Division by zero,seems flawed

Discussion in 'Pseudoscience Archive' started by Secret, Sep 21, 2011.

  1. Secret Registered Senior Member

    Messages:
    299
    Defining Division by zero,seems flawed

    I'm very interested in the number 0
    And I was not very satisfied when divison of zero is undefined (except in projective geometry, but the "division sign" does not mean the division in a simple sense)

    One day I came up with this
    Given
    0N=0 ---(*) (where N is any real number)
    (But we will consider N=0 separately below unless specified as it seemed to does not work)

    If we define a new relation (0[sup]-1[/sup]) which act as the inverse of "multiplied by 0" (Note 0[sup]-1[/sup] =/= 1/0), then the above becomes
    N=0[sup]-1[/sup] ---(1)

    Now given
    0/N=0 (where N is any real number) ---(2*)
    (However unless specified, we only consider N=/=0)

    (0)(1/N)=0
    Since 1/N=/=0, it is just a constant
    Using (1), it now becomes
    1/N=0[sup]-1[/sup] ---(2a)

    Furthermore using (1)

    Therefore (1)=(2a)
    and ()

    Therefore I get two bizarre results
    1/N=N
    1/0[sup]-1[/sup])=0[sup]-1[/sup])
    (N=/=0)


    Now consider
    1/0
    By (*) we get
    1/0=1/0N
    =(1/0)(1/N)
    Using (3), it now becomes
    (1/0)(N)
    =N/0
    Therefore another bizarre result
    1/0=N/0---(4)
    Since N is any real number therefore
    1/0=2/0=3/0=1.5/0=root(2)/0 ... EXCEPT 0/0

    Now consider when N=0 in (*), i.e.
    (0)(0)=0
    Using the relation invented in (1)
    0=0[sup]-1[/sup] (N=0) ---(5)

    Unfortunately, I still get nowhere when dealing with 0/0
    0/0
    =0(1/0)
    Using (4)
    =0(N/0)
    When N=0
    Using (5)
    =0[sup]-1[/sup](N/0[sup]-1[/sup])


    For Tl;dr
    Red means some possibly flawed steps and green means steps that are proved true (NOTE if one step is flawed that means the subsequent steps are also flawed, however I only highlighted the steps that seemed to assume something)

    Please tell me what's wrong in this "proof" and debunk if necessary
     
    Last edited: Sep 21, 2011
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  3. James R Just this guy, you know? Staff Member

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    39,397
    You started with

    \(0 \times N = 0\)

    But to retain this equation when you introduce \(0^{-1}\) you need to multiply both sides by the inverse:

    \(0^{-1} \times 0 \times N = 0^{-1} \times 0\)

    It does not follow from this that \(N=0^{-1}\).

    You haven't specified what the value of \(0^{-1} \times 0\) is.

    You can't have \(0^{-1} \times 0 = 1\), because then you'd have \(N=1\), but since N can be an arbitrary number this is a contradiction that tells you that your definition of the "multiplicative inverse of zero" must be wrong.

    This is why \(0/0\) is undefined, by the way.
     
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  5. Secret Registered Senior Member

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    299
    I'm not sure whether I've got the right knowledge, or interpret correctly of the properties of inverses

    What I know about inverses:
    They are symmetrical about y=x
    f[sup]-1[/sup](f(x))=f(f[sup]-1[/sup])(x))
    Only 1-1 f(x) have inverses as functions (without restricting domains)

    By this bolded property, is it correct to assume (0)0[sup]-1[/sup] just cancel each other out?

    EDIT: Nvm, reading your edited statement
    EDIT 2: Ah I understand now, if I tried to use the inverse on both sides, then N=1. I can't just left that zero on the other side untouched
     
    Last edited: Sep 21, 2011
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  7. Secret Registered Senior Member

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    299
    This also raise more questions related to 0

    1. If I say f and f[sup]-1[/sup] cancel each other's actions out, is it means that I am actually saying f(f[sup]-1[/sup])=1?

    2. Actually what is the detailed prove of 0N=0 ,especially 0x0=0?

    3. 0/0, 1/0, 2/0 etc. are undefined (in standard definition of "division"). They LOOKED different, is it because of how we WRITE them? (compare with 1=0.9999999...)

    4. Is other undefined e.g. d/dx|x| when x=0, sin[sup]-1[/sup](3),log0 etc. differ from the divisions by zeros i)algebrically, ii)geometrically?

    5. Is it possible to invent a new no. line where divisions by zeros are defined, in a similar way when mathematicians imagine/define the no. i=root(-1) (where complex numbers turns out to have practical applications)?If yes, where will this line/number system be located relative to other lines/number systems? (e.g. the imaginary line is perpendicular to the real line)

    6. I understand why 0/0 is undefined, but why 0[sup]0[/sup] is however, commonly defined to be 1 yet it cause no problems (even it is technically undefined)?

