point with the property that it’s distance to every other point is infinite.

Projective geometry continues to boggle my mind.

There are many, many ways you can define and work with infinity. In this context, I’ve had success by considering what we want numbers to do and be, and then trying to ask if infinity is the same.

For example, real numbers live on the number line, and if you move one spot to the right (i.e., add 1) you get a larger number. Is this true of infinity? Infinity in this context is the way we talk about how many numbers there are. So if 1 + infinity is larger than infinity, then it doesn’t live on the number line, and isn’t a “number” in the traditional sense.

But how do you show infinity plus one is actually bigger than infinity? I like Steve Strogatz’s discussion at http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/.

Hope that’s helpful!

I don’t think of infinity as a number, but it’s a complicated question. Best answer I see online is: http://www.cut-the-knot.org/WhatIs/WhatIsInfinity.shtml

There are lots of different infinities in different contexts. So I don’t think there’s a simple answer to “is infinity a number” — it is an ordinal number, it is a cardinal number, it is not a real number, it is not a counting number, …

In the context of limits, I usually definite (positive) infinity as meaning “increases without bound” — but that opens to a bit of confusion for things like x + 2 sin x, which certainly is unbounded but not always increasing. So then I have to get a bit more specific, like “eventually passes and stays beyond any value you name”.

When I picked my mathematics study up again (mathematicus interruptus?), I had a wonderful Russian professor who said something about infinity that continues to echo in my heart. He said, “‘Infinity’ [as in, the infinity symbol] is not a number — it’s an indication that there is no limit in that direction.”

I have found this idea to be both poetic and resonant for young adolescents. I think a good definition should resonate in the mind and stay with you for a good long time. And it should make you hungry to explore more.

– Elizabeth (aka @cheesemonkeysf)