Does .99999… = 1?
At supper today my daughter was discussing her maths (she's 13) - she had been studying putting decimal numbers into what she called standard form $A*10^n$, and what were the possible values for $A$.
She proudly said that all her classmates had said that $A$ should be between 1 and 10, while she had said that it should be between 1 and 9.9999 ... (recurring)
What interested me was that this is the first realistic attempt to distinguish between the two that I've ever really encountered. It led to an interesting discussion about open and closed intervals.
I've reformulated the next bit to clarify what I was trying to get at
So my question is whether there are any other places where a distinction between the two might have the vestige of a mathematical rationale - in the sense that there was, in my daughter's formulation, an interestingly intelligent attempt to get at the idea of the openness of the (end of) interval by distinguishing two different ways of writing the same number.
Our mathematical notation and ideas do not follow this intuition - indeed there can be a strong resistance to it, not least because it is such hard work to teach some students that the two numbers really are the same.
But I am interested in whether there are other naïve ways of looking at the distinction between the two formulations which are grasping at real mathematical content (rather than confusion) and which could help my daughter to explore the mathematical concepts she is on the edge of understanding.