    7. Is it possible to have a number system of base 0 or negative (or other non interger numbers)?

    8. a/0 where a=/=0 can be defined as +-infinity or unsigned infinity depending on the maths field you are considering. But for 0/0 is it undefined for all known fields of maths?
     
    Last edited: Sep 21, 2011
  8. Pincho Paxton Banned Banned

    Messages:
    2,387
    I got this...

    ON/1=0/N

    Added it to some soup, and it was very tasty!
     
  9. AlphaNumeric Fully ionized Registered Senior Member

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    6,702
    1/0 doesn't exist in the Reals, so treating it like a number is like saying "What is the x which satisfies x + elephant = table?", it's nonsense. If you treat it like a normal number then you can 'prove' all sorts of false things.

    Remember, a number like 1/3 is not "1 divided by 3", it is "The UNIQUE number which when multiplied by 3 gives 1", 1/3 is just a nice notation. Likewise with any rational number a/b, it is the UNIQUE number which when multiplied by the integer b gives the integer a. For instance, 1/3 and 3 form a pair, where one is the unique number which multiplied by the other gives 1. Does this happen with 1/0? Suppose you said 0 multiplies 1/0 to give 1. What multiplies 2/0? Well 2/0 = 1/(0/2) = 1/0 so it's partner is 0 again. But then that's not a unique pairing. So allowing there to be a partner to 0 makes for all sorts of issues. That is why there is no multiplicative inverse of an additive identify in such types of fields.

    I'd suggest reading up on rings, groups, principle ideal domains and fields. There's all sorts of fundamental reasons why the Reals are what they are and don't include some 1/0 ~ infinity element (for one that would make them compact under the usual topology, ie morphic to a sphere, and there's a can of worms you don't want to let out of the bag).
     
  10. James R Just this guy, you know? Staff Member

    Messages:
    39,397
    In general

    \(f(f^{-1}(x)) = f^{-1}(f(x))=x\)

    But that's not exactly what you're doing here. We already have as a given that the inverse function of multiplication is division.

    I think it's actually an axiom of arithmetic that requires no proof. It's a definition of what we mean by "0". That is, it is a given that:

    \(0 + x = x + 0 = x\)
    \(0x = x0 = 0\)

    If they are undefined, then there's no way to compare them to say if they are different or the same.

    Geometry is approximately reducable to algebra, so if something is algebraically undefined then it is probably geometrically undefined too.

    You can't compare two undefined things to say whether they are the same or different.

    It is only a convention to graph the imaginary axis perpendicular to the real axis (that's called an Argand diagram). There's no sense in which imaginary numbers are "really" perpendicular to real numbers.

    I'm not sure about defining systems where division by zero is allowed.

    In general \(x^0 = 1\) for all x. As you say, \(0^0\) is undefined, by you can define it to remove the discontinuity in the function \(f(x) = x^0\) if you want to.

    A base 0 number system is impossible. I'm not sure what use a negative-base number system would be.

    Probably not. Not my area of expertise, though.
     
  11. Emil Valued Senior Member

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    2,801
    Secret,

    Divide by zero equal with :spank:
     
  12. Stryder Keeper of "good" ideas. Valued Senior Member

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    13,105
    There is a "Point" often missed. Zero was actually a number created to fill the void, initially a "radix point" was used to reference the absence of a value. So really it shouldn't be dividing by zero, but dividing by a radix point.
     
  13. raydpratt Registered Senior Member

    Messages:
    89
    In the realm of positive integers, x/y means to count the number of times that y can be subtracted from x before x is reduced to 0 (including any fractional part of y that must be subtracted to reach 0).

    Thus, x/0, where x>0, equals ∞, for the subtactions will never reduce x to 0 despite ∞ number of subtractions.

    And, x/0, where x=0, equals 1, for we will count one required process of subtraction and the remainder will equal 0.

    In these two contexts, I have no trouble with division by 0.
     
    Last edited: Oct 11, 2011
  14. Dywyddyr Penguinaciously duckalicious. Valued Senior Member

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    19,252
    No.
     
  15. Big Chiller Registered Senior Member

    Messages:
    1,106

    0/0 is undefined yet \(0*0=0\) ?
     
  16. Me-Ki-Gal Banned Banned

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    4,634
    interesting post there sport . Likey Likey . Stupid people can understand that one . Very good
     
  17. prometheus viva voce! Registered Senior Member

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    2,045
    x / 0 is undefined for any x because x * 0 = 0 for any x.
     
