Finite number divided by infinity

[Rodney182]

TL;DR

How does this make sense - ''Any finite number divided by infinity is to near as to nothing as makes no odds, so the average population of the Universe can be said to be zero.''

Beginner math person here. I was reading Hitchhiker's Guide to the Galaxy when I ran into this:

''It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is to near as to nothing as makes no odds, so the average population of the Universe can be said to be zero.''

The bold part is where the narrator lost me. I have to wonder why that would be zero. And have to wonder why would you need to divide the numbers to get to this answer at all?

''As makes no odds?'' This part confuses me completely.

I tried to reason my way through, but I can't. How do you divide a finite number by an infinite one? And if you can't, why would the answer be zero? And why can't we?

Is there an answer to this? Or is this just a thing Douglas Adam made up?

I'm really enjoying these books, they spawn lots of questions I'd never wondered before.

 

[PeroK]

Technically, infinity is not a number. The rules of arithmetic (addition/subtraction and multiplication/division) apply to numbers. You can't add, subtract, multiply or divide using infinity.

Douglas Adams is playing around with some apparent paradoxes that arise when you imagine that you can divide by infinity - or treat it like a number.

PS The concept of infinity plays an important role in mathematics, but you have to be rigorous and define carefully what you mean by "infinity" or "infinite".

 

[Rodney182]

Technically, infinity is not a number. The rules of arithmetic (addition/subtraction and multiplication/division) apply to numbers. You can't add, subtract, multiply or divide using infinity.

Douglas Adams is playing around with some apparent paradoxes that arise when you imagine that you can divide by infinity - or treat it like a number.

PS The concept of infinity plays an important role in mathematics, but you have to be rigorous and define carefully what you mean by "infinity" or "infinite".

Can't believe that didn't occur to me. Thanks!

 

[A.T.]

''It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is to near as to nothing as makes no odds, so the average population of the Universe can be said to be zero.''

The bold part is where the narrator lost me.

What about the 2 sentences before the part you bolded:

"However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds."

Does this really follow?

 

[DaveC426913]

It doesn't require infinities for the average population the universe to be zero. It just depends on now much precision you want.

Similarly, there is logic out there somewhere (*citation needed) that lays out how an event with zero probability can still occur:

eg.: Choosing a particular real number at random from the interval 0 to 1: The probability of selecting any one exact number, such as 0.5, is zero. Yet you must select some number in that interval, and 0.5 is one of the possibilities.

 

[A.T.]

Similarly, there is a statement out there somewhere (*citation needed) that lays out how an event with zero probability can still occur.

Yes, for example when you have infinitely many equally probable outcomes. Like choosing a random real number between 0 and 1.