Infinity minus Infinity

  1. Does subtracting infinity from infinity leave you with zero?

    Or could you subtract infinity from infinity and still have infinity?
  2. jcsd
  3. Subtracting infinity from infinity has no mathematical meaning. Unless we are talking about limits and orders of magnitude.
  4. JasonRox

    JasonRox 2,303
    Homework Helper
    Gold Member

    You can't subtract but you can add infinity from infinity.

    Let N be the natural numbers and R be the real numbers. And card(X) denote the cardinality of X.

    card(R) + card(N) = card(R)
    card(N) + card(N) = card(N)
  5. Hurkyl

    Hurkyl 15,998
    Staff Emeritus
    Science Advisor
    Gold Member

    What "infinity" are you talking about? What notion of subtraction are you talking about? Your question really cannot be answered unless these are specified. (Though we can guess at what you meant, in order to give an answer)

    Some examples:
    The extended real line contains two points "at infinity": [itex]+\infty[/itex] and [itex]-\infty[/itex]. The extended real line also comes with a notion of subtraction that's defined for most, but not all arguments. [itex](+\infty) - (-\infty) = +\infty[/itex] and [itex](-\infty) - (+\infty) = -\infty[/itex], but [itex](+\infty) - (+\infty)[/itex] and [itex](-\infty) - (-\infty)[/itex] are undefined. (And any combination involving at least one finite number is defined)

    In the cardinal numbers, there are lots of infinite cardinals. (literally, too many to count) Subtraction makes little sense for them, because if [itex]\alpha \leq \beta[/itex] and [itex]\beta[/itex] is infinite, then [itex]\alpha + \beta = \beta[/itex].

    The hyperreal line contains many infinite and infinitessimal numbers, and in a certain sense, the hyperreals behave exactly like the reals. (e.g. you can subtract any hyperreal from any other hyperreal)
    Last edited: Mar 25, 2007
  6. HallsofIvy

    HallsofIvy 41,050
    Staff Emeritus
    Science Advisor

    But, in order to do that, you have to be talking about "cardinal numbers", not the regular real numbers- which I'm pretty sure is what the OP was talking about. "Infinity", in any sense, is not a real number and so neither addition nor subtraction (nor, for that matter multiplication or division) is defined for "infinity".
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