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How come when I put .3 repeated in a calculator times 3, it becomes 1, but on others it is .9?

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  • H
    hhh M0★ M1★ M2 M3★ M4★
    last edited by debbie Jun 24, 2020, 4:21 AM Jun 22, 2020, 6:30 PM

    How come when I put .3 repeated in a calculator times 3, it becomes 1, but on others it is .9?

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    • T
      The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
      last edited by The Blade Dancer Jun 22, 2020, 6:36 PM Jun 22, 2020, 6:34 PM

      That's a matter of interpretation. Some matters of math are debated. For example, if you multiply 0.3 repeated 3 times, it's 0.9 repeated, which is soooooo close to 1, but not quite, hence other calculators counting it as 0.9 (sort of an abbreviation in math terms, they wouldn't go on with the 9's forever). However, if you convert 0.3 repeated into fraction form, it is 1/3, and 3 x 1/3 is 1. It all depends on how you look at it.

      Hope this helped.

      The Blade Dancer
      League of Legends, Valorant: Harlem Charades (#NA1)
      Discord: Change nickname if gay#7585

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      • P
        Potato2017 M5★
        last edited by Jun 22, 2020, 8:34 PM

        Actually, most people agree that 0.9999999999999999... is 1.
        Here's a way you can look at this. If two numbers are different, you should be able to find a number between them. Can you find a number between 0.999999999999999999... and 1? If not, then they can't be different numbers.
        Another way:

        1/3=0.333333333333333...Multiply both sides by 31=0.999999999999999...\begin{aligned} 1/3&=0.333333333333333... \\ \text{Multiply bo}&\text{th sides by 3}\\ 1&=0.999999999999999... \end{aligned} 1/3Multiply bo1​=0.333333333333333...th sides by 3=0.999999999999999...​

        One final way:

        Let x=0.9999999999999999...Then 10x=9.9999999999999999...Subtract the first equation from the first.9x=9x=1\begin{aligned} \text{Let }x&=0.9999999999999999...\\ \text{Then }10x&=9.9999999999999999...\\ \text{Subtract the }&\text{first equation from the first.}\\ 9x&=9\\ x&=1\\ \end{aligned} Let xThen 10xSubtract the 9xx​=0.9999999999999999...=9.9999999999999999...first equation from the first.=9=1​

        The best Potato
        aops: Potato2017
        yt: http://bit.ly/potatosubscribe
        discord: Potato2017#1822 (it's tent#0001 now)
        tetr.io: https://ch.tetr.io/u/potato2017
        -Potato2017

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        • N
          nastya MOD M0 M1 M2 M3 M4 M5
          last edited by debbie Jun 22, 2020, 11:22 PM Jun 22, 2020, 11:14 PM

          Hi @userusername,
          @The-Darkin-Blade and @Potato2017 are both correct and provided great mathematical proofs of this fact. Thank you all! 🙂

          Thinking about it, it is true that some calculators are "cleverer" than others, and they "know" that 0.333...‾×3=1.\overline{0.333...}\times 3=1.0.333...×3=1. It could be for different reasons: some calculators have this fact stored as basic initial information for other calculations, while other calculators can transform some of the repeating digits of the fractions, so when they receive 0.333...‾×3\overline{0.333...}\times 30.333...×3 they think of it as 1/3×3,^1/_3\times 3,1/3​×3, and that's why we get 111 as an answer. The other, less "clever", calculators just multiply these numbers and we get 0.333...‾×3=0.999...‾\overline{0.333...}\times 3=\overline{0.999...}0.333...×3=0.999... as a result. Or, maybe, they are more sophisticated than we think, and they want to clearly show what the actual answer is. 😉

          But, in the end, for us, the answer 0.999...‾\overline{0.999...}0.999... and 111 are the same. The easiest way to realize this and put it in your head as not like something that you just "know", but something that you "understand", is to realize that the number 0.999...‾\overline{0.999...}0.999... has infinitely many 9′s,9's,9′s, so it is infinitely close to 1,1,1, and there is no actual difference between 0.999...‾\overline{0.999...}0.999... and 1.1.1. At least, I think of it in this way 🙂

          Hope this helped you to find your own understanding of what is going on!

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          • W
            Walnut M0 M1 M2
            last edited by Jun 23, 2020, 3:24 PM

            so 1/3 is 0.3 repeating, right? 1/3 *3 is 1, but 0.3 repeating *3 is 0.9 repeating. So, we know that 0.9 repeating is 1.

            Walnut He

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            • T
              The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
              last edited by Jun 23, 2020, 4:04 PM

              All a matter of interpretation

              The Blade Dancer
              League of Legends, Valorant: Harlem Charades (#NA1)
              Discord: Change nickname if gay#7585

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              • P
                Potato2017 M5★
                last edited by Potato2017 Jun 24, 2020, 4:20 PM Jun 23, 2020, 5:17 PM

                If you put 1/3⋅31/3 \cdot 3 1/3⋅3into almost any calculator, 0.99999999 will come out. This is because when it calculated 1/3, it rounded 0.3333333333333333333... into 0.333333333333. When it calculated 0.3333333333333*3, it forgot that 0.33333333333333 was rounded. Good calculators will give you 1.

                Edit: the asterisks made it italicized 😛 i fixed it

                The best Potato
                aops: Potato2017
                yt: http://bit.ly/potatosubscribe
                discord: Potato2017#1822 (it's tent#0001 now)
                tetr.io: https://ch.tetr.io/u/potato2017
                -Potato2017

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                • T
                  The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
                  last edited by Jun 23, 2020, 5:21 PM

                  Personally I like to think of 0.99 repeating as a real small tad less than 1.

                  The Blade Dancer
                  League of Legends, Valorant: Harlem Charades (#NA1)
                  Discord: Change nickname if gay#7585

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                  • H
                    hhh M0★ M1★ M2 M3★ M4★
                    last edited by Jun 23, 2020, 7:22 PM

                    Oh, and why when I do .3 times 2 it is 0.66666667 instead of 0.66666666?

                    N 1 Reply Last reply Jun 23, 2020, 10:40 PM Reply Quote 0
                    • N
                      nastya MOD M0 M1 M2 M3 M4 M5 @hhh
                      last edited by Jun 23, 2020, 10:40 PM

                      Hi @userusername,
                      It is because your calculator rounds the number 0.66666666...‾\overline{0.66666666...}0.66666666... to the number 0.66666667.0.66666667.0.66666667.
                      The same thing happens when you want to round the number 0.316495...‾\overline{0.316495...}0.316495... to its hundredths. Instead of 0.310.310.31 you will get 0.32,0.32,0.32, since the next digit after 111 is 666 that is ≥5,\geq 5,≥5, so the rounding goes up.

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