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Thread: Mathematic Paradox

  1. #1
    LSJ's Avatar Protector Domesticus
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    Default Mathematic Paradox

    Just something to think aboot. Not really science, but would get buried in thema devia in 1 minute.

    This is something I discovered for myself many years ago, and still find it amusing.
    In math, only 1 is equal to 1, anything higher or lower is, of course, higher or lower, and definately not 1.

    The paradox, which I will give the simplest example of later, comes from the flawed system of fractions and decimal equivalents.

    1 divided by 2 is 1/2, or 0.5. If we multiply 1/2 by 2, we get 1. That makes sense.
    But fractions also use impossible equations, where multiple infinite numbers can equal a whole number. It makes no sense.
    0.44444444444444444444444444444444 and so on multiplied by any number cannot equal anything but an infinite number.

    We can divide 1 by 3, and get 1/3, or 0.3333333'''
    If we multiply 1/3 by 3, we get 1. Yet 3 thirds in demicals equals 0.999999"" (0.333333"" x 3)
    So, if (1/3 x 3 = 1) then (0.3333333"" x 3 = 1)
    Yet 0.333333"" x 3 equals 0.999999"".....

    So a number with infinitely trailing decimals somehow equals 1? But 1 is greater than 0.99999999999.....

    Impossible fractions. It also works with every other number - you just have to find a fraction of it that equals an infinite decimal.
    Last edited by LSJ; September 14, 2006 at 07:02 PM.

  2. #2
    therussian's Avatar Use your imagination
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    Default Re: Mathematic Paradox

    Actually, 1 is equal to 1.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 and so on. Same with every rational number. The all go to infinite.

    You have to idealize every once in a while. You have to round eventually, or otherwise mathematics would be impossible due to the concept of infinity. Don't think of them as decimals, firstly. Think of them only as fractions. So if 1/1 divided by 3/1 is equal to 1/3, then 1/3 multiplied by 3/1 is equal to 3/3 = 1/1.


    you're really complicating a very complicated science. But I see what you're saying. I'll have to ask my Calculus teacher about that

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    Default Re: Mathematic Paradox

    Quote Originally Posted by DarkProphet
    In math, only 1 is equal to 1, anything higher or lower is, of course, higher or lower, and definately not 1.
    Correct. However, as you have discovered, the number 1 does have multiple decimal expansions, as do many numbers.
    Quote Originally Posted by DarkProphet
    But fractions also use impossible equations, where multiple infinite numbers can equal a whole number. It makes no sense.
    0.44444444444444444444444444444444 and so on multiplied by any number cannot equal anything but an infinite number.
    Perhaps it's disorienting, but the sum of an infinite sequence of numbers need not be infinite. This touches on what all of calculus is based on: limits. 0.4444... is defined as follows: take 0.4, then take 0.44, then 0.444, and keep on going. The limit is the unique number that fulfills two conditions:
    1. No matter how many 4's you tack on (that is, a finite number of 4's), the resulting decimal number will never become greater than the limit.
    2. For any number less than the limit, it's possible to put together enough 4's (again, a finite number) that the resulting decimal number is greater than the number you chose.

    What does this mean? Well, point 1 means that your expanding decimal will never become larger than the limit, but point 2 means that it will be larger than any number smaller than the limit. Based on this, plus the fact that without allowing an infinite series of decimals there's no way to represent most rational numbers in decimal, mathematicians decided it would be sensible to define the infinite series as being equal to a finite number, given that such a number exists. As it happens, such a number does exist for any repeating decimal, a fact I can roughly prove to you if you're interested. (Try checking if 1 is a limit of 0.999... under the above two criteria if you like.)
    Quote Originally Posted by DarkProphet
    We can divide 1 by 3, and get 1/3, or 0.3333333'''
    If we multiply 1/3 by 3, we get 1. Yet 3 thirds in demicals equals 0.999999"" (0.333333"" x 3)
    So, if (1/3 x 3 = 1) then (0.3333333"" x 3 = 1)
    Yet 0.333333"" x 3 equals 0.999999"".....

    So a number with infinitely trailing decimals somehow equals 1?
    Correct.
    Quote Originally Posted by DarkProphet
    But 1 is greater than 0.99999999999.....
    It is not, as you just demonstrated. 1, like many numbers, has multiple decimal expansions. It's true that 1 is greater than any number of finite nines arranged like that, but so too is one-third greater than any finite number of threes arranged after a decimal point. It's a matter of limits: what's true of a finite number of digits strung together is not always true of an infinite number of such digits.

    Think of it this way, if you like: if 1 is greater than 0.999..., then what's 1.000... − 0.999...? Well, first you have to carry the one to the right, to get to the rightmost digit. Any intermediate digits get turned into 9's, of course, as we were all taught in grade school (try doing 100 − 99 in long subtraction if you've forgotten: first you have to turn the rightmost 0 into a 10, but that requires you to cross off the leftmost 1 and turn the intervening 0 into a 9). The thing is, though, there is no rightmost digit, so you have to carry the one off to infinity . . . turning every digit into a 9.

    It is, of course, possible to construct an internally consistent system in which 1.000... ≠ 0.999.... But mathematicians have felt that it's not a particularly worthwhile course to pursue, since ultimately there's no other sensible value for 0.999... to have, if it's to be defined at all. It's not sensible to define it as being either greater than or less than 1.
    Quote Originally Posted by DarkProphet
    It also works with every other number - you just have to find a fraction of it that equals an infinite decimal.
    Actually, it only works with terminating decimals, I believe (although this, alas, I can't say with certainty, at least without further thought). So 1(.000...) and 0.5(000...) can be represented as .999... and 0.4999..., but 0.333... is the only representation of one-third that I can think of. I can think about this more and see if I can come up with a rough proof if you're interested.

    One number is certainly does not work is zero. Any decimal expansion containing a nonzero digit must be either positive or negative: a sum of uniformly positive or uniformly negative numbers is positive or negative, respectively. Therefore, the only infinite decimal expansion for zero is 0.000....
    Quote Originally Posted by Bulgaroctonus
    You have to idealize every once in a while. You have to round eventually, or otherwise mathematics would be impossible due to the concept of infinity.
    I don't think "rounding" is the correct term for limits at all. It implies a loss of accuracy or fidelity, which isn't the case here.
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