Question about PI and infinity

[uperkurk]

How comes \frac{\pi}{\pi}= 1 yet \frac{∞}{∞} is indeterminate? I mean \pi is infinite... so it's essentially just another type of infinity.

If I said that \frac{3,4,5,6,7...∞}{3,4,5,6,7...∞} = 1 would I be correct? Or again would this be the same as \frac{∞}{∞} ?

 

[Stephen Tashi]

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 \pi does not have infinite size. It is not "infinite" in that sense. We can imagine \pi represented by an infinite sequence of digits, but that does not make it an "infinitely large" number.

 

[Mute]

How comes \frac{\pi}{\pi}= 1 yet \frac{∞}{∞} is indeterminate? I mean \pi is infinite... so it's essentially just another type of infinity.

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 "$\infty/\infty$".

If I said that \frac{3,4,5,6,7...∞}{3,4,5,6,7...∞} = 1 would I be correct? Or again would this be the same as \frac{∞}{∞} ?

You'd have to define your notation first. What is "$3,4,5,6,7,\dots,\infty$" even supposed to represent?

 

[DrewD]

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.

 

[uperkurk]

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?

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.

 

[HallsofIvy]

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 "\frac{\infty}{\infty}".

Your difficulty is that you are trying to treat "\infty" as if it were a regular "real number", like \pi, that you could do arithmetic with- and it isn't.

 

[dodo]

Hi, uperkurk,
division, as you used it in \frac{\pi}{\pi}, is an operation between two numbers, but \lbrace 3,4,5,6,7...\rbrace 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 \lbrace \frac 3 1, \frac {31}{10}, \frac {314}{100}, \frac {3141}{1000}, \frac {31415}{10000}, \frac {314159}{100000}, ... \rbrace, 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 \lbrace 3,4,5,6,7... \rbrace 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 \frac 3 3 = 1, and \frac 4 4 = 1, and \frac 5 5 = 1, ... what happens as you go on. The best you can say is that\lim_{n \to \infty} \frac n n = 1that is, that the fraction \frac n n tends to 1 as n 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 \frac 6 3, \frac 8 4, \frac {10} 5, ... that is, \frac {2n} n, tend to a different value (2) as n 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.