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does .9 repeating equal one?

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  • does .9 repeating equal one?

    This will be hard to show, doing math on a post is kind of hard.

    I never really thought about what .9 repeating was, I just said it all the time like "99.9999999999% sure I'm right." But I said that once and remembered that when a number is a repeating you write the digits that repeat over x amounts of nines (x being the amount of digits the numerator has). For example 1/9=.1 repeating, because 1 divided by nine equals .1 repeating. Same thing goes for .16 repeating. It is 16/99 because 16 divided by 99 equals .16 repeating. So that means that .9 repeating would by nine over nine, which is one.

    Then I was playing around with some numbers the other day, and here's what I got:

    let n=0.9 repeating
    10n=9.9999...
    -n=-.09999...
    ---------------------
    9n=9
    n=1

    [Twilight zone music]

    Is what I've proved true?
    "The first man who, having fenced off a plot of land, thought of saying, 'This is mine' and found people simple enough to believe him was the real founder of civil society. How many crimes, wars, murders, how many miseries and horrors might the human race had been spared by the one who, upon pulling up the stakes or filling in the ditch, had shouted to his fellow men: 'Beware of listening to this imposter; you are lost if you forget the fruits of the earth belong to all and that the earth belongs to no one." - Jean-Jacques Rousseau

  • #2
    Absolutely not.
    "The issue is there are still many people out there that use religion as a crutch for bigotry and hate. Like Ben."
    Ben Kenobi: "That means I'm doing something right. "

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    • #3
      Yes.

      Decimal representations and the real numbers are not bijective...
      12-17-10 Mohamed Bouazizi NEVER FORGET
      Stadtluft Macht Frei
      Killing it is the new killing it
      Ultima Ratio Regum

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      • #4
        0.99999... ~ 1, but not = 1
        It's assumed to be 1 by many people, but it's not really 1.

        Am I missing something?
        "The issue is there are still many people out there that use religion as a crutch for bigotry and hate. Like Ben."
        Ben Kenobi: "That means I'm doing something right. "

        Comment


        • #5
          No, 0.9999... = 1 exactly. They're two different representations of the same real number.
          12-17-10 Mohamed Bouazizi NEVER FORGET
          Stadtluft Macht Frei
          Killing it is the new killing it
          Ultima Ratio Regum

          Comment


          • #6
            Come on, you compsci geek. This is elementary Analysis.
            12-17-10 Mohamed Bouazizi NEVER FORGET
            Stadtluft Macht Frei
            Killing it is the new killing it
            Ultima Ratio Regum

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            • #7
              It is true, 0.9999.... is, in the real system of numbers, equal to 1. There are a lots of proofs of this, some of them using analytic concepts of series, other working directly with some basic properties of the real number.

              Frogger: decimal representeations and the real numbers are actually bijective (i.e., they have the same cardinal) even though one of them is complete(the real numbers) and the other not.

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              • #8
                Originally posted by Frogger
                No, 0.9999... = 1 exactly. They're two different representations of the same real number.

                Come on, you compsci geek. This is elementary Analysis.
                We don't do the pure stuff like that.

                We're taught 0.9 repeating is 0.9 repeating.
                "The issue is there are still many people out there that use religion as a crutch for bigotry and hate. Like Ben."
                Ben Kenobi: "That means I'm doing something right. "

                Comment


                • #9
                  Definitely not. Both 1.0 and .99999..... have seperate locations on the number line.

                  Interestingly, Cantor chowed that there are infinitely more irrational numbers (Aleph 1) than there are rational numbers (Aleph 0). (Technically this is the Continuum hypothysis IIRC)

                  Or to put it another way, the entire space taken up on the number line by rational numbers (Integers, and fractions) is zero.

                  Some interesting Cantorian mathematics

                  Xo + Xo = Xo
                  X1 + Xo = X1
                  Xo - Xo = Xo
                  Xo * Xo = Xo

                  (Xo*Xo = Xo+Xo+ ....)

                  X0 = Aleph null
                  X1 = Aleph 1 etc

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                  • #10
                    IoT is thinking along the same lines as me then.
                    "The issue is there are still many people out there that use religion as a crutch for bigotry and hate. Like Ben."
                    Ben Kenobi: "That means I'm doing something right. "

                    Comment


                    • #11
                      I meant to say, actually, that the representation is not bijective, since it is onto, but not 1-1.

                      Please explain your use of word "complete" in this situation.
                      12-17-10 Mohamed Bouazizi NEVER FORGET
                      Stadtluft Macht Frei
                      Killing it is the new killing it
                      Ultima Ratio Regum

                      Comment


                      • #12
                        Originally posted by IoT
                        Definitely not. Both 1.0 and .99999..... have seperate locations on the number line
                        No they don't.
                        12-17-10 Mohamed Bouazizi NEVER FORGET
                        Stadtluft Macht Frei
                        Killing it is the new killing it
                        Ultima Ratio Regum

                        Comment


                        • #13
                          The reason why there's disagreement is because math is such a stupid field.

                          Like it seems the most common approach these days is to use the logic that something like 1/3 is 0.3 repeating, so 3/3 must be 0.9 repeating, but it's also 1, therefore they're equal...
                          "The issue is there are still many people out there that use religion as a crutch for bigotry and hate. Like Ben."
                          Ben Kenobi: "That means I'm doing something right. "

                          Comment


                          • #14
                            I have philosophical qualms about assuming that infinitely many .9's equals 1.
                            "The issue is there are still many people out there that use religion as a crutch for bigotry and hate. Like Ben."
                            Ben Kenobi: "That means I'm doing something right. "

                            Comment


                            • #15
                              I learned it the following way:

                              In .333333..., no matter how many threes you list, the threes you haven't listed have a value one third of the way equal to the decimal place before...that is, one third of the way to making the next decimal place a four.

                              In .666666....no matter how many sixes you list, the sixes you haven't listed have a value two thirds of the way equal to the decimal place before...that is, two thirds of the way to making the next decimal place a seven.

                              Thus, in .999999....no matter how many nines you list, the nines you haven't listed have a value three thirds of the way equal to the decimal place before, the chain of nines is broken, and the number is exactly equal to one.
                              "Although I may disagree with what you say, I will defend to the death your right to hear me tell you how wrong you are."

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