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Math Help - 0.99999999999........=1

  1. #1
    Junior Member Singular's Avatar
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    Question 0.99999999999........=1

    my teacher give me a problem

    PRoof that 0.999999........ equals 1

    The proof :
    0.9999.........=p
    9.9999.........=10p
    9+0.999.......=10p
    9+p =10p
    9 =9p
    p =1

    But I still don't believe that 0.99999........................ = 1 ????!!!!!!
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  2. #2
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    Nothing wrong.

    The definition of a decimal is the convergent value. Meaning the value it approaches. Since .99999... converges (approaches) to 1 the value is 1.
    Note,
    1.00000... also converges to 1.
    Thus,
    .9999.... = 1.000...

    There is nothing wrong. You are assuming that decimal representations are unique. They are not.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Singular View Post
    my teacher give me a problem

    PRoof that 0.999999........ equals 1

    The proof :
    0.9999.........=p
    9.9999.........=10p
    9+0.999.......=10p
    9+p =10p
    9 =9p
    p =1

    But I still don't believe that 0.99999........................ = 1 ????!!!!!!
    0.999.. is defined by:

    0.999..=\sum_{n=1}^{\infty}\frac{9}{10^n}

    The right hand side is a geometric series whose sum is 1.

    RonL
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  4. #4
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    I find it simple enough like this

    \frac{1}{9}=0.1111111...

    0.1111111...*9=0.999999....

    but in fact...

    \frac{1}{9} * 9 = 1 as \frac{9}{9}=1


    Hope that helps you believe!
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  5. #5
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    All of the previous are totally correct proofs. The first time one sees it, however, they can seem like deceiving tricks.

    Maybe this will help to convince you. Two real numbers are different only if they have a non-zero difference. So try to answer yourself this question:

    What is 1 - .99999.... ?

    Convince yourself that, no matter what positive number you pick, the difference is always smaller than that positive number. Thus the difference must be zero.
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  6. #6
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    Quote Originally Posted by anthmoo View Post
    I find it simple enough like this

    \frac{1}{9}=0.1111111...

    0.1111111...*9=0.999999....

    but in fact...

    \frac{1}{9} * 9 = 1 as \frac{9}{9}=1


    Hope that helps you believe!
    The problem with all the symbol manipulations like this that can be used
    to demonstrate the 0.999..=1, is that the operations being performed are
    not defined (as far as the reader knows at any rate) for these "infinite"
    decimals.

    These are extrapolations of what happens with rationals and finite decimals
    and have to be given a proper meaning before you have a proof.

    RonL
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  7. #7
    Eater of Worlds
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    Here's a response I seen on Danica McKellar's math site. I thought it was a good
    answer:


    "First, we start by labeling:

    x= .999999999999..... (repeating infinitely)

    Then if you multiply both sides by 10, you get:

    10x = 9.99999999999..... (repeating infinitely)

    Well, what if we subtracted the top equation from the bottom one? We'll get another true statement. Before I do that, let me review why this will work. Say that a = b , and c = d. Then do you agree that "a-c = b-d" is also a true statement? After all, a = b , and c = d. So once you agree with that, let's continue and go ahead and subtract the top statement from the bottom one. We get, for the left hand side of the equations:

    10x -x = 9x

    And for the right hand side:

    9.99999..... - .99999999...... = 9

    And we conclude that "9x = 9" is also a "true statement." Divide both sides by 9, and you get x = 1. Since we started with x = .9999999 (repeating infinitely), we've now shown that the "infinite decimal of .9999 repeating" equals the whole number 1. Yes, I did it much faster on the TV show.

