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I hate to say it but I still don't believe this, it just goes against all intuition that I have, but people much smarter than I have proven it so I take it on f

I hate to say it but I still don't believe this, it just goes against all intuition that I have, but people much smarter than I have proven it so I take it on faith for doing things like calculus etc just my lizard brain won't let me accept something that looks like less than 1 being 1 the same way that the limit of 1/x as x goes to infinity is zero but it doesn't seem like ti should be. The number gets infinitesimally small but it's still some non-zero number -- I dunno this is probably proving my ignorance it's just what it is.

ben509 7y ago
Maybe try this: do longhand division of 7 by 7 or 9 by 9 or whatever, but instead of putting 1 in the first place, start with 0. 0.99 7 | 7.00000 0 7 0 6 3 70
- gigatexal 7y ago
  I follow that but then I could see it just continues to .9999 forever never converting to 1 it’s always that tiny fraction less than 1...
MrManatee 7y ago
Don't feel too bad. The notation "0.999..." looks non-threatening, which tricks people into believing that they understand what it means. We could make "0.999... = 1" look scarier by writing it as [n ↦ 1 - 10^(-n)] = [n ↦ 1], where [n ↦ a_n] denotes the equivalence class of a Cauchy sequence of rational numbers. These statements mean the same thing, but with the scarier notation much fewer people would mistakenly believe that they understand what it says. I would expect mathematics majors to learn what 0.999... means during their undergraduate university courses. But then there's still the question of why mathematicians chose to define it that way. To really understand that, you need to be able to come up with alternative definitions and to investigate the consequences of those definitions. And for most undergraduates, it might still take a few years to build that level of mathematical maturity. For anyone who is not a math major, I certainly don't want to discourage any curiosity about this subject. Just don't be discouraged if you feel you can't fully understand what's going on. Understanding what 0.999... means and why mathematicians chose to define it that way is quite subtle.
- gigatexal 7y ago
  Thank you. I had totally expected to get dunked on and find your empathy refreshing. I’ll keep trying to understand it.
  - gigatexal 7y ago
    It has to do with infinities (infinite sums) which might explain why it’s so interesting

ben509 7y ago

Maybe try this: do longhand division of 7 by 7 or 9 by 9 or whatever, but instead of putting 1 in the first place, start with 0. 0.99 7 | 7.00000 0 7 0 6 3 70

gigatexal 7y ago
I follow that but then I could see it just continues to .9999 forever never converting to 1 it’s always that tiny fraction less than 1...

MrManatee 7y ago

Don't feel too bad. The notation "0.999..." looks non-threatening, which tricks people into believing that they understand what it means. We could make "0.999... = 1" look scarier by writing it as [n ↦ 1 - 10^(-n)] = [n ↦ 1], where [n ↦ a_n] denotes the equivalence class of a Cauchy sequence of rational numbers. These statements mean the same thing, but with the scarier notation much fewer people would mistakenly believe that they understand what it says. I would expect mathematics majors to learn what 0.999... means during their undergraduate university courses. But then there's still the question of why mathematicians chose to define it that way. To really understand that, you need to be able to come up with alternative definitions and to investigate the consequences of those definitions. And for most undergraduates, it might still take a few years to build that level of mathematical maturity. For anyone who is not a math major, I certainly don't want to discourage any curiosity about this subject. Just don't be discouraged if you feel you can't fully understand what's going on. Understanding what 0.999... means and why mathematicians chose to define it that way is quite subtle.

gigatexal 7y ago
Thank you. I had totally expected to get dunked on and find your empathy refreshing. I’ll keep trying to understand it.
- gigatexal 7y ago
  It has to do with infinities (infinite sums) which might explain why it’s so interesting