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.

 

[Matterwave]

The take-away is probably don't try to learn math from a science fiction book.

''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.

The former is not a consequent (deduced from) of the latter. One could imagine an infinite space with only a finite number of worlds.

However, not every one of them is inhabited.

Probably true.

Therefore, there must be a finite number of inhabited worlds.

This again does not follow from logic/math. For example, the natural numbers are infinite and not every one of them are odd, and yet there are an infinite number of odd numbers.

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.''

Division by infinity is not defined in our usual arithmetic because our usual arithmetic works on fields (often the real numbers) that don't include infinity as a number, as @PeroK already mentioned.

However, it is true that if you take the limit of a finite constant term over a variable that grows to infinity, the limit is zero. But this is not the same as "division by infinity".

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?

There are number systems in which you can, that's why I had to caveat with "our usual arithmetic" earlier. The hyper reals and the extended reals are two examples.

But this gets into topics that are beyond high school level. For "ordinary math" as we simply know it, you can't divide by infinity.

 

[Matterwave]

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

A more brain bending one might be that the probability of choosing a rational number from a uniform distribution over [0,1] is 0.

 

[Rodney182]

The take-away is probably don't try to learn math from a science fiction book.

I'm not. I just enjoy trying to answer questions from a science-fiction book, that's all.

The former is not a consequent (deduced from) of the latter. One could imagine an infinite space with only a finite number of worlds.

That's a good point.

Probably true.

That would be the lore of the book, I suppose. The worlds are inhabited because that's what they are in the lore and the author of the Hitchhiker's Guide presumably knows this.

This again does not follow from logic/math. For example, the natural numbers are infinite and not every one of them are odd, and yet there are an infinite number of odd numbers.

Could you explain this to someone who doesn't know the terms 'natural numbers' or why they're infinite, and why there's an infinite number of odd numbers?

Division by infinity is not defined in our usual arithmetic because our usual arithmetic works on fields (often the real numbers) that don't include infinity as a number, as @PeroK already mentioned.

However, it is true that if you take the limit of a finite constant term over a variable that grows to infinity, the limit is zero. But this is not the same as "division by infinity".

Just checking if I follow - the limit is zero because it's impossible to put a limit on infinity?

There are number systems in which you can, that's why I had to caveat with "our usual arithmetic" earlier. The hyper reals and the extended reals are two examples.

But this gets into topics that are beyond high school level. For "ordinary math" as we simply know it, you can't divide by infinity.

Yep, that sounds more advanced, haha.

 

[Rodney182]

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?

I was going over this a few times as well. If the amount of worlds is infinite, but not every one of them is inhabited, then the number of inhabited worlds is finite - my guess would be because it falls 'below' infinity. But then again what does that even mean? I can't tell you.

 

[Rodney182]

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.

Oh, I see. That's actually fascinating. But I'm failing to understand how that connects to this:

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