• Kalkaline
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      331 year ago

      It’s so dumb and it makes perfect sense at the same time. There is an infinitely small difference between the two numbers so it’s the same number.

      • @[email protected]
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        1 year ago

        There is no difference, not even an infinitesimally small one. 1 and 0.999… represent the exact same number.

        • @clutchmatic
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          81 year ago

          They only look different because 1/3 out of 1 can’t be represented well in a decimal counting system.

          • @funnystuff97
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            1 year ago

            Right, it’s only a problem because we chose base ten (a rather inconvenient number). If we did math in base twelve, 1/3 in base twelve would simply be 0.4. It doesn’t repeat. Simply, then, 1/3 = 0.4, then (0.4 × 3) = (0.4 + 0.4 + 0.4) = 1 in base twelve. No issues, no limits, just clean simple addition. No more simple than how 0.5 + 0.5 = 1 in base ten.

            One problem in base twelve is that 1/5 does repeat, being about 0.2497… repeating. But eh, who needs 5? So what, we have 5 fingers, big whoop, it’s not that great of a number. 6 on the other hand, what an amazing number. I wish we had 6 fingers, that’d be great, and we would have evolved to use base twelve, a much better base!

            • @clutchmatic
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              11 year ago

              I mean, there is no perfect base. But the 1/3=0.333… thing is to be understood as a representation of that 1 split three ways

          • @[email protected]
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            131 year ago

            An infinitesimal is a non-zero number that is closer to zero than any real number. An infinitesimal is what would have to be between 0.999… and 1.

            • @[email protected]
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              91 year ago

              You are correct and I am wrong, I always assumed it to mean the same thing as a limit going to infinity that goes to 0

              • @[email protected]
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                1 year ago

                It’s a weird concept and it’s possible that I’m using it incorrectly, too - but the context at least is correct. :)

                Edit: I think I am using it incorrectly, actually, as in reality the difference is infinitesimally small. But the general idea I was trying to get across is that there is no real number between 0.999… and 1. :)

                • @[email protected]
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                  21 year ago

                  I think you did use it right tho. It is a infinitesimal difference between 0.999 and 1.

                  “Infinitesimal” means immeasurably or incalculably small, or taking on values arbitrarily close to but greater than zero.

                  • Kogasa
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                    51 year ago

                    The difference between 0.999… and 1 is 0.

                    It is possible to define a number system in which there are numbers infinitesimally less than 1, i.e. they are greater than every real number less than 1 (but are not equal to 1). But this has nothing to do with the standard definition of the expression “0.999…,” which is defined as the limit of the sequence (0, 0.9, 0.99, 0.999, …) and hence exactly equal to 1.

            • Ghoelian
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              21 year ago

              Wait what

              I always thought infinitesimal was one of those fake words, like gazillion or something

              • @[email protected]
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                31 year ago

                It sounds like it should be, but it’s actually a real (or, non-real, I suppose, in mathematical terms) thing! :)

      • iAmTheTot
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        261 year ago

        No, it’s not “so close so as to basically be the same number”. It is the same number.

        • @nachom97
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          51 year ago

          They said its the same number though, not basically the same. The idea that as you keep adding 9s to 0.9 you reduce the difference, an infinite amount of 9s yields an infinitely small difference (i.e. no difference) seems sound to me. I think they’re spot on.

          • iAmTheTot
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            61 year ago

            No, there is no difference. Infitesimal or otherwise. They are the same number, able to be shown mathematically in a number of ways.

            • @nachom97
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              61 year ago

              Yes, thats what we’re saying. No one said it’s an infinitesimally small difference as in hyperbolically its there but really small. Like literally, if you start with 0.9 = 1-0.1, 0.99 = 1-0.01, 0.9… n nines …9 = 1-0.1^n. You’ll start to approach one, and the difference with one would be 0.1^n correct? So if you make that difference infinitely small (infinite: to an infinite extent or amount): lim n -> inf of 0.1^n = 0. And therefore 0.999… = lim n -> inf of 1-0.1^n = 1-0 = 1.

              I think it’s a good way to rationalize, why 0.999… is THE SAME as 1. The more 9s you add, the smaller the difference, at infinite nines, you’ll have an infinitely small difference which is the same as no difference at all. It’s the literal proof, idk how to make it more clear. I think you’re confusing infinitely and infinitesimally which are not at all the same.

