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$$\lim_{n \to \infty} \frac n n = 1$$that 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.