# The difference between 10 and 9.99999 … (recurring) [duplicate]

Possible Duplicate:
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.

• There is no difference between $10$ and $9.9999\ldots$; both decimal representations correspond to the same number. – Arturo Magidin Jan 11 '12 at 20:32
• @ArturoMagidin: There's no difference between the real numbers they represent (they represent the same real number), but they are different representations of that number. – Isaac Jan 11 '12 at 20:35
• Can someone identify in a comment the question this duplicates - I did try to find one and evidently failed. What I was trying to do btw, and why I tagged intuition, was not to identify that the numbers were the same (I know that), but to identify why people (thinking about intelligent 13-year-old children, for example) might think that they were different. I want to teach my daughter good maths, and I was looking for ways of exploring the same idea with her. – Mark Bennet Jan 11 '12 at 20:51
• @MarkBennet: I'm not sure why a comment with the link wasn't generated upon the first close-as-duplicate vote, but the link is now at the top of the question. – Isaac Jan 11 '12 at 20:54
• @MarkBennet See the earlier answer by Isaac. I expect your daughter would enjoy this: store.doverpublications.com/0486210960.html by Albert Beiler. Repeating decimals are chapter 10, pages 73-82, especially for $1/p$ for $p$ prime. The simplest property is when 10 is a primitive root $\pmod p,$ you get the repeating part of $a/p$ a cyclic permutation of that for $1/p.$ So, ( 10^18 - 1 ) / 19 = 52631578947368421 and 2 * ( 10^18 - 1 ) / 19 = 105263157894736842 and 3 * ( 10^18 - 1 ) / 19 = 157894736842105263, and you can see where the extra 0 goes in the actual decimal 1/19. – Will Jagy Jan 11 '12 at 21:47