An Infinite Question

Discussion in 'Science and the Universe' started by JJM, Oct 1, 2004.

  1. JJM

    JJM Well-Known Member

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    Why does infinity times 0 = 1 I realize the two are inverse and that when you multiply inverse they equal one but still no mater what rate you increase something even if never stop increasing it if you start with nothing you'll end up with nothing.



    This concept is beyond me. Can anyone explain?
     
  2. iBrian

    iBrian Peace, Love and Unity Admin

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    That's actually a very profound forumlation! I'm afraid maths was never my strong point, so I'd be as happy to see an answer, too. :)
     
  3. brucegdc

    brucegdc Moderator

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    Actually infinity times zero is indeterminant - not one. Ask Dr Math covers it at http://mathforum.org/library/drmath/view/53660.html . The trick to dealing with infinity in math is to use a function that approximates each term (converges to that value, in math-jabber) when the input value of the function gets closer to zero. That way you can deal with it algebraically.

    If there's a real answer, any substitution of functions that work for the correct limits should give the same answer when the resulting function is expanded indefinitely ( the "limit" of the function). For infinity * 0, the resulting function's limit will vary depending on which function is used for the substitution, which means that there is no real answer.

    I hope I kept the math jargon to a minimum above, but three years of majoring in Math may have permanently corrupted my brain.

    Oh yea - 1/0 (inverse of zero) is not infinity, by the way - it's another indeterminant.
     
  4. JJM

    JJM Well-Known Member

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    Ok then but isn't the slope of a vertical line infinity, or is my friend’s Geometry teacher making things up (wouldn't surprise me;)), and it is also -1/0 so I assumed they could be one in the same. Sorry. But If it's undefined then why in algebra when you see X^0 you assume it's 1. What if X is 0? Wouldn’t it not work then?


    Here’s another question. 1/9 is .11 repeated right, 7/9 is .77 repeated so logically 9/9 would be .99 repeated but it's not it's one. Why is that? Does .99 repeated = 1 or does .99 repeated just not exist?
     
  5. brucegdc

    brucegdc Moderator

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    Quick answer - it's math, it doesn't have to make sense ;-)

    Slope of a line approaches infinity as it approaches vertical, but infinity isn't a number, so you can't use the equation to define infinity - the equation is indefinite (all dividing by zeros are).

    Strangely enough infinity to the 0 power is also indeterminant, as is zero. Every other real number to the 0th power is one - those two are just plain wierd.

    .99 repeated infinitely is mathematically indistinguishable from 1 - one of those fun items that you can use to 'prove' 1 = 2 (10x - x = 9x... 10*.99... = 9.99.... subtract .99.... and you get 9x = 9 x => x= 1 so .999.... is effectively 1)

    If you think this is bad, try some of the theoretical algebra where x = y does not imply y=x and suchlike (rings, fields, *shudder*) which is why I only majored in Math for 3 years..... to quote Monty Python's King Arthur "Run AWAY!!!!!!"
     
  6. JJM

    JJM Well-Known Member

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    thanks:)
     
  7. Ciel

    Ciel in essence

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    "If an angel were to tell us about his philosophy..........many of his statements might well sound like 2 x 2 = 13."
    Georg Christophe Lichtenberg, Aphorisms.
     
  8. Enkidu

    Enkidu Member

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    'Infinity' can mean a number of different things. A very basic use is in the area of induction, which gives rise to the natural numbers (1,2,3,4,....). Induction is an axiom that basically states, 'given x, you can get x+1'. The natural numbers then use two basic axioms: '1 exists' and 'induction'. In this context, infinity is then simply the statement that the inductive process can be repeated indefinitely.

    [If I recall correctly, 0 is not part of this scheme, and would require its own axiom; however, my recollection on this one is a bit hazy - its a slightly different point, but goes to the actual definition of 0 as an additive identity, with 1 being a multiplicative identity. Philosophically speaking, '1 exists' is also a much easier axiom than '0 exists'.].

    A more rigorous statement to that effect is that for any given number, X, the above inductive process will eventually generate a number Y such that Y>X. Since X can be arbitrarily large, the natural numbers are infinite.
    Generalising, 'Infinity' is shorthand for saying that the answer to a given process* is arbitrarily large. Or, alternatively: for any given number, no matter how large, at some stage the process will generate an answer that exceeds the number.

    *process in this sense is usually a countable series expansion. An example of a countable series expansion is 1+1/2+1/4+1/8+1/16+.... Some countable expansions 'converge' [i.e. the answer is finite, such as the one given], whereas others do not converge and therefore become arbitrarily large. 'Countable' means that a 1 to 1 mapping exists between a given set and the set of Natural numbers, {1,2,3,...}. It is possible to have 'Uncountable' sets, but the explanation of those would be very tangential to this post

    When considering 1/0, its possible to use a limit process to determine that the answer must be arbitrarily large. This can be done by substiting a (small) value y into the expression, which becomes 1/y. For any given X, it is then possible to find a value of y such that 1/y > X. In other words, the expression 1/y tends to infinity as y tends to 0, often shortened [technically incorrectly] to 1/0 = infinity.

    The final part to this fallacy is then to say 1/0=infinity; therefore infinity x 0 is equivalent to 1/0 x 0. Cancelling the denominator with the numerator, you get 1/0 x 0 = 1.

    The problem here is that the statement 1/0 x 0 is actually meaningless. Sure, you could use the limit process above and say that 1/y x y = 1 for any y, no matter how small (i.e. let y tend to 0) but the problem is that 0 is not well-defined in this case; I could just as easily say that 1/0 x 0 is equivalent to 1/y x 2y as y tends to 0. 2y tends to 0 just as y does, and the answer here would then be 1/0 x 0 = 2 [or any other number you care to choose]. Note also, that if I don't use a limit process for the 0 multiplier, then I get a result of 1/0 x 0 = 0.
     

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