View Poll Results: Repeating decimal equations

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  • 0.333... == 1/3 and 0.999... == 1 (both true)

    64 60.95%
  • 0.333... == 1/3 and 0.999... != 1

    21 20.00%
  • 0.333... != 1/3 and 0.999... == 1

    0 0%
  • 0.333... != 1/3 and 0.999... != 1 (both false)

    20 19.05%
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Thread: 0.999... == 1: why is it disturbing?

  1. #1
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    0.999... == 1: why is it disturbing?

    This is a common "debate" on internet forums. I tried to see if it was on these forums, but I couldn't find it. Anyway, what always surprises me is how much it bothers people, especially because equations like 0.333... == 1/3 don't seem to. I want to know why that is.

    My theory is that people learn about repeating decimals in grade school. Repeating decimals come out of the division algorithm they're taught. That algorithm "is division" and whatever it produces is the right answer. The algorithm works for numbers like 1/3, but doesn't work for 1/1 (it terminates immediately instead of generating a repeating decimal). Since the algorithm fails to produce 0.999... from 1/1, the equation seems wrong and unnatural.

    What do you think?
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  2. #2
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    Re: 0.999... == 1: why is it disturbing?

    Use the sum of infinite series to find the fractional value of .999....
    .999... is equal to .9 + .09 + .009 + .0009...
    This can be represented as a geometric sequence, where a1 = .9 and the ratio = .1
    Using the formula for infinite series (a1/(1-r)), we can see that it comes out to be (.9/(1-.1)), the denominator is 1-.1 which is .9, and .9/.9 = 1.
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  3. #3
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    Re: 0.999... == 1: why is it disturbing?

    To the limit 0.999... equals 1, it feels rather natural to me.
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  4. #4
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    Re: 0.999... == 1: why is it disturbing?

    I don't see why this should be disturbing. It's just mathematics. The trick is in the fact that the numbers repeat infinitely 0 that makes 0.99999..... mean something else than what people think.
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  5. #5
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    Re: 0.999... == 1: why is it disturbing?

    Quote Originally Posted by zmjjmz View Post
    Use the sum of infinite series to find the fractional value of .999....
    .999... is equal to .9 + .09 + .009 + .0009...
    This can be represented as a geometric sequence, where a1 = .9 and the ratio = .1
    Using the formula for infinite series (a1/(1-r)), we can see that it comes out to be (.9/(1-.1)), the denominator is 1-.1 which is .9, and .9/.9 = 1.
    Happy?
    I know it's true. But many people vehemently believe that's false, and it's surprising to me. I guess I'm kind of looking for a meta-discussion here.
    Help yourself: Search the community docs or try other resources.
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  6. #6
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    Re: 0.999... == 1: why is it disturbing?

    Quote Originally Posted by jpkotta View Post
    I know it's true. But many people vehemently believe that's false, and it's surprising to me. I guess I'm kind of looking for a meta-discussion here.
    So, here is why it makes sense to me. 1/infinity equals .000...1, right? and .999... is 1 - (1/infinity). Since 1/infinity is basically zero (it is the smallest number possible), then you are basically doing 1-0, which is one.

    The reason people believe it is false is because they haven't gotten far enough in math. They haven't learned about limits and such, which help in understanding this.
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  7. #7
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    Re: 0.999... == 1: why is it disturbing?

    I assume it's just the mathematical education someone received.
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  8. #8
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    Re: 0.999... == 1: why is it disturbing?

    Quote Originally Posted by smartboyathome View Post
    The reason people believe it is false is because they haven't gotten far enough in math. They haven't learned about limits and such, which help in understanding this.
    I don't really know anything about limits
    I just learned the infinite series thing in trig.
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  9. #9
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    Re: 0.999... == 1: why is it disturbing?

    Quote Originally Posted by zmjjmz View Post
    I don't really know anything about limits
    I just learned the infinite series thing in trig.
    In trig, you get an introduction to limits, so you learned enough to know about this. That infinite series will help you learn limits in the future.
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  10. #10
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    Re: 0.999... == 1: why is it disturbing?

    I have the answer.

    42.
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