Magnus Anderson wrote:Is it because the latter evaluates to \(\infty\) or because you think the former equals to a finite number or because of something else?

Nope, you've pretty much got it. If it evaluates to a finite number, you can use it in arithmetic. If it evaluates to \(\infty\), you can't.

Magnus Anderson wrote:And how did you come to your conclusion?

Because arithmetic deals with numbers, quantities. \(\infty\) is not a quantity.

Magnus Anderson wrote:Why is it okay to say that \(10 \times (0.9 + 0.09 + 0.009 + \cdots) = 9 + 0.9 + 0.09 + \cdots\) but not okay to say that \(10 \times (1 + 1 + 1 + \cdots) = 10 + 10 + 10 + \cdots\)?

Well, you can, but here's where the other rendition I talked about comes in: if you want to do arithmetic on \(\infty\), you have to play by different rules. \(10 \times (1 + 1 + 1 + \cdots) = 10 \times \infty = \infty\). <-- In what case, with normal arithmetic using normal numbers, can you say 10 x n = n?

Note that those weird rules don't apply to the case of \(0.\dot9\): \(10 \times (0.9 + 0.09 + 0.009 + \cdots) = 10 \times 0.\dot9 = 9.\dot9\).

Magnus Anderson wrote:If I understand you correctly, we can do arithmetic with \(0.9 + 0.09 + 0.009 + \cdots\) because it's not equal to \(\infty\).

Why is that so?

Because \(0.9 + 0.09 + 0.009 + \cdots\) is a quantity and you can do arithmetic with quantities. \(\infty\) is not so you cannot.

Magnus Anderson wrote:So don't do arithmetic with \(\infty\) or accept that different rules apply. <-- Take your pick.

But you didn't explain why.

You can only do arithmetic with quantities. If you choose to do arithmetic with things other than quantities, you gotta figure out a different set of rules that makes sense with those things.

For example, suppose you wanted to do arithmetic with 'red'. One good starting point might be to ask: what does adding 1 to red mean? red + 1 = ??? You gotta figure out what that could possibly mean. Are you adding 1 unit of saturation to red? If so, then maybe the rule is: red + 1 = brighter red. Maybe adding 1 means adding 1 more square foot of red (like if you were painting). Then red + 1 = more surface area of red.

^ It would be a different kind of arithmetic, not the traditional one that everyone is used to.

As I said, \(\infty\) is a property of sets, the property of being endless. I keep asking you what \(\infty\) + 1 means. I'm asking what it means to add something to an infinite set such that it's property of being endless changes somehow. I'm asking this in much the same vein as asking what does red + 1 mean. I need to understand what's being conveyed by the statement "red + 1" in order to know what it equals. Same with \(\infty\) + 1. What am I supposed to be imagining. The best I can do is to imagine adding something to a collection of things that's already infinite and asking myself: does this change the property of being endless. And the answer seems to be: no, it's still endless. If you say, it makes it

more endless, then I have to ask about

that in the same vein: what am I imagining when I imagine something being more endless than something else that's endless?