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Thread: 10 divided by 3

  1. #1 10 divided by 3 
    Forum Professor leohopkins's Avatar
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    10 / 3 = 3.333333333333R.............

    However..

    3 * 3.3333333333333333R...... = 9.99999999999R........

    Where has the 0.11111111111R.....

    disappeared to ?


    The hand of time rested on the half-hour mark, and all along that old front line of the English there came a whistling and a crying. The men of the first wave climbed up the parapets, in tumult, darkness, and the presence of death, and having done with all pleasant things, advanced across No Man's Land to begin the Battle of the Somme. - Poet John Masefield.

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  3. #2 Re: 10 divided by 3 
    . DrRocket's Avatar
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    Quote Originally Posted by leohopkins
    10 / 3 = 3.333333333333R.............

    However..

    3 * 3.3333333333333333R...... = 9.99999999999R........

    Where has the 0.11111111111R.....

    disappeared to ?
    It is not 0.111111111111111... that disappeared but rather

    0.000000000000.............................1

    And it did not go anywhere.

    0.999999999999999999999999999999.................. . = 1


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  4. #3 Re: 10 divided by 3 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by leohopkins
    10 / 3 = 3.333333333333R.............

    However..

    3 * 3.3333333333333333R...... = 9.99999999999R........

    Where has the 0.11111111111R.....

    disappeared to ?
    It is not 0.111111111111111... that disappeared but rather

    0.000000000000.............................1

    And it did not go anywhere.

    0.999999999999999999999999999999.................. . = 1
    How on earth can 0.9999999999.......... = 1 ???

    Okay so your calculator will not go on to infinity but do this......

    0.99999999999 + 0.11111111111 = ? guess ?
    The hand of time rested on the half-hour mark, and all along that old front line of the English there came a whistling and a crying. The men of the first wave climbed up the parapets, in tumult, darkness, and the presence of death, and having done with all pleasant things, advanced across No Man's Land to begin the Battle of the Somme. - Poet John Masefield.

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  5. #4 Re: 10 divided by 3 
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    Quote Originally Posted by leohopkins

    Okay so your calculator will not go on to infinity but do this......

    0.99999999999 + 0.11111111111 = ? guess ?
    My "guess" is that it equals 1.11111...

    In any event, it is quite true that .999999... = 1. All this means is that 1 has more than one decimal expansion. This is true for every rational number.

    A decimal expansion is simply a way of writing a number as an infinite sum of powers of ten, where the coefficients of each power of ten take values between 0 and 9. For rational numbers (like 1, 3.24, etc) it turns out there's more than one way to write it as such a sum.

    In this case, we can write 1 = 1+0*10^-1+0*10^-2+...

    or 1 = 0+9*10^-1+9*10^-2+...

    Both infinite series can be shown to converge to the same number (and the first series very clear is the number 1).

    To make all this rigorous however requires some knowledge of real analysis.
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  6. #5 Re: 10 divided by 3 
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    Quote Originally Posted by leohopkins
    Quote Originally Posted by DrRocket
    Quote Originally Posted by leohopkins
    10 / 3 = 3.333333333333R.............

    However..

    3 * 3.3333333333333333R...... = 9.99999999999R........

    Where has the 0.11111111111R.....

    disappeared to ?
    It is not 0.111111111111111... that disappeared but rather

    0.000000000000.............................1

    And it did not go anywhere.

    0.999999999999999999999999999999.................. . = 1
    How on earth can 0.9999999999.......... = 1 ???

    Okay so your calculator will not go on to infinity but do this......

    0.99999999999 + 0.11111111111 = ? guess ?

    x=0.999999999999999999999..................

    10x=9.999999999999999999999999999


    10x-x = 9.9999999999999......... - 0.999999999999..........
    =9

    9x=9

    x=1
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  7. #6  
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    1/9 = 0.1111111111111...

    9 * 1/9 = 0.9999999999999...

    9 * 1/9 = 1

    Therefore,

    1 = 0.9999999999999...
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  8. #7  
    Forum Professor leohopkins's Avatar
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    Yes I see your point....

    So I would be correct in also saying that. 4.9999999 ... = 5 ?
    The hand of time rested on the half-hour mark, and all along that old front line of the English there came a whistling and a crying. The men of the first wave climbed up the parapets, in tumult, darkness, and the presence of death, and having done with all pleasant things, advanced across No Man's Land to begin the Battle of the Somme. - Poet John Masefield.

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  9. #8  
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    Quote Originally Posted by leohopkins
    Yes I see your point....

    So I would be correct in also saying that. 4.9999999 ... = 5 ?
    Yes

    4.999999999.......... = 4 + 0.999999........ = 4 +1= 5
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  10. #9 Re: 10 divided by 3 
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    Quote Originally Posted by leohopkins
    How on earth can 0.9999999999.......... = 1 ???

    Okay so your calculator will not go on to infinity but do this......

    0.99999999999 + 0.11111111111 = ? guess ?



    And since any constant expression is a function, than 0.999... = 1 (that is, there can only be one output for x, therefore making 0.999... and 1 non-unique numbers, relatively).
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  11. #10  
    Forum Freshman IAlexN's Avatar
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    Would that mean then that if your would write 0.9' as an answer to a question, It would actually mean that the answer was 1. Hence, the proof that I'm familiar with, that DrRocket wrote in a post, or is that open for interpretation regarding the question?

    So if i would write 9.9' as an answer, when the answer actually was 10, would I be correct?
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  12. #11  
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    Quote Originally Posted by IAlexN
    Would that mean then that if your would write 0.9' as an answer to a question, It would actually mean that the answer was 1. Hence, the proof that I'm familiar with, that DrRocket wrote in a post, or is that open for interpretation regarding the question?

    So if i would write 9.9' as an answer, when the answer actually was 10, would I be correct?
    If by 9.9', you mean 9.999..., then yes, you'd be correct.

    But if I were grading it, I'd contemplate -1 point for writing the answer in a less simple form (graders eyes get tired after a while, so try to help them out).
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  13. #12  
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    .99 +.11 =1.1

    Clearly .99999999999999999999...... +.111111111111111111111111111.....

    Is more than one.
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  14. #13  
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    Quote Originally Posted by Duelix
    .99 +.11 =1.1

    Clearly .99999999999999999999...... +.111111111111111111111111111.....

    Is more than one.
    Yep

    .999999........ + .1111111111....... = 1 + 1/9 = 10/9 > 1
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  15. #14  
    Forum Sophomore An inconvenient lie's Avatar
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    humans made up numbers... In nature there is no such symmetry as a 100 trillion digits of a centimeter probably not even in quarks but it is something i have always been bothered by!
    http://whatreallyhappened.com/WRHART...alwarming.html
    Global warming is an inconvenient lie!

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  16. #15  
    Forum Sophomore An inconvenient lie's Avatar
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    maybe continuum numbers don't really exist meaning perhaps they have to be written out as fractions to be correct?
    http://whatreallyhappened.com/WRHART...alwarming.html
    Global warming is an inconvenient lie!

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  17. #16  
    Forum Radioactive Isotope MagiMaster's Avatar
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    There is a faint possibility that the universe really is discrete on the smallest scales, but that still wouldn't make the real numbers any less useful.
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  18. #17  
    Forum Sophomore An inconvenient lie's Avatar
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    lol of course not!

    People made up numbers to comfort themselves who knows maybe a long time ago some cavemen kids were sitting around with 2 rock cups and the mom went and poured a little more water in one of the caveman kids cups and the caveman kid got pissed and made up a way of measuring it lol just a funny example its like atheists say about religion
    http://whatreallyhappened.com/WRHART...alwarming.html
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