6 ms·

I believe the confusion is spread across several layers. For those in the know, infinite decimal expansion is syntactic sugar for a limit over a particular infi

by enord 9y ago

I believe the confusion is spread across several layers. For those in the know, infinite decimal expansion is syntactic sugar for a limit over a particular infinite series of sums, but this convention sidesteps how people are taught syntactic manipulation and how it's intetpteted in a particular algebra. 0.999... is presented as a term, that can be syntactically replaced with an actual infinite decimal expansion, upon which the normal syntactic opetations can be applied, specifically "=" as in "0.999...=1". This is ofcourse impossible, because even if the rules are valid you would be applying them forever. This is the apparent absudity of the equation. In defence of people who hold this intuition: it's a valid concern in many branches of math(discrete, constructive, infinitesimal etc.), just not in informal expositions of the properties of real numbers.

cgmg 9y ago
Yes, '0.999...' formally stands for the sum from n = 1 to infinity of 9 / 10^n.
- KGIII 9y ago
  Heh... You kinda prove my point for me. If you could step up to the board and actually expand the ... for us, we'd be much obliged. You can keep writing 9 forever and you will never reach a whole number. But, it does reach a whole number and there is proof! So, they made a rule. The rule is, of course, silly. Infinity itself is silly. Not even atoms are infinite. Not even atoms can be infinity divisible. It's a horrible idea, really. (I have karma to burn.) By the way, it's a damned good thing infinity doesnt exist. If you had a tape measure that was infinitely long and folded it in half, we aren't really sure what would happen. Hell, I wasn't using that karma for anything better. While I'm here... When you're done finding an infinite tape measure, I'd like it if you could bring me -1 apple. No, not a slip of paper representing an owed apple, but -1 apple. Math and reality are only loosely coupled. Mathematics is a language. It can tell lies. 1 + 1 = 74 See? That's pretty obviously a lie. It can tell stories, physicists are pretty fond of using it for that. It can be so very descriptive and elegant. It can be so deceptive, as we see with the 'well, let's just let that be represented by 9/10' silliness. It's pretty horrible, really. Like I said, it's best not to think about it too much. It's kind of contrived in a few areas.
  - cgmg 9y ago
    > If you could step up to the board and actually expand the ... for us, we'd be much obliged. It seems you don't know what a sequence is. A sequence is just a function whose codomain is the natural numbers, i.e. you give any natural number you choose, and I give you the corresponding digit. I don't need to 'expand' the ellipses to define that function, any more than I have to write down all perfect squares in order to define what a perfect square is. Seriously, this is basic mathematics. You should study it before publicly embarrassing yourself by calling mathematicians 'liars'.
    - KGIII 9y ago
      I do understand it. In fact, I understand it quite well. I can give you the textbook definitions. I've studied them completely. I hold a Ph.D., you know. You are accepting rote still, as opposed to concerning yourself with the Philosophy of Mathematics. The ... represents an infinitely repeating numeral, does it not? It is a shorter way of writing 0.999999999999 (into infinity). Now, you can do that forever and you will never reach a whole number. Ever... Go ahead, try it. I'll wait. So, we've all happily agreed on some very contrived rules that you'll quote verbatim and gleefully declare to be basic mathematics and I the embarrassment. You'll commit no more thought to the process and, as I've already stated, that's a very good choice. (Karma for days...)
      - cgmg 9y ago
        > Now, you can do that forever and you will never reach a whole number. Ever... Go ahead, try it. I'll wait. What in the world are you rambling about? Do you know what a limit is? > I hold a Ph.D., you know. Yeah, and I'm the king of France. If you really have a Ph.D., I'd wager it's not in mathematics, given the content of your posts.
        Simon_says 9y ago
        I, on the other hand, would bet KGIII's Ph.D. is in math. I have a Ph.D. in math, too, and I've met, shall we say, a wide variety of fellow Ph.D. holders.
- enord 9y ago
  Oh, well then could you point me to a formal definition of the syntax and semantics for the unary "..." operator then? What is then domain and codomain, for example? Much obliged.
  - cgmg 9y ago
    ... is not an operator.
KGIII 9y ago
Rightfully, they just voted me down and refuse to think about it. I can't blame them, really. They were probably taught mathematics by rote and maybe stopped at arithmetic.

by enord 9y ago

cgmg 9y ago

Yes, '0.999...' formally stands for the sum from n = 1 to infinity of 9 / 10^n.

KGIII 9y ago
Heh... You kinda prove my point for me. If you could step up to the board and actually expand the ... for us, we'd be much obliged. You can keep writing 9 forever and you will never reach a whole number. But, it does reach a whole number and there is proof! So, they made a rule. The rule is, of course, silly. Infinity itself is silly. Not even atoms are infinite. Not even atoms can be infinity divisible. It's a horrible idea, really. (I have karma to burn.) By the way, it's a damned good thing infinity doesnt exist. If you had a tape measure that was infinitely long and folded it in half, we aren't really sure what would happen. Hell, I wasn't using that karma for anything better. While I'm here... When you're done finding an infinite tape measure, I'd like it if you could bring me -1 apple. No, not a slip of paper representing an owed apple, but -1 apple. Math and reality are only loosely coupled. Mathematics is a language. It can tell lies. 1 + 1 = 74 See? That's pretty obviously a lie. It can tell stories, physicists are pretty fond of using it for that. It can be so very descriptive and elegant. It can be so deceptive, as we see with the 'well, let's just let that be represented by 9/10' silliness. It's pretty horrible, really. Like I said, it's best not to think about it too much. It's kind of contrived in a few areas.
- cgmg 9y ago
  > If you could step up to the board and actually expand the ... for us, we'd be much obliged. It seems you don't know what a sequence is. A sequence is just a function whose codomain is the natural numbers, i.e. you give any natural number you choose, and I give you the corresponding digit. I don't need to 'expand' the ellipses to define that function, any more than I have to write down all perfect squares in order to define what a perfect square is. Seriously, this is basic mathematics. You should study it before publicly embarrassing yourself by calling mathematicians 'liars'.
  - KGIII 9y ago
    I do understand it. In fact, I understand it quite well. I can give you the textbook definitions. I've studied them completely. I hold a Ph.D., you know. You are accepting rote still, as opposed to concerning yourself with the Philosophy of Mathematics. The ... represents an infinitely repeating numeral, does it not? It is a shorter way of writing 0.999999999999 (into infinity). Now, you can do that forever and you will never reach a whole number. Ever... Go ahead, try it. I'll wait. So, we've all happily agreed on some very contrived rules that you'll quote verbatim and gleefully declare to be basic mathematics and I the embarrassment. You'll commit no more thought to the process and, as I've already stated, that's a very good choice. (Karma for days...)
    - cgmg 9y ago
      > Now, you can do that forever and you will never reach a whole number. Ever... Go ahead, try it. I'll wait. What in the world are you rambling about? Do you know what a limit is? > I hold a Ph.D., you know. Yeah, and I'm the king of France. If you really have a Ph.D., I'd wager it's not in mathematics, given the content of your posts.
      - Simon_says 9y ago
        I, on the other hand, would bet KGIII's Ph.D. is in math. I have a Ph.D. in math, too, and I've met, shall we say, a wide variety of fellow Ph.D. holders.
enord 9y ago
Oh, well then could you point me to a formal definition of the syntax and semantics for the unary "..." operator then? What is then domain and codomain, for example? Much obliged.
- cgmg 9y ago
  ... is not an operator.

KGIII 9y ago

Rightfully, they just voted me down and refuse to think about it. I can't blame them, really. They were probably taught mathematics by rote and maybe stopped at arithmetic.