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Question about PI and infinity

  1. Mar 8, 2013 #1
    How comes [itex]\frac{\pi}{\pi}= 1[/itex] yet [itex]\frac{∞}{∞}[/itex] is indeterminate? I mean [itex]\pi[/itex] is infinite... so it's essentially just another type of infinity.

    If I said that [tex]\frac{3,4,5,6,7...∞}{3,4,5,6,7...∞} = 1[/tex] would I be correct? Or again would this be the same as [itex]\frac{∞}{∞}[/itex] ?
     
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  3. Mar 8, 2013 #2

    Stephen Tashi

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    Perhaps the first thing to clarify is the difference between the size of a number and the size of a representation of the number. The number [itex] \pi [/itex] does not have infinite size. It is not "infinite" in that sense. We can imagine [itex] \pi [/itex] represented by an infinite sequence of digits, but that does not make it an "infinitely large" number.
     
  4. Mar 8, 2013 #3

    Mute

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    No, it's not. ##\pi## is a finite, specific number. The fact that its decimal expansion does not terminate has absolutely nothing to do with its evaluation in fractions. 1/3 = 0.33333... is also a non-terminating decimal expansion. Would you think (1/3)/(1/3) = 1 should imply "##\infty/\infty##" = 1?

    At any rate, infinity is not a number. "##\infty/\infty##" is not 1 because on its own the fraction doesn't mean anything (in the usual number system). The expression is indeterminate because it could be the result of a limit process that can have many different possible values. For example, consider the functions f(x) = x and g(x) = e^x. As you take x to infinity, both f(x) and g(x) tend to infinity. However, as x goes to infinity the following fractions all have the indeterminate form "##\infty/\infty##", but give different results: ##\lim_{x \rightarrow \infty} f(x)/g(x) = 0##, ##lim_{x \rightarrow \infty} g(x)/f(x) = \infty##, ##lim_{x \rightarrow \infty} f(x)/f(x) = 1##. This is why you can't assign a fixed, certain value to "##\infty/\infty##".

    You'd have to define your notation first. What is "##3,4,5,6,7,\dots,\infty##" even supposed to represent?
     
  5. Mar 8, 2013 #4
    Pi is not infinite. It has an unending decimal representation, but it is finite and a real number with a well defined operation of mult/div. Infinity is quite different. There is no real number to use for the calculation.

    Remember that you can measure out a distance of pi, but not infinity.
     
  6. Mar 8, 2013 #5
    Thanks for clearing that up. Sorry I thought I made it quite obvious but 3,4,5,6,7...∞ means 3,4,5,6,7 ect all positive whole numbers in order and never ending. To infinity basically.
     
    Last edited: Mar 9, 2013
  7. Mar 9, 2013 #6

    HallsofIvy

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    That was not the point. The question was "what does the product of all those things mean?" Normally, a sequence extending to "infinity" would mean the limit but the limit of the product 1(2)(3)(4)... does not exist. If you rewrote it as (1/1)(2/2)(3/3)... then the limit would be 1 but that has nothing to do with "[itex]\frac{\infty}{\infty}[/itex]".

    Your difficulty is that you are trying to treat "[itex]\infty[/itex]" as if it were a regular "real number", like [itex]\pi[/itex], that you could do arithmetic with- and it isn't.
     
  8. Mar 9, 2013 #7
    Hi, uperkurk,
    division, as you used it in [itex]\frac{\pi}{\pi}[/itex], is an operation between two numbers, but [itex]\lbrace 3,4,5,6,7...\rbrace[/itex] is a set. If you want to "divide two sets", you would need to define what you mean by that.

    Not a big crime, actually, since in analysis courses the real numbers are defined as sequences of fractions, like for example [itex]\lbrace \frac 3 1, \frac {31}{10}, \frac {314}{100}, \frac {3141}{1000}, \frac {31415}{10000}, \frac {314159}{100000}, ... \rbrace[/itex], that may converge to a "hole" where no actual fraction is (even if some are very close, none is at the actual spot); then operations are defined among these sequences. But your example sequence [itex]\lbrace 3,4,5,6,7... \rbrace[/itex] does not get closer to anything: you can always mention a number, a million, a quadrillion, and your sequence will always surpass that number. It is unbounded.

    Perhaps what you had in mind is that, if [itex]\frac 3 3 = 1[/itex], and [itex]\frac 4 4 = 1[/itex], and [itex]\frac 5 5 = 1[/itex], ... what happens as you go on. The best you can say is that[tex]\lim_{n \to \infty} \frac n n = 1[/tex]that is, that the fraction [itex]\frac n n[/itex] tends to 1 as [itex]n[/itex] grows arbitrarily large (not surprisingly, as it was 1 all along), but even that depends on how the numerator and denominator grow; for example, the fractions [itex]\frac 6 3[/itex], [itex]\frac 8 4[/itex], [itex]\frac {10} 5[/itex], ... that is, [itex]\frac {2n} n[/itex], tend to a different value (2) as [itex]n[/itex] grows large.

    You will gradually meet these issues as/if you approach college. Hope this helps with some ideas to toy with in the meantime.
     
    Last edited: Mar 9, 2013
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