As we know 0.999... = 1 I wonder in which contexts is it acceptable to use the recurring decimal "0.999..." as it is a representation of the number 'one' can it be used as an alternative to the number 'one', do we draw a distinction between one as a natural number and real number? (is it acceptable to say '0.999...' is a Natural number)
Asked
Active
Viewed 86 times
3

2If your universe consists of only the natural numbers then you must stay away from writing that decimal representation, because otherwise you are leaving your universe. If your universe consists of real numbers, for which there is a notion of series convergence, then you can go ahead and complicate 1 by writing it like that. – Nicolas Bourbaki Jan 18 '22 at 10:55

Can we bring this to a very simple case, when would our universe only be natural numbers? Could we allow our universe to consist of real numbers but only be able to use natural due to context in a physical scenario ,such as 'there are n coils on this wire', how incorrect would it be to write 'there are "0.999..." coils on this wire'? – user1007028 Jan 18 '22 at 17:23

1Does this answer your question? [Does it make any sense to prove $0.999\ldots=1$?](https://math.stackexchange.com/questions/1600940/doesitmakeanysensetoprove0999ldots1) – amWhy Jan 23 '22 at 21:54

https://math.stackexchange.com/questions/287311/howcaniexplain0999ldots1?noredirect=1&lq=1 – amWhy Jan 23 '22 at 21:56

Most of all, see [this duplicate](https://math.stackexchange.com/questions/287311/howcaniexplain0999ldots1?noredirect=1&lq=1) – amWhy Jan 23 '22 at 21:57
1 Answers
3
Since one is a natural number and since $0.99999\ldots$ is a way of representing the number one, then, yes, $0.99999\ldots$ is a natural number. It is hard to imagine a context in which someone is working only with natural numbers and in which that choice is a reasonable one, but that's another problem.
José Carlos Santos
 397,636
 215
 245
 423

is there any backing to the idea that the 'natural number, one' and the 'real number, one' could be thought as being in some way different such that the use of '0.999...' is poorly defined, for example we know they are the same number but is there some contextual idea where we want to represent a cardinality for example, as saying 'there is 0.999... apple' is clearly incorrect, or is that more about language with respect to concrete ideas? – user1007028 Jan 18 '22 at 17:27

No. You should distinguish between the *number* one and the several ways we have *representating* it. One such repesentation is $1$. Another one is $0.99999\ldots$. Still another one is “one”. But it is always the number one. – José Carlos Santos Jan 18 '22 at 17:30

When dealing with various representations, is there contexts where it is perhaps incorrect to use '0.999...' or is it simply based on common sense like how a ruler has 2cm and not '4/2 cm' – user1007028 Jan 18 '22 at 17:43

I would say that it's just common sense. Your example is fine. – José Carlos Santos Jan 18 '22 at 17:44

Jose, I know you know very well that the question is a duplicate, and yet you answered anyway? – amWhy Jan 23 '22 at 21:53

@user1007028 it is equivalent to 1, it equals 1. Please see the clearer responses in the links I posted under your question. – amWhy Jan 23 '22 at 21:58

@amWhy Asking whether or why $0.999999\dots=1$ has appeared here many times, yes. But the question here was whether it is acceptable to use the expression $0.999999\dots$ to represent the number one. – José Carlos Santos Jan 23 '22 at 21:59

Yes. The two are equivalent, given the proofs in the links I've provided. – amWhy Jan 23 '22 at 22:02

You admit that yourself. Equivalent things can replace their equivalent. What you point out is generally reserved for philosophy of language. This is a math site, and it is likely you have merely confused this user. – amWhy Jan 23 '22 at 22:05

Thank you for links @amWhy sometimes I can have overly pedantic questions and this one cleared it up for me in the most simple way in fairness. – user1007028 Jan 24 '22 at 09:54