phloomp wrote:Meta-observation:. I've TAed a bunch of math, and my biggest takeaway is that when you're confused, it is very important to ask yourself "what do these words even mean?" The reason people don't do this, and aren't taught to do this, is that the answer is often very difficult! Real numbers are quite intuitive, but what does "real number" even mean? If you haven't seen completions of metric spaces (or Dedekind cuts) you might enjoy to read about them, if only to see how much work needs to be done. Same goes for "infinity" or even "divide".

This is my number one advice for students when they're struggling to solve a math problem, at all levels. Before you can solve the problem you have to know what every word and symbol in it means. Not in a vague, everyday sense, but in the precise sense they're being used in the question.

dowan wrote:the problem isn't whether infinity is a number (it's certainly not a real number), the problem is 1/0. When mathematics runs into this, it breaks the formula. It's one of the big obstacles in the way of a Grand theory of everything.

The square root of -1 is also not a number, and you can't get it by multiplication or division (of numbers). Luckily, someone didn't let this get in their way, which is why we have i.

Mathematicians don't have a problem with division by zero being undefined in arithmetic. If anything the current situation is ideal, because when division by zero occurs in a formula that applies to a physical situation it's a sign that something has broken. If you hacked your arithmetic rules and let people divide by zero to get some spurious 'infinity' value it might fool physicists into thinking they're doing legitimate calculations.

As for 0.99999. Let S = 0.999..., then 10S = 9.999..., so 10S -S = 9S = 9.999... - 0.999... = 9, and so S = 1. You could, if you wanted, define 0.999 to be equal to something else, but the rules governing the arithmetic operations would then be broken, and you'd have to special case them to avoid being able to prove that S = 1 in the way I demonstrated. It's not clear what benefit anyone would get by doing this. 0.999.. isn't a number, it's a symbolic way of writing a number. It turns out that if you want this symbol to obey the same manipulation rules as the terminating decimals then the number this symbol represents must be the same as the number commonly represented by 1.

As for infinitesimal values, it is possible to define them in a way that is consistent with the conventional arithmetic of real numbers. However, the existing numbers and the decimals representing them keep the same values and interpretations as they do in conventional models. They have to, as the whole point of these non-standard constructions is that they obey the same (first order) rules as the conventional numbers.