A Monster for Every Season: Summer 2
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    Firbolg in the Playground
     
    Bohandas's Avatar

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    Default A more concise and intuitive demonstration that 0.9999...=1

    Most of the explanations I've seen of why 0.999...repeating equals 1.0 have seemed a little esoteric, but I just thought of a really simple demonstration showing that they're equal

    Start with the number 1.1111....repeating
    Subtract 1 (leaving you with 0.1111...repeating)
    add 0.9999....repeating

    and you see immediately that it gives you back the original number

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    Dragon in the Playground Moderator
     
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    Default Re: A more concise and intuitive demonstration that 0.9999...=1

    Esoteric?

    0.333.... = 1/3.

    3 x 1/3 = 1, so 3 x 0.333.... = 1, so 0.999.... = 1.

    What's esoteric about that?
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    Troll in the Playground
     
    Griffon

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    Default Re: A more concise and intuitive demonstration that 0.9999...=1

    Quote Originally Posted by Bohandas View Post
    Most of the explanations I've seen of why 0.999...repeating equals 1.0 have seemed a little esoteric, but I just thought of a really simple demonstration showing that they're equal

    Start with the number 1.1111....repeating
    Subtract 1 (leaving you with 0.1111...repeating)
    add 0.9999....repeating

    and you see immediately that it gives you back the original number
    Quote Originally Posted by Peelee View Post
    Esoteric?

    0.333.... = 1/3.

    3 x 1/3 = 1, so 3 x 0.333.... = 1, so 0.999.... = 1.

    What's esoteric about that?
    It's worse than that, it's wrong Jim.

    0.1111 + 0.9999 = 1.1111. Which may be what was meant by "the original number", but it wasn't immediately clear to me that that was what was meant.
    Last edited by halfeye; 2022-01-17 at 05:00 PM.
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    Firbolg in the Playground
     
    Bohandas's Avatar

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    Default Re: A more concise and intuitive demonstration that 0.9999...=1

    Quote Originally Posted by halfeye View Post
    0.1111 + 0.9999 = 1.1111. Which may be what was meant by "the original number", but it wasn't immediately clear to me that that was what was meant.
    Yes. It starts with 1.1111.... and ends with 1.1111....

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    Bugbear in the Playground
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    Default Re: A more concise and intuitive demonstration that 0.9999...=1

    Quote Originally Posted by Bohandas View Post
    Start with the number 1.1111....repeating
    Subtract 1 (leaving you with 0.1111...repeating)
    add 0.9999....repeating

    and you see immediately that it gives you back the original number
    I don't think this is a good proof. It looks to me that you're assuming what you're trying to prove on line 3. If .9 repeating doesn't equal 1, then adding it wouldn't result in the original number.
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    Orc in the Playground
     
    SwashbucklerGuy

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    Default Re: A more concise and intuitive demonstration that 0.9999...=1

    Quote Originally Posted by Bohandas View Post
    Most of the explanations I've seen of why 0.999...repeating equals 1.0 have seemed a little esoteric, but I just thought of a really simple demonstration showing that they're equal

    Start with the number 1.1111....repeating
    Subtract 1 (leaving you with 0.1111...repeating)
    add 0.9999....repeating

    and you see immediately that it gives you back the original number
    From what I can tell, proving that 0.9999... + 0.1111.... = 1.1111.... requires*** some way of working with sums of infinite geometric series, and if you can do that it's more straight forward to just directly prove that 0.9999... = 1.

    ***Maybe it doesn't but I suspect other methods would be even more esoteric
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    Barbarian in the Playground
     
    ClericGuy

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    Default Re: A more concise and intuitive demonstration that 0.9999...=1

    (X 10 - X)/9 = X
    Correct?

    Now use 0.9999... for X

    0.9999... 10 = 9.9999...
    9.9999... - 0.9999 = 9
    9 / 9 = 1

    Simple, right?
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    Dragon in the Playground Moderator
     
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    Default Re: A more concise and intuitive demonstration that 0.9999...=1

    Quote Originally Posted by Rogan View Post
    (X 10 - X)/9 = X
    Correct?
    Uncertain. Need proof.
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    Dwarf in the Playground
     
    BardGuy

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    Default Re: A more concise and intuitive demonstration that 0.9999...=1

    Quote Originally Posted by Rogan View Post
    (X 10 - X)/9 = X
    Correct?

    Now use 0.9999... for X

    0.9999... 10 = 9.9999...
    9.9999... - 0.9999 = 9
    9 / 9 = 1

    Simple, right?
    It may be because it's been more than a decade since I've done algebra, but I'm getting x=x
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    Titan in the Playground
     
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    Default Re: A more concise and intuitive demonstration that 0.9999...=1

    Between any two distinct numbers, there are infinite other numbers. (e.g. between 1.3 and 1.4 is 1.31, 1.32, 1.34593875834754985398475, etc)

    If .999... does not equal 1, you should be able to name at lest one of the numbers in between.

    You can't, therefore they must be the same number, written in two different ways.

    Nothing esoteric about that, I feel.

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