  18. raydpratt Registered Senior Member

    Messages:
    89
    Anyone who at least passed an algebra class knows that you are correct, but the real question is whether you are correct by decree rather than by necessity. Specifically, let's recognize that the numbers zero and infinity are unlike any other numbers, one with no count and the other with no countable end. So, why should they obey all the rules of other numbers?

    I think it's important to experimentally and logically work out what various operations with these numbers might mean in each individual context in which they occur. The word "undefined" is merely one such view. Other views might someday prove profitable.
     
  19. Pincho Paxton Banned Banned

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    2,387
    Zero is a man made number on its own. All things have an opposite, and the Universe is made from opposite particles. Zero is a result of two opposites, but on its own it is being used as a cause, and not an effect.. zero is never a cause, it is always an effect. So zero on its own doesn't exist.

    Infinity however isn't a number, so infinity doesn't have the same problem. If infinity is just used as an approximation of top end, with no known top end.. so infinity is ?. You are allowed to use infinity on its own. There are some mistakes made with infinity however.. infinite black holes.. black holes, aren't infinite, they have a top end which isn't really that high at all. Some other mistakes that I can't think of at the moment.
     
  20. AlphaNumeric Fully ionized Registered Senior Member

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    6,702
    Because the integers form a ring but not a field.
     
  21. dumbest man on earth Real Eyes Realize Real Lies Valued Senior Member

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    3,523
    "0" is only a "place holder" I was taught (me thinks) - not actually a "number". Possibly the "Radix" mentioned earlier means the same. I have no real number education, heck I still have not figured out "interest on a loan" - I mean how can it be worth more, yet buy less ?!! Geezey Peezey, I digressed!
     
  22. Aqueous Id flat Earth skeptic Valued Senior Member

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    6,152
    questions of mathematical rigor for me are best approached by the two interminable mathematical applications of my life, food and money, always gnawing at me, always forcing me to stop hoping the obvious and the impossible will merge and I will wake up as the center of the universe instead of its edge.

    Food works best for division problems, money for subtraction.

    So if I start with a pie, I prefer pecan, but they are hard to slice, so maybe leave that for integer division. say I have a key lime pie and I wanna divide it by 0. okay I'm gonna start with my infinitely thin knife so I don't have to account for truncation error. so of course this is very theoretical since I cant afford the key lime anyway so the infinitely thin knife is completely reasonable. I will for a moment at least temporarily believe that I have created a monster because such a device will never impart the force required but wait there's hope because I would use infinite force.

    now you're sitting here at the table with me ogling my pie and so I feel kind of obligated to ask you if you want some. you sense a discourtesy of some kind, and say, well, just a teeny tiny slice. More out of decorum than actual magnitude of your hunger.

    So I gleefully set off in pursuit of the slicing operation that produces the infinitessimal for you, leaving me with an amount approaching unity. You notice maybe that every slice I remove I then slice in half again. It occurs to you that this pie will become radioactive with age before you get a chance to savor it.

    So, tapping at your Rolex, you inquire into the status of your slice. Am I done yet? Well let's see, let me add up my remainders and see if I arrive at unity. So I go: 1/2 for me 1/2 for you, slice, 1/2 + 1/4 for me, 1/4 for you, slice, etc, pretty soon the band has packed and left, then the barflies, so you say hey, I won't live to eat it, this won't work.

    So we simply agree upon the possibility of an infinitely small piece, somewhere out in the infinite future. So then you say, man, it would be much easier to start with two infinitely thin knives, applied with infinite force, separated by an infinitely small gap. I say, sure, why not, so I give you nothing in one operation, and gladly consume the rest.

    Over coffee, the elixir of ponderment, you ask, how many pieces of infinitely small pie could I get with that little gadget, you know, being a patent attorney, so you get these dollar signs in your eyes, you might just buy Rolex Corp. cuz you'll be sitting on an infinite stack of Benjamins.

    So I ponder this, and conclude that indeed you will indeed get an infinity of zero width slices but i'm not sure of the par value in, say, pennies per slice. Or gold atoms per slice. Whatever.

    So you then stumble upon Ponzi squared, which goes like this: for each zero width slice you can subdivide them again, each producing an infinite subset of infinitely infinitely small slices. But then you can do Ponzi to the power infinity, then that to infinity, and so on and so on...

    So yeah, I think you're onto something there. *burp*
     
    Last edited: Oct 30, 2011
  23. Big Chiller Registered Senior Member

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    1,106

    So then 0 doesn't represent nothing in itself (definitely not 0 is a figure), perhaps then \({nothing/nothing}\,\neq\,{undefined}\).
     
    Last edited: Nov 1, 2011

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