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I'd say the key point is to understand the difference between a number, and the decimal representation of a number. 0.99999... is one possible representation of

I'd say the key point is to understand the difference between a number, and the decimal representation of a number. 0.99999... is one possible representation of number 1. 1 is another one. Once one understand the definition of the decimal representation, it's just a simple proof to show that 0.99999... = 1.

kypro 1y ago
I'm not a mathematician, but this is always the way I've looked at it too. We can't represent values like 1/3 precisely in the decimal number system, the best we can do is represent in a way that it's clear what's implied with minimal error. The representation isn't really suppose to be interpreted as an infinite decimal series, and depending on how you interpret 3.333... you could argue it's a slightly different value. And that's plainly obvious – 3.333... != 1/3
throwaway31131 1y ago
I think one other piece is one needs to understand the number is not being built, the whole representation exists all at once. When I tutor students the confusion is they think of each 9 like a brick being added to a wall and for them the wall is never done, that’s their argument why 0.999 doesn’t equal 1. Then when you explain numbers don’t have a time dimension they usually get it.
constantcrying 1y ago
>I'd say the key point is to understand the difference between a number, and the decimal representation But there is no such distinction. In fact the decimal representation is "closer" to a real number, then just 1. >is one possible representation of number 1. Why? You are just asserting things. You do not even give an argument why that should be the case. Why is 0.999... a representation of 1 and not 0.123?
- yodsanklai 1y ago
  > But there is no such distinction Of course there's a distinction. A decimal representation is a sequence of digits, not a number > Why? It boils down from the definition of the decimal representation and the limit of a geometrical sequence. https://en.wikipedia.org/wiki/Decimal_representation https://en.wikipedia.org/wiki/Decimal_representation
  - constantcrying 1y ago
    >Of course there's a distinction. A decimal representation is a sequence of digits, not a number Oh and what is a real number? Might a real number be a sequence of rationals? Or more correctly an equivalence class of cauchy series. >It boils down from the definition of the decimal representation and the limit of a geometrical sequence. No, it doesn't. You are assuming the conclusion.

kypro 1y ago

I'm not a mathematician, but this is always the way I've looked at it too. We can't represent values like 1/3 precisely in the decimal number system, the best we can do is represent in a way that it's clear what's implied with minimal error. The representation isn't really suppose to be interpreted as an infinite decimal series, and depending on how you interpret 3.333... you could argue it's a slightly different value. And that's plainly obvious – 3.333... != 1/3

throwaway31131 1y ago

I think one other piece is one needs to understand the number is not being built, the whole representation exists all at once. When I tutor students the confusion is they think of each 9 like a brick being added to a wall and for them the wall is never done, that’s their argument why 0.999 doesn’t equal 1. Then when you explain numbers don’t have a time dimension they usually get it.

constantcrying 1y ago

>I'd say the key point is to understand the difference between a number, and the decimal representation But there is no such distinction. In fact the decimal representation is "closer" to a real number, then just 1. >is one possible representation of number 1. Why? You are just asserting things. You do not even give an argument why that should be the case. Why is 0.999... a representation of 1 and not 0.123?

yodsanklai 1y ago
> But there is no such distinction Of course there's a distinction. A decimal representation is a sequence of digits, not a number > Why? It boils down from the definition of the decimal representation and the limit of a geometrical sequence. https://en.wikipedia.org/wiki/Decimal_representation https://en.wikipedia.org/wiki/Decimal_representation
- constantcrying 1y ago
  >Of course there's a distinction. A decimal representation is a sequence of digits, not a number Oh and what is a real number? Might a real number be a sequence of rationals? Or more correctly an equivalence class of cauchy series. >It boils down from the definition of the decimal representation and the limit of a geometrical sequence. No, it doesn't. You are assuming the conclusion.