            • @[email protected]
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              21 year ago

              Technically you’re both right as there are no infinitesimals in the real number system, which is also one of the easiest ways to explain why this is true.

        • Dandroid
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          -11 year ago

          That’s what it means, though. For the function y=x, the limit as x approaches 1, y = 1. This is exactly what the comment of 0.99999… = 1 means. The difference is infinitely small. Infinitely small is zero. The difference is zero.

    • Nioxic
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      71 year ago

      There was also a veritasium video about this.

      It was interesting.

      • @[email protected]
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        1 year ago

        The tricky part is that there is no 0.999…9 because there is no last digit 9. It just keeps going forever.

        If you are interested in the proof of why 0.999999999… = 1:

        0.9999999… / 10 = 0.09999999… You can divide the number by 10 by adding a 0 to the first decimal place.

        0.9999999… - 0.09999999… = 0.9 because the digit 9 in the second, third, fourth, … decimal places cancel each other out.

        Let’s pretend there is a finite way to write 0.9999999…, but we do not know what it is yet. Let’s call it x. According to the above calculations x - x/10 = 0.9 must be true. That means 0.9x = 0.9. dividing both sides by 0.9, the answer is x = 1.

        The reason you can’t abuse this to prove 0=1 as you suggested, is because this proof relies on an infinite number of 9 digits cancelling each other out. The number you mentioned is 0.9999…8. That could be a number with lots of lots of decimal places, but there has to be a last digit 8 eventually, so by definition it is not an infinite amount of 9 digits before. A number with infinite digits and then another digit in the end can not exist, because infinity does not end.

        • SorceressAddie
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          51 year ago

          That is the best way to describe this problem I’ve ever heard, this is beautiful

        • @wumpoooo
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          21 year ago

          Maybe a stupid question, but can you even divide a number with infinite decimals?

          I know you can find ratios of other infinitely repeating numbers by dividing them by 9,99,999, etc., divide those, and then write it as a decimal.

          For example 0.17171717…/3

          (17/99)/3 = 17/(99*3) = 17/297

          but with 9 that would just be… one? 9/9=1

          That in itself sounds like a basis for a proof but idk

          • @quicksand
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            21 year ago

            Yes that’s essentially the proof I learned in high school. 9/9=1. I believe there’s multiple ways to go about it.

        • DrMango
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          21 year ago

          This is the kind of stuff I love to read about. Very cool

        • @TeddE
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          31 year ago

          If the “…” means ‘repeats without end’ here, then saying “there’s an 8 after” or “the final 9” is a contradiction as there is no such end to get to.

          There are cases where “…” is a finite sequence, such as “1, 2, … 99, 100”. But this is not one of them.

          • @[email protected]
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            31 year ago

            I’m aware, I was trying to use the same notation that he was so it might be easier for him to understand

      • @[email protected]
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        61 year ago

        Your way of thinking makes sense but you’re interpreting it wrong.

        If you can round up and say “0,9_ = 1” , then why can’t you round down and repeat until “0 = 1”? The thing is, there’s no rounding up, the 0,0…1 that you’re adding is infinitely small (inexistent).

        It looks a lot less unintuitive if you use fractions:

        1/3 = 0.3_

        0.3_ * 3 = 0.9_

        0.9_ = 3/3 = 1

      • @kadu
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        51 year ago

        deleted by creator

          • @Cortell
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            61 year ago

            Think carefully

            What does 0.99…8 represent to you exactly

            If it’s an infinite amount of 9s then it can’t end in an 8 because there’s an infinite amount of 9s by definition so it’s not a real number

      • @[email protected]
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        41 year ago

        No, because that would imply that infinity has an end. 0.999… = 1 because there are an infinite number of 9s. There isn’t a last 9, and therefore the decimal is equal to 1. Because there are an infinite number of 9s, you can’t put an 8 or 7 at the end, because there is literally no end. The principle of 0.999… = 1 cannot extend to the point point where 0 = 1 because that’s not infinity works.

      • @[email protected]
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        31 year ago

        There is no .99…8.

        The … implies continuing to infinity, but even if it didn’t, the “8” would be the end, so not an infinitely repeating decimal.