    What I didn't talk about on the show is the somewhat philosophical issue this proof brings to light. Our mathematical system was invented by the human mind, which assumes that space (and the number line) is infinitely divisible and that there *is* such thing as a theoretical "point" that takes up no space, and has no volume or mass. So, this little proof shows that - according to a math system that assumes such things - indeed, .999999.... (repeating infinitely) equals 1. There are those who would say this isn't true, but then they are forgetting that we are dealing with a "language" that was invented by the human mind, and also that infinity is a slippery concept. "
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  8. #8
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    Quote Originally Posted by BubbleBrain_103 View Post
    All of the previous are totally correct proofs. The first time one sees it, however, they can seem like deceiving tricks.

    Maybe this will help to convince you. Two real numbers are different only if they have a non-zero difference. So try to answer yourself this question:

    What is 1 - .99999.... ?

    Convince yourself that, no matter what positive number you pick, the difference is always smaller than that positive number. Thus the difference must be zero.
    That a number is smaller than any finite number does not proove that it
    is zero. If we are in the standard reals it does, but not in the non-statandard
    reals. The problem here is that this is all handwaving, define what we mean
    by the recuring decimal and there is no problem.

    RonL
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  9. #9
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    Hello,

    your problem is a few thousand years old.

    Have a look here: Zeno of Elea - Wikipedia, the free encyclopedia

    EB
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  10. #10
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    Quote Originally Posted by earboth View Post
    Hello,

    your problem is a few thousand years old.

    Have a look here: Zeno of Elea - Wikipedia, the free encyclopedia

    EB
    ^For an essay

    0.999... - Wikipedia, the free encyclopedia

    The real thing...but seriously stay at Math Help Forum to understand what wikipedia means!
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  11. #11
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    Quote Originally Posted by CaptainBlack View Post
    That a number is smaller than any finite number does not proove that it
    is zero. If we are in the standard reals it does, but not in the non-statandard
    reals. The problem here is that this is all handwaving, define what we mean
    by the recuring decimal and there is no problem.

    RonL
    I'm familiar with the so called hyper-reals. However, the user I'm assuming was inquiring about real numbers. And even if we were operating in the hyperreal number system, the difference between two reals, 1 and .9999..., is again a real number, and the argument still applies.

    Showing that a real number is zero by showing that it is both non-negative and smaller than any positive is quite a standard argument in analysis. I really wouldn't call it handwaving, often it's the only reasonable approach.
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  12. #12
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    Quote Originally Posted by anthmoo View Post
    ...

    The real thing...but ...
    Hello,

    thanks for the link.

    To introduce infinite periodic decimals I used the paradoxon of "Achille and the tortoise" which was described by Zenon. There are other paradoxa described by this philosopher which will give very nice introductions to problems of analysis.

    Therefore - and only therefore - I refered to this site of wikipedia.

    EB
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  13. #13
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    Quote Originally Posted by BubbleBrain_103 View Post
    I'm familiar with the so called hyper-reals. However, the user I'm assuming was inquiring about real numbers. And even if we were operating in the hyperreal number system, the difference between two reals, 1 and .9999..., is again a real number, and the argument still applies.
    There is no diffrerence even in the non-standard reals because of
    what the notation 0.999.. is definded to mean, that is it is the sum
    of the appropriate series.

    Showing that a real number is zero by showing that it is both non-negative and smaller than any positive is quite a standard argument in analysis. I really wouldn't call it handwaving, often it's the only reasonable approach.
    In analysis because we know that we are talking about the standard
    reals for which it is true. I doubt the OP has anything but the hazyest
    idea of what the reals are so an argument should hold whatever variant
    of the reals they have in mind (in fact anyone who maintains that 0.999..
    does not equal 1 has some non-standard interpretation of the reals in
    mind, but that is irrelevent for the purposes of this argument)

    RonL
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  14. #14
    Member Ranger SVO's Avatar
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    I just wanna add my 10-cents worth
    Attached Thumbnails Attached Thumbnails 0.99999999999........=1-equal.jpg  
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  15. #15
    Grand Panjandrum
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    Quote Originally Posted by Ranger SVO View Post
    I just wanna add my 10-cents worth
    Tell us how you do the addition:

    0.666.. + 0.333..

    to get 0.999..?

    RonL
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