3 ms·

There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero.

gspr 6y ago
> There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero. Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.
- brlewis 6y ago
  You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1. I think people intuitively see that infinitely zero equals zero.
  - gspr 6y ago
    > 1 - 0.999... = 0.000... proves that 0.999... = 1. If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point
    - brlewis 6y ago
      If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...
      - gspr 6y ago
        > If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
        brlewis 6y ago
        They have to know that 0.999... means you never stop writing nines. Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.
        gspr 6y ago
        > They have to know that 0.999... means you never stop writing nines. How do they know that that's a real number?
        brlewis 6y ago
        They don't have to know that it's a real number. Knowing that you never stop writing nines is sufficient to perform the calculation.
        gspr 6y ago
        You're asking them to perform subtraction. They probably know how to do that with real numbers, but problably not with much else. So they'll have to know that they're real numbers (or whatever numbers you are demanding that they be – you're still unclear on this point if it's not actually the reals).
        brlewis 6y ago
        Internally I'm saying they're real numbers. In what I say to the person trying to intuit that 0.999... = 1, I'm deliberately avoiding talking about number systems. I'm assuming this person thinks of numbers as sequences of digits, possibly with a decimal point.
        gspr 6y ago
        And that is where it all goes wrong.

gspr 6y ago

> There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero. Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.

brlewis 6y ago
You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1. I think people intuitively see that infinitely zero equals zero.
- gspr 6y ago
  > 1 - 0.999... = 0.000... proves that 0.999... = 1. If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point
  - brlewis 6y ago
    If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...
    - gspr 6y ago
      > If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
      - brlewis 6y ago
        They have to know that 0.999... means you never stop writing nines. Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.
        gspr 6y ago
        > They have to know that 0.999... means you never stop writing nines. How do they know that that's a real number?
        brlewis 6y ago
        They don't have to know that it's a real number. Knowing that you never stop writing nines is sufficient to perform the calculation.
        gspr 6y ago
        You're asking them to perform subtraction. They probably know how to do that with real numbers, but problably not with much else. So they'll have to know that they're real numbers (or whatever numbers you are demanding that they be – you're still unclear on this point if it's not actually the reals).
        brlewis 6y ago
        Internally I'm saying they're real numbers. In what I say to the person trying to intuit that 0.999... = 1, I'm deliberately avoiding talking about number systems. I'm assuming this person thinks of numbers as sequences of digits, possibly with a decimal point.
        gspr 6y ago
        And that is where it all goes